# Talk:Geodesic

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## Straight line?

"In Euclidean geometry, the geodesic are the straight line, but in more general spaces they need not be" -- not sure about this. A geodesic is what we mean by a "straight line" -- what else can a straight line be if not a geodesic? The fact that great circles don't appear straight is just a nasty side-effect of the Mercator projection mindset. -- Tarquin 14:54 Oct 28, 2002 (UTC)

Yes, I agree; I reformulated a bit. Also, geodesics by definition really only give locally the shortest paths, not necessarily globally. AxelBoldt 19:26 Oct 28, 2002 (UTC)

It's been a (somewhat) long time since I studied that, but aren't geodesics straight with respect to the local curvature in some sense ? That should be added to the article. --FvdP

## Einstein Equation

Article said:

This curvature is in turn determined by the energy and mass distribution; this is the content of the Einstein equation.

Moved from article:

; this is the content of the Einstein equation
• If that refers to E=Mc**2, that is part of special rel & curvature of space-time is general rel . (If eqn implies this phenom, say more how and abt why, and explain at least here on talk in what sense it is "the content", a metaphysical-sounding expression that doesn't belong in physics w/o much explanation.)
• If not, needs clarification of the term Einstein equation, i.e. write a stub for your dead link.

--Jerzy 17:34, 2004 Jan 26 (UTC)

## Geodesic are straight

I removed "In mathematics, a geodesic is a curve which is everywhere locally straight." I belive it can not be usefull, it pretends to be little mathematical, but does not have more sense than the old formulation, contrary "Geodesic stay for the curves which are "straight" in a sense." at least explaign the meaning of the word and no mathwords used.

Tosha 22:39, 22 Feb 2004 (UTC)

I reverted that, because it makes absolutely no sense as a statement in the English language. Anthony DiPierro 22:54, 22 Feb 2004 (UTC)

I do not know English, its true. Ok, is it better now?

Tosha 23:18, 22 Feb 2004 (UTC)

Not much. What does "stay for the curves" mean? I'll let someone else, who knows more about geodesics than I do, fix it. Anthony DiPierro 23:32, 22 Feb 2004 (UTC)

I rewrote it before, as the simplest expression that implies in the general case. I recognize some of what Tosha is trying to say. I've worked with a great number of Russian scientists who are very good and know the difficult some have of learning English. But the way it is written is simply jibberish right now. Tosha, if you feel the definition needs clarifying, please discuss it here. I would be more than happy to collobarate with you on parts of the article. In the meantime, I am going to change it back. -- Decumanus 23:36, 22 Feb 2004 (UTC)

Ok, I hope now everybody happy(?) I wanted to get rid from word locally here (see above)

Tosha 01:24, 23 Feb 2004 (UTC)

Well the phrase "in some sense" is just too vague. It means nothing here and is misleading. I'm curious: why do you object to the word "locally". It is the correct term here. It does not "pretend" to be mathematical. It is a standard way of stating this property. -- Decumanus 02:01, 23 Feb 2004 (UTC)

What you do is much worse, if you want to grab an idea of geodesic then it is "straight in some sense" to do math you should define what is the "sense" and infect there are many different senses for this on the same space depending on structure you have/choose, now you have this strange curvature ... what does it mean for metric space for instance...

I will not change it back, I'm tired (hope someone will do it)

Tosha 03:13, 23 Feb 2004 (UTC)

Are you telling me you are not aware of the concept of geodesic curvature? -- Decumanus

There is no meaning for geodesic curvature in metric space, but even if you make one the curves with zero geod.curvature might not be geodesic.

Tosha 03:22, 23 Feb 2004 (UTC)

Your statement "there is no meaning for geodesic curvature in metric space" tells me a lot about your background in mathematics. Up until now I was prepared to give you the benefit of the doubt. No I am not so sure. -- Decumanus

BTW geodesic curvature is nearly defined in the article, so one could just add one line in subsection "Riemannian and pseudo-Riemannian manifolds" instead of giving ref.

Tosha 04:22, 23 Feb 2004 (UTC)

Tosha, i can't tell whether you or Decumanus is the more authoritative editor for this article, but i'd urge you to work out the wording you have in mind on this page.

For instance, after staring for a while at the first paragraph of your first edit of Geodesic, which reads

In general geodesic stay for the curves which are "straight" in a sense

, i begin to suspect that you intended the meaning of that sentence to be close to

In each of the various contexts in which the term Geodesic is used, it stands for any one of the curves which "straight", in the sense of "straight" that is appropriate to that context.

In proposing that reading, i note that the concepts of "stay" and "stand" are related (in fact, the probably come from the same ancient word) even tho a native speaker would never consider using "stay" to cover the metaphorical sense of "stand" that occurs in my suggested interpretation.

(And to me, that sounds like a good introductory "motivating the idea" approach for an opening paragraph, and one that is consistent with shifting to a much more rigorous approach, such as Decumanus seems to be pushing for, in discussing the specific meaning of geodesic in the various contexts.

(By the way, i also note that your use of "in a sense", which has caused some objections here, is not something you suddenly added to the article, but rather a variation or elaboration of

In more general spaces the geodesics can be more complicated, but one often still thinks of them as "straight" in a sense.

which was introduced by User:AxelBoldt in an edit of 18:55, 2002 Oct 28, as far as i so far notice, remaaining in the article without objection for over a year. And part of my point to you is that fact does not seem to have come out here. IMO you're going to have to work hard not just at the technical content, but also both in communicating it clearly in this weird language of English (that you so bravely have undertaken to learn to an impressive degree), and finding out why other editors are being so seemingly stupid in not following your reasoning.)

You may have things to add to the article that no one else on WP is prepared to contribute, but right now, we can't tell whether or not that's the case.

I don't want to try and address the question of whether Decumanus has tried as hard as they should to understand what you're saying, but IMO there will be no hope of finding that out without your taking time to make sure that the ideas you are bringing forward are clearly understood. IMO, that will take a lot of patience on your part to help Decumanus and others understand your meanings. --Jerzy 05:24, 2004 Feb 23 (UTC)

I've done a copy edit here. I'm with Tosha on this - he's a valuable contributor in this field, and I believe he knows exactly what he is talking about. I have similarly copy edited other pages of his.

On the geodesic curvature matter; the definition now standing on that page needs work; for one thing it isn't obvious to me that it is compatible with the link from the Gauss-Bonnet theorem page (though it may be in fact). I think that point could be addressed there. For the time being, I felt linking geodesic to geodesic curvature wasn't clarifying, and I took out the link.

Charles Matthews 08:53, 23 Feb 2004 (UTC)

It is the same as in the Gauss-Bonnet theorem, integrated along the curve. I have no problem with Tosha (and anyone else), as per my original offer.. -- Decumanus 20:53, 23 Feb 2004 (UTC)
Forgive me. It is of course the scalar magnitude that appears in the Gauss-Bonnet theorem. This may have been the source of a little confusion. The statement about it vanishing along a geodesic of course applies to either the scalar or the vector.
Also I was in the midst of doing battle with a certain user I won't name over at Talk:Quasar, and was abnormally testy. Please accept my apologies if I impugned your mathematical knowledge, Tosha,-- Decumanus 21:53, 23 Feb 2004 (UTC)

Many things come to my mind when I think of geodesics, and "straight" is definitely not one of them. As discussed above, "straight" could be taken to depend on context, but the most intuitive notion of "straight" to me comes from lines in Euclidean space. These are in direct contrast with the mental image of a 2-sphere embedded in Euclidean 3-space in the natural way with spherical geodesics drawn upon it. Granted, these lines on the sphere only appear to be curved as an artifact of the Euclidean viewpoint, but I would argue that this is the most intuitive viewpoint for the reader new to the material. What do you all think about this? - Gauge 03:18, 15 Oct 2004 (UTC)

The clearest argument I can think of for "straightness" re. geodesics is the exponential map, at least for (pseudo-)Reimannian manifolds. The exponential map gives a very clear sense of what it means to move in a direction, and continue to do so, without deviating from one's "straight" path. (because it links the tangent space to the curve, it says "keep going that way, only even more"). exp is not just for lie groups. Also, "straight" is captured even more strongly by parallel transport, which says that not only can you go straight, but you can do so without spinning. Err, well, ahhh.. you know what I mean :). Spinor ... susy ... hmmm.. I'll try to add words to this effect sometime later. linas 04:57, 30 September 2005 (UTC)

## Hilbert problem

I removed 4th Hilbert problem (link to Hilbert Problems: http://aleph0.clarku.edu/~djoyce/hilbert/problems.html) from this page, it is clearly relevant, but not at all the first problem which should be mentioned here, also the note after that is not quite correct, it was solved at least in dimension 2.

Tosha 17:04, 8 Jun 2004 (UTC)


## higher dimensions

The article applies "geodesic" only to lines. It can be also be applied to higher-dimensional submanifolds. I could write something on this, but I'm not sure whether there's a difference between geodesic surfaces and totally geodesic surfaces (see e.g. [1], p.6). Does anyone know? Fpahl 15:48, 14 Sep 2004 (UTC)

## more precisely?

The second sentence under "For metric spaces" begins with "More precisely". It seems to me that the precise definition is actually something new, and that it is a non-trivial statement that a curve satisfying that definition "is everywhere locally a distance minimizer". Also, the definition seems to be too narrow in that it requires the curve parameter to be proportional to the arc length; wouldn't any reparametrization of such a geodesic still count as a geodesic? Fpahl 15:56, 14 Sep 2004 (UTC)

## Revert by Linas of edits by pdn

Linas, you make the statement: "masses follow timelike geodesics, period. no ifs-ands or buts". It is my understanding that only the world lines of free particles are geodesics of spacetime. Thus, here on the surface of the Earth, I feel the 'weight' of acceleration and my world line(s) is certainly not a geodesic. Alfred Centauri 14:27, 6 Jun 2005 (UTC)

Isn't the problem one of distinguishing between paths and geodesics ? Free particles (and photons) definitely follow geodesics (and they are timelike (null for photons) geodesics), but if the particle is not free, then the concept of geodesic doesn't apply - although the concept of path (or worldline) still does. My conclusion:
• Worldlines are paths in spacetime.
• Worldlines of free particles are called geodesics of the spacetime.
• Particles that are not free obviously still have wordlines (but they are not geodesics), but one can still calculate tangent vectors at any point on the worldline and determine if they are spacelike, timelike or null (assuming we know the metric). For material particles, they are timelike at each point of the worldline, for photons etc. they are null at each point.

Suggestions/comments ? Mpatel 15:58, 6 Jun 2005 (UTC)

Sorry about the revert, maybe I was having a bad day. I admit I misread the statements about timelike; (I misread them as saying that worldines of massive particles aren't timelike...oops). On the other hand, I completely failed to understand what was being said about the spacelike geodesics .. that they're "graphs of filaments"? My brain fritzed at that point, so I just reverted.
Maybe its time to start a new article, say, geodesic (general relativity) ? The current geodesic article is getting long, and the GR stuff is kind of buried at the bottom. There's lots of interesting things that can be said about GR geodesics, and that article isn't really structured to say them. The current article could continue to stand as the "math formalities" of geodesics. linas 23:51, 6 Jun 2005 (UTC)
Acted upon Linas' suggestion by creating the suggested new article: geodesic (general relativity). ---Mpatel 16:56, 13 Jun 2005 (UTC)

## I is not defined before it is used.

Domain ${\displaystyle I}$ is not introdcued before it is used in the first definition. It should be introduced to make the article self-contained and accessible to people who know mathematics but do not use geometry daily. Wazow 10:54, 11 August 2005 (UTC)

Fixed. linas 22:16, 15 August 2005 (UTC)

## geodesics need not be locally length minimizing

This statement is incorrect in the preamble. Though this is often used to characterize them, they are not inherently defined this way. Perhaps we need to write an explanation of variation of arclength and the Jacobi equation. Cypa 04:27, 28 September 2005 (UTC)

I'd rather not change the introduction, since the intro is really meant to provide a simple, "layman's" description, for which "minimizing" is good enough. Or rather, I don't think the intro can be reworded to say "extremizing" without making it harder to understand. I believe the article does try to be more precise as to the definition in the later sections; if not, it should be fixed.
Secondly, I am personally not even aware of any examples where geodesics are not locally length minimizing. I cannot imagine that there are any, on finite-dimensional manifolds? Or have I forgotten something "obvious" in GR? Can you provide an example? linas 22:53, 28 September 2005 (UTC)

Perhaps the intro should not be changed. Or maybe my understanding is flawed. Anyway, the definition is recursive in the sense of a Jacobi field. Consider a one parameter family of geodesics with fixed endpoints and variational parameter d. Then the Jacobi field explicitely characterizes the family up to variation in d. Now consider the geodesic segment from (0,d) in some segment of the Jacobi field, and reiterate the construction ... the conclusion follows directly. Cypa 00:26, 30 September 2005 (UTC)
I'm sorry, I did not understand that. What's recursive? You can define geodesics without any reference to a Jacobi field (indeed, the article does so). On the contrary, I have the impression that it would be hard to define the Jacobi field without first defining the geodesics. I don't think you need a jacobi field to use variational calculus (i.e. to minimize the lagrangian/hamiltonian). So I don't know what the conclusion is, or what its following from.
The early section which first mentions the geodesic equation should be re-written to emphasize that the equation just comes from the minimization of the Lagrangian. The bit about "self-tangent" can be cut out or moved to later in the article. So, yes, the article could be clarified in that area. linas 01:21, 30 September 2005 (UTC)

I just re-wrote the section that held the geodesic eqn. I'm hoping this makes it easier to understand. The fear-inducing mention of parallel transport has been banished to a later section. linas 03:51, 30 September 2005 (UTC)

okay, well, there's no real big deal with regards to it all. This is enough in the talk section just so that the interested student has a sense that there is more to know. The construction on this page seems to be strongly weighted towards physics and GR, which is certainly very useful to have. However, I think you can arrive at a slightly more general definition using the relevent differential geometric approach. In this case the geodesic is perhaps more suitably thought of as a curve with "constant speed." In a sequel to this page, perhaps it would be of use to distinguish these definitions. So, perhaps a derivation of the geodesic in GR using the Euler-Lagrange approach, followed by a definition of the geodesic using a restriction of the first variational principle. Perhaps later on I will have more time and try to incorporate this myself. Cypa 04:59, 30 September 2005 (UTC)
I stll don't get where you're going with this. The first variational principle is exactly the same thing as the Euler-Lagrange. I mean, I just re-wrote that section to try to say this as clearly as possible. Am I missing something? Yes, I guess one could actually add a short section stepping through the steps to actually derive this ... I dunno. The article geodesic (general relativity) does this; I'm not sure we need it here, or that its even edifying or encylopedic, its not comoplicated and all the textbooks do it.
As to constant speed, yes, need discussion of that, I think it would go well with a section discussing the exp map. That, and straightness, see discussion above and below.
As to being physics-centric, the GR guys lit off to write geodesic (general relativity) because this article was "too mathematical". Everything on this page holds for arbitrary dimension, Riemann or pseudo-.
As to "more general approaches" that might be valid for non-reimannian manifolds, I do not know what those are, or what the examples would be. We need to start with an example manifold for which the variational principle breaks down or is somehow pathological. linas 05:20, 30 September 2005 (UTC)

## Wishlist, todo list

It would be nice to have, in this article, or in a related one,

Further Edification:

• Discussion of exponential map and its use for solving geodesic equation, in general, and not just on Lie groups. Also, curves of "constant speed".
• Discussion of straightness in this context and very very breif mention of parallel transport with regards to straightness.

Exotic geometry:

• There are scads of metric spaces. Which ones have bizzare geodesics? Or are otherwise geodesically patholigical in interesting ways?
• Examples of metric manifolds that are not geodesically complete (there are examples in GR).
• Examples of manifolds for which geodesics terminate. Schwarzchild singularities provide a 1-D example, Kerr provides a 2D. (Point reader to article geodesic (general relativity)) A slightly more exotic example is the three-body problem in Newtonian gravity. How about examples where the terminus is fractal, e.g. the Hopf whatever-its-called-disaster from chaos theory? Cauchy horizon.
• Examples of manifolds for for which shortest geodesics are not unique. (Is there a name for this class of manifold??)
• Mention of Finsler manifold.
• Discussion of geodesy on infinite-dimensinal manifolds, e.g. topological vector spaces.
• Vastly expanded coverage of sub-Riemanian geometry.
• geodesics in algebra.

Mention of whiz-bang uses and applications:

linas 04:45, 30 September 2005 (UTC)

I think we should not expand it too much, Finsler and sub-Riemannian cases are too special and might be done in a separate article (as geodesic (general relativity)) Tosha 05:19, 20 October 2005 (UTC)

## I'm late

I did not watch this article for quite a while, now it has many problems: bit wrong intro is just a little thing, main thing is completely wrong section Riemannian geometry and the geodesic equation.

### Intro

On intro: the curent intro is has partly wrong def, it is not sutabale for all cases, in particular it is very wrong for pseudo-riemanniann manifold (and for general gravity) as well as for spaces with connection. It was a long discussion before we converge to the one I like and I think we have to keep it.

What was wrong with the current intro? Several people thanked me for replacing the old intro with the current one; the old intro was poorly written and did not make much sense.

Did you read what I wrote? Tosha 03:22, 21 October 2005 (UTC)

Yes, but you did not answer my question. What was wrong with the current intro? I did read the old discussion about the old intro, and it was clear that several people in that conversation did not know what they were talking about.

Main problem: it does not work for pseudo-riemanniann manifold (and for general gravity) as well as for spaces with connection. Note that I did not change it so far, but I will do it later Tosha 15:30, 21 October 2005 (UTC)

### (pseudo-)Riemannian geometry

On Riemannian geometry and the geodesic equation One can not define it trough lenght, at least yoyu should ask for constant speed and it does not work for pseudorieamannian case.

I don't think I put anything in there about constant speed, that may have been left over from the old version of the article. Lets fix that.

I do not blame you, I'm just correcting mistake Tosha 03:22, 21 October 2005 (UTC)

What was the mistake? 09:02, 21 October 2005 (UTC)

It is explained above, if you do not see the mistake, you do not know what is geodesic, why then you edit this article? Tosha 15:30, 21 October 2005 (UTC)

### geodesic flow

In addition, the geodesic flow section can be removed almost completely, and refer to Hamiltonian mechanics

No, because the geodesic flow is fundamental to what it means to be geodesic. It is a much better and more modern presentation than the Euler-Lagrange equations, (which is the old-fashioned approach). If one wanted to be absurd, one could just say that geodesics are a special case of Lagrangian mechanics, and just redirect this article to that. But that would be wrong.

So why not leave all technical stuff there there is no need to repeat it here Tosha

What technical stuff, where? The definition that a geodesic is a curve that parallel transports its own vector is a terrible definition and is very difficult to understand unless one already knows this topic very very well. I would like to remove that section entirely. By contrast, a derivation of the geodesic equation from either lagrangian or hamiltonian principles is the usual way of introducing the concept, and is simple and easy to understand. The hamiltonian method is simpler than the Euler-Lagrange equations because it gives a pair of first order differential equations which can actually be solved simply, for real-life mainfolds. You can actually write down something that is solvable, e.g. on a computer, by working with these. It is much much harder to convert the parallel-transport equation into something useful for practical calculations.

the Hamiltonian can be mentioned in def of geodesic (I'm not against anything which is correct) note that there is no technical detales about lagrangian as well. There is reason to do this def for Riemannian geometry and most books do it this way (and it is not becouse they are stupid, main thing: it is more usefull). Also, as it is stated in the end metric def coinsided with Riemaniann

This section had many things not really related to the topic now at least it is clear what geodesic flow is Tosha 15:30, 21 October 2005 (UTC)

### General

So I revert this secton, and thin a bit about intro Tosha 01:16, 20 October 2005 (UTC)

And I revert back. Lets try to work through this in a more gentlemanly fashion. linas 00:58, 21 October 2005 (UTC)

Not this way, the article had serious problems, see above, so before reverting you should describe the reason for making it wrong! Tosha 03:22, 21 October 2005 (UTC)

What were the serious problems? What was wrong? All you did was to delete material and make the article more difficult to understand. linas 09:02, 21 October 2005 (UTC)

Just read carefully what I wrote, I will not repeat it again! Tosha 15:30, 21 October 2005 (UTC)

Guys. Firstly, the issue of whether to use the variational approach as the main tool, or not. Here there is a cultural difference, no question. Theoretical physics uses variational ideas routinely. Pure mathematics, for reasons we should write up elsewhere, routinely replaces them with something else if possible. This has been going on since Riemann's use of the Dirichlet principle, and the troubles that caused. We cannot settle this on the page.

More over in my edit it is not even in def there is only comment about lagrange and we can add one about Hamilton. Tosha 15:44, 21 October 2005 (UTC)

So, I agree with Tosha if he says that parallel transport gives the clean approach. On the other hand, I don't necessarily agree that the page should attempt to do the pseudoRiemannian case from the outset. Surely start with the Riemannian case, to be kind to the readers. Do it the clean way. Do it the physics way after that. Introduce the psR case, by reference to GR. Point out any serious difficulties now that 'length' is indefinite as a concept. This would be longer, but could be better. Charles Matthews 15:33, 21 October 2005 (UTC)

There is metric def, so why should we repeat it for Riem case? the given def is what people use in Riem. Geom, it is used in most of books, clearly metric def gives a motivation. And sinse it works just as well for pseudo-Riemannian case why not top leave it this way?

Ok, I will not revert it my-self, I think my edit is much better, and curent is wrong. Hope someone will do it for me Tosha 15:41, 21 October 2005 (UTC)

To throw in my own two cents: I think one should give as many characterizations of a geodesic as possible. I believe the definition as curves parallel transporting their own tangent vectors is the most common among mathematicians and should probably come first. But let us certainly include the variational definition.
Tosha, we value your contributions here. Don't throw in the towel yet. I'm sure we can work out a compromise with Linas on the presentation. -- Fropuff 16:36, 21 October 2005 (UTC)

My feeling is there are some problems with arranging material in this article, some of which I corrected plus the section on (pseudo-)Riemannian geometry is almost completely wrong. Sinse I promised to not revert I will not, but everybody else is wellcome to compare and revert Tosha 22:06, 21 October 2005 (UTC)

### Proposal

Tosha, when you say the section on (pseudo-)Riemannian geometry is almost completely wrong, do you mean the section called Riemannian geometry and the geodesic equation? If we remove the word "pseudo" from that section, does it become almost completely right?

But still is not right, see above Tosha 01:26, 22 October 2005 (UTC)
Up above, you never actually explain what is wrong. Please describe what it was that you think was incorrect. linas 19:14, 24 October 2005 (UTC)

Overall, I'd like the article to start with something as simple as possible, and move to more correct, more general definitions. This would include a discussion of why the simple definition fails, e.g. null geodesics, lack of existance proofs.

That is nice if you do it correctly, if you have to make statements which are not quite correct as in your intro you should tell that you are cheating to make it simpler.

Here's a proposal I'd like to make:

• The Euler/Largrange or Hamiltonian approach provides a simple, intuitive way to derive the geodesic equations on Riemanninan manifolds. As far as I know, the variational approach is valid in the riemannian case. Right?
as well as for pseudo-Reim (if you do it correctly)
• In the Riemannian case, to the best of my knowledge, one really can prove that there exist solutions, that these solutions are unique, and that they are in some sense a "complete set" of geodesics. Right?
ok
• In the Riemannian case, the Hamiltonian approach has two advantages over the Euler-lagrange variational approach: 1) it yeilds first order differential equations, which are thus easy to solve, 2) it uses the language that shows up in symplectic geometry, and so is useful for helping the reader make contac with tat world.
You can think this way, some times lagrange is better, but both are bad as defs! and both should be mentioned after def.
Can you add text to the article to explain why these are bad defs? Are they bad defs because they are not general enough, or are they bad defs for some other reason? This definition is commonly given in many textbooks, and so if they're bad, this should be explained in this article.
• In the pseudo-Riemannian case, the variational principle fails, and needs to be replaced with something else, e.g. the notion of parallel transport of ones own tangent vector. However, this is a far more complicated notion, since the reader needs to first understand what parallel transport is, and this can cause a lot of difficulty.
No, it works (if you do it correctly)
So what is the problem?
• The parallel-transport formulation has other problems: it requires the reader to understand what a covariant derivative is, whereas the the variational approach does not require this. Many readers who might be able to understand the variational approach will not yet understand what a covariant derivative is. In fact, this is how every book I've read (class I've taken) does it: first one introduces the geodesics, then one introduces the covariant derivative as that thing that makes the geodesic work, and then one shows that the covariant derivative has many interesting properties, and then one shows how parallel transport works. My concern is that by posing the more difficult, more general definition first, the reader will be turned off, and will not read any further.
If you do Riemannian geometry it is not a prob, if you do not want to do it why read this subsection. That is the most useful def, many important properties follow right from it. in any case there is no need to repeat def from metric subsection (but it can be used as a motivation)
The article should be something that should be understandable. Giving a difficult formal definition first is the wrong way to write an encylopedia article. This is why I dislike the edits that you are making.
• After the above sections, discussion of further generalizations would be good. e.g. formulations based on Jacobi field, Again, this is "advanced" since it requires more sophisticated methods.
I think it is better to leave Jacobi field in Jacobi field, «see also» is the best place for it
• With only the slightest of encouragement, I will copy the section called "geodesic flow" to its own unique article, and merge Tosha's flow definition there as well. That will unclutter this article. For the Riemannian case, is the section called "geodesic flow" mostly right, or do you see errors there?
I did not see errors, but it is almost irrelevent. we should not copy here everything from wikipedia which is bit relevent, a ref is enough. It also creating problem with editing (one will need to correct mistake few times)
What's irrelevant about it? You beleive that no one studies geodesic flows? Tosha, I eally have great difficulty understanding your viewpoints.

Would that work? linas 00:42, 22 October 2005 (UTC)

I think at the moment, the best we can do is revert my edit. Tosha 01:26, 22 October 2005 (UTC)

If I may, I see a recurring theme in this discussion that I've run into before. So, there's this difficulty in providing a encyclopedia entry for something titled "geodesic." For example, I have at least 4 disparate definitions of geodesic before me right now, which could be parsed as "Riemannian geodesic" "Finslerian Geodesic," "constant speed Finslerian geodesic," and "spray geodesic." Now this is all fine and good, so which one are we talking about? Well, to some, it seems most in this dscussion, clearly we are talking about Riemannian goedesics, or geodesics in GR. That's fine. But the topic is quite expansive, and topics which are expansive in their basic mathematical natures should not, it seems to me, be presented in an over-simplified manner (for example what is a geodesic on a Riemannian varifold?). In other words, it is better to give a vague definition that points in the direction of the expansive meaning, than to provide "only" a closed definition which really applies in the simplest and most ordinary textbook exercise-type cases. I'd like to quote Thurston here:
"mathematics is a huge and highly interconnected structure. It is not linear. As one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of the structure, not just a familiarity with a particular tour through the structure." -- William Thurston
I for one would love if we could try to approach this article with more of this flavor. I have spent, clearly Linas has spent, and clearly Tosha has spent, too many thousands of hours on mathematics to be disregarded with a wave of the hand about being wrong. There are cases of this from time to time, but usually, there are internal contexts, expansive and elaborate constructions, etc, which are direct consequences of a certain aspect of definitions in a certain context etc. To have an encyclopedia which is incapable of embracing this elasticity (nonlinearity) that is inherent to the subject, is problematic in my estimate. Look at planetmath, some words have several entries to address this particular nuance. With the unique structure of wikipedia, i think we have the opportunity to embrace a spirit of convolution that may ultimately lead to an unparalleled intuition machine for these topics. What we need is a desire to learn from each other and to make the most useful/correct page that we can in a "collaborative" sense. My 2 cents. Cypa 02:02, 22 October 2005 (UTC)
My two cents. Few of us are Thurston. Charles Matthews 07:29, 22 October 2005 (UTC)

At least it seems that there is no objections to revert it back so I do it Tosha 19:45, 22 October 2005 (UTC)
Tosha, your edits are not making this article "better" or "easier to understand". They are making the article more distant and less approachable. There are many students in engineering and physics who will not have a deep background in mathematics, and these folks should get an article that is understandable at that lower, simpler level, rather than formally correct at a math grad-student level. (May I remind you that geodesics show up in robot-arm mechanical engineering, and also in space-flight engeineering. I get the impression that even full professors in these areas have at best a weak background in Riemannian geometry.) linas 19:14, 24 October 2005 (UTC)

Luna, with all respect, I do not want to keep a wrong article, even if it is really easy to read. BTW, if you noticed I did make some changes, which you were mentioned in the above discussion. Tosha 02:26, 25 October 2005 (UTC)

Sigh. You keep asserting that something was wrong, but you haven't actually made the effort to explain what was wrong. It will be very hard to not repeat my mistakes if you do not actually explain what these were. linas 14:46, 25 October 2005 (UTC)

a curve c(t)=(t^3,0) in the plane minimize length but is no a geodesic, isn't it? you can correct def, by making it const.spped but then it will repeat def in metric geometry part. Also the statements about ways to solve it were completely wrong... Tosha 22:58, 25 October 2005 (UTC)

Thank you, you are quite right, the discussion should have avoided talk of length. Mention of Lagrange multipliers for the velocity makes the affair too complicated.linas 13:13, 26 October 2005 (UTC)

My two cents as an undergraduate maths student: please forgive me if I say anything stupid (because I admit I know little about geodesics compared to all the experts around). With geodesics I learnt them the "parallel transport" way first and the "variational" way afterwards, and yet I know plenty of classmates who learnt them the other way round, or one and not the other, even though we are doing the same degree at the same time. So I think having separate sections that describe them one way independently of the other way would be great, and then a section can follow on how they are related to each other. I was looking up "geodesic" here myself hoping to understand more the connection between the two approaches that I've learnt. -- KittySaturn 00:03, 23 October 2005 (UTC)

## Riemannian geometry and the geodesic equation with comments

Given a (pseudo-)Riemannian manifold M, a geodesic may be defined as a solution to a differential equation, the geodesic equation. The geodesic equation may be derived, using variational principles, as the Euler-Lagrange equations of motion applied to the length of a curve.

wrong you can not use length


Given a smooth curve

${\displaystyle \gamma :I\to M}$

that maps an interval I of the real number line to the manifold M, one writes the length of the curve as the integral

${\displaystyle l(\gamma )=\int _{I}{\sqrt {g({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt,}$

where ${\displaystyle {\dot {\gamma }}(t)}$ is the tangent vector to the curve ${\displaystyle \gamma }$ at point ${\displaystyle t\in I}$. Here, ${\displaystyle g(\cdot ,\cdot )}$ is the metric tensor on the manifold M.

Using the length given above as if it were an action, and solving for the Euler-Lagrange equations, one gets the geodesic equation.

wrong again it is not what you will get.


In terms of the local coordinates on M, this is

${\displaystyle {\frac {d^{2}x^{a}}{dt^{2}}}+\Gamma ^{a}{}_{bc}{\frac {dx^{b}}{dt}}{\frac {dx^{c}}{dt}}=0}$

Here, the xa(t) are the coordinates of the curve γ(t) and ${\displaystyle \Gamma ^{a}{}_{bc}}$ are the Christoffel symbols. Repeated indices imply the use of the summation convention.

Rather than working with the length, one may obtain exactly the same equations by minimizing the energy functional

${\displaystyle E(\gamma )={\frac {1}{2}}\int _{I}g({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,dt,}$

In physics, this energy functional is called the action, although this term is not commonly used in mathematics. Working with the action instead of the length has the minor advantage of avoiding a pesky square-root and the major advantage of using the same conceptual language used in other important areas of geometry and topology.

foggy, but wrong again


The geodesic is a second order ordinary differential equation. Equivalently, geodesics may be defined to be the solutions to a pair of first order differential equations; these equations, the Hamilton-Jacobi equations, are obtained by working with the Hamiltonian formulation instead of the Lagrangian formulation, and are given in the next section.

Right, OK, thank you. linas 13:13, 26 October 2005 (UTC)

### Correction to energy functional

I changed

${\displaystyle E(\gamma )={\frac {1}{2}}\int _{I}g({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,dt,}$

to

${\displaystyle E(\gamma )={\frac {1}{2}}\int _{I}g(\gamma (t),{\dot {\gamma }}(t))\,dt,}$

which I believe to be a correct. But as I am no expert of on Riemannian manifolds, I figured I would alert others to the change.

Jrohrs 18:04, 1 August 2006 (UTC)

## Completeness

Tosha, you also removed the following section:

Given a point p in M and a vector V in TpM, the exponential map will map the vector tV to a geodesic in M, where t is a real number, scaling the vector V.
This is a strage statement, it does not define exp.map, or at least give very non-complete def.


The Hopf-Rinow theorem states, among other things, that any two points on a Riemannian manifold are joined by a geodesic.

I guess one should add "complete"

In general, for manifolds that are not Riemannian, geodesics need not exist, need not be unique, and the manifold may not be geodesically complete.
Well if not Riemannian then geodesics might be not defined, which spaces are ment here? pseudo-Riemanian? Finsler? metric spaces? what is the meaning of non-uniqueness and so on, it is just too foggy...


I am guessing that you removed this because the article failed to define the meaning of the exponential map in the context of a Riemann surface? linas 13:13, 26 October 2005 (UTC)

(I thought a section on completeness would be interesting, because some systems aren't, e.g. constrained mechanical systems, such as robot arms, aren't geodesically complete. Ditto for GR.)linas 15:11, 26 October 2005 (UTC)

I think it enough to mention Hopf-Rinow theorem in see also, or at most give a def og geodesically complete space here and refer to Hopf-Rinow.Tosha 18:57, 26 October 2005 (UTC)

Robot arms and satellite orbits are given by sub-Riemannian manifolds. The metric only works in some directions, not all. (one of the prototype problem is the holonomy (Berry phase) for the "falling cat" which can turn around while falling to land on its feet). The variational equations aren't always for "shortest energy/length", they're often for "shortest time from here to there, as limited by maximum torque of the motors", so instead of being constant-velocity, they try to be constant-acceleration. I've never studied this stuff closely, I know that there are books on this stuff, and I know that questions of existance are important. Its more or less that same kind of math, w/ metrics, connections, parallel transport, geodesics, slightly modified, since its usually easier to work in the tangent space and the hamiltonian formulation. Not sure why, it has something to do with the incomplete metric. (whereas the inner product on the tangent space is still well defined). linas 00:57, 27 October 2005 (UTC)

## Geodesic equation

There should be a section (or at least a subsection) on the geodesic equation, as this is certainly something that a general reader should see stand out. Looking at the table of contents, the concepts mentioned are important, but geodesic equation should be there too. MP (talk) 09:39, 19 February 2006 (UTC)

There should be a derivation of the geodesic equation from the action since it is pretty important...if no one objects ill add a section on it. — Preceding unsigned comment added by 207.161.187.38 (talk) 00:41, 15 August 2014 (UTC)

The derivation of the geodesic equation in relativity theory is already presented at geodesics in general relativity. The present article already does describe how to obtain the geodesic equation from the calculus of variations, although it is generally expected that someone looking for detailed derivations of results would consult any of a number of standard textbooks on the subject, rather than relying on a general-purpose encyclopedia. Sławomir Biały (talk) 10:58, 15 August 2014 (UTC)

## "Planetodesy"

While at first glance it sounds hokey, it is a legitimate word: See Google search. While it may not be widely used, I think it is worth mentioning——I'm not attempting to hijack this article with the term (though, generally, in articles talking about "geodesy" and "geodetic" concepts not specifically referring to Earth, I think "planeto-" should be used instead of "ge-"), I'm just pointing out the term's origin and more generic form. ~Kaimbridge~19:22, 30 July 2006 (UTC)

As the root "ge-" in geodesic, geodesy, geology, etc., refers to Earth, there is a more generalized root for any planetary body (including moons, stars and other celestial bodies), "planet-": Thus "geodesic" becomes "planetodesic", "geodesy" "planetodesy", "geology" "planetology", etc. (one glaring exception is "geometry" which, while etymologically does mean "Earth measure", has assimilated into the mathematical vernacular in its own right).
It certainly is a valid word; I'm not sure that it belongs on this page, though. I've edited it down to a size that I think is more appropriate for the introduction. "Planetodesy" might merit its own page, however... --MOBle 21:26, 1 August 2006 (UTC)
I want to see a citation or five that uses the word "planetodesic". mdf 18:32, 13 October 2006 (UTC)
Here are a few: [2], [3], [4](PDF). There are even more for planetodetic and (especially) planetographic. This Encyclopedia of Astrobiology, Astronomy, and Spaceflight article, "planetocentric and planetographic coordinates", gives a decent explanation. These generalized terms aren't used much because most discussion of these concepts deal with Earth, which uses the "ge-" prefix (but they are certainly valid——and the proper terms for general planetary discussion).  ~Kaimbridge~19:00, 14 October 2006 (UTC)
Yeah, "planetocentric" and "planetographic" are well known, and probably need no attribution. However, the other references above only demonstrate "planetodesy" (which, prior to coming to this article, was unique for me). None of them are an actual use of "planetodesic" though. I asked because a "smoke test" google search on the word only calls up Wikipedia and its derivatives. Has anyone used the word at all? If not, I'd suggest that this may be because the word "geodesic" is now as abstract as "geometry" is, to the point that even planetodetic specialists use the word. mdf 12:47, 16 October 2006 (UTC)
All right then, if you agree that "planetodesy" is a valid word, then "planetodesic" is too——they are just different grammatical forms of the same word: Just as you would say "the length of the geodesic" (not "the length of the geodesy"), you would say "the length of the planetodesic" (not "the length of the planetodesy")! P=) I agree it sounds odd and contrived (like translating something into Spanish by just adding "el ...-o": "let's have some 'el fisho' for supper"), but that's just because its not heard and used much (which will, hopefully, change as planetary exploration increases). One immediate distinction I see is that most (if not all) geodetic formularies——Helmert, Sodano, Vincenty, etc.——are specifically created and limited for Earth and other similarly elliptic bodies (i.e., they are poor for even reasonably specific accuracies for a theoretical celestial body with an aspect ratio of even .75-.8, and completely useless for .5 or less), thus a general purpose "planetodetic formula" should be open-ended, such as (Gaussian) quadrature or a series expansion (with the series term's structure identified).  ~Kaimbridge~15:03, 16 October 2006 (UTC)

"Planetodesic" yields zero books.google.com hits, and the only google hits are to this article and this talk page. So I went ahead and removed it. If someone wants to wedge the term "planetodesy" back in I suppose that's okay, although (a) it probably doesn't belong in the lead and (b) it's so obscure it's probably not necessary at all. —Steve Summit (talk) 13:10, 7 February 2007 (UTC)

## Autoparallel versus Geodesic

In a general space, there is a difference between an autoparallel curve (which parallel transports its tangent vector) and a geodesic (which is defined as a curve which extremizes length, regardless of which type of space you're working in). In the specific case of a (pseudo-)Riemannian manifold with torsion-free and metric-compatible connection, the difference vanishes. I've tried to establish this distinction more clearly. For a nice treatment, I recommend the first dozen pages of Ortin's "Gravity and Strings". Note that Wald assumes torsion-free and metric-compatible very early on, so he never separates the two cases. --MOBle 21:26, 1 August 2006 (UTC)

## geodesic definition

Hello,

I posted this in another page, so this is just a repost.

There a few things in the definition of geodesic provied I do not quite agree with. Firstly, when one talks about points, I think points are "in" a space, not on it, as this assumes some kind of ambient space. We live in R^3, not on it.

I also think that one cannot support the statement that geodesic are defined as the shortest paths. Locally or not, because of reparametrisation, for example. I like the definition given in "John M. Lee. Riemannian Manifolds: An Introduction to Curvature. Number 176 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1997. " where geodesic are simply the curves whose acceleration is zero (well, the covariant derivative of the velocity). Geodesics are, locally, the shortest paths, but the opposite is not true. From the point of view of accelaration, it is natural then to see that the lenght of the velocity vector of a geodesic is constant.

Regards

Krzysztof Krakowski

129.180.1.224 02:40, 19 April 2007 (UTC)

With respect to the distinction between "in" and "on", I think that this is simply a matter of personal taste and the flexibility of the English language, and not connected to the question of an ambient space. Whether one chooses to say "x in M" or "x on M", it is equivalent to the set-theoretic notation "x ∈ M", which certainly does not require that M be a subset of anything.
On the other hand, I agree that defining a geodesic to be a (locally) shortest path is an intuitive definition at best. It should probably be the first one given in the article, at least as motivation or a preliminary definition. It is, after all, the one most likely to be understood by a reader little background in differential geometry. I suggest that the "zero acceleration" definition should be introduced after the more heuristic definition has been covered. Sullivan.t.j 05:26, 19 April 2007 (UTC)

In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space.

Uh, or the longest? "Extremal" does not mean "shortest". --76.224.88.42 (talk) 21:03, 20 December 2007 (UTC)

## i removed "geodesic grid"

"Geodesic" (adj) in "geodesic grid" refers to geodesy (n), the shape of the Earth. The points in such a grid are geodetic (adj): they lie on the surface of the Earth; unfortunately -- for it makes for confusion -- it is also common to say they are geodesic (adj) points. The "geodesic" in "geodesic grid" does not refer to geodesics (n), the subject of this article. (And when I said "inflection", I wasn't referring to the mathematical concept, but the grammatical one.) It would've helped to follow the link to the geodesic grid article to see what it was about.

Silly rabbit, indeed.  :P 128.83.68.169 02:21, 12 July 2007 (UTC)

I don't think you're going to have much luck distinguishing between the two mathematical notions of geodesic on grammatical grounds. Geodesic may be used either as a noun or an adjective to refer to quantities which involve geodesics (n). Likewise, I have seen geodetic (adj) used in this way — although it is uncommon (see, for instance, Penrose and Rindler "Spinors and Space-time"). It appears that geodesic grid is just fine as a see also: I had checked it before reverting. It seems to discuss geodesics in the sense this article talks about. Maybe I am missing something, since you assure me that the geodesics over there are different from the geodesics here. Perhaps you need to remove the link from that article as well? Silly rabbit 12:28, 12 July 2007 (UTC)
"I don't think you're going to have much luck distinguishing between the two mathematical notions of geodesic on grammatical grounds." Right, that's exactly my point; the inflected forms are the same and can be confusing.
"It appears that geodesic grid is just fine as a see also: I had checked it before reverting. It seems to discuss geodesics in the sense this article talks about." Ummm, no. It talks about how to generate points for use in a grid for finite differences. The points are generated from the vertices of triangular refinements of an icosahedron (projected onto a sphere); the points do not lie on a regular system of great circles (the geodesics of a sphere). This becomes all the more obvious when -- to better model the shape of the Earth -- an oblate sphere is used (and the former model is simply dilated in the directions perpendicular to the axis).
"Perhaps you need to remove the link from that article as well?" Oops, missed that one before. I just changed it to point to the geodesy article. Thanks.
128.83.68.169 19:58, 12 July 2007 (UTC)

I find this article to be too focussed on mathematics. If you are mentioning "metrics" and the "affine connection" in the first paragraph you are already in trouble. While I can appreciate Riemannian geometry and so forth, I remind you all that "geodesic" has "geo" for "earth" in front of it, hence remind you of its origins. That is, in geodesy. I suggest two changes to the article: (1) that the introduction be reordered so that the simpler "path over the earth" definition/discusson is given as the first paragraph, followed by the ramp up to the mathematics (you are scaring people away now), and (2) the inclusion of a new section, one coming right after the introduction, on earth geodesics. The new section would talk about such things as geodesics on a sphere, and spheroids such as WGS84, with some simple examples, perhaps even a figure... In short...you math people have to back off! :) The article might even benefit from a split into two articles, one being the simple, earth-centric discussion, the other being the more complex math-centric discussion.

(Consider the poor high-schooler writing his report the night before it is due who just wants to know what geodesic means...and he goes to his favorite reference wikipedia, only to find this article...Yikes!)

As an example of a figure, see: http://909ers.apl.washington.edu/~dushaw/perth_bermuda2.jpg The upper path is the geodesic on a sphere, the lower path is a geodesic on the wgs84 ellipsoid. I am the author of the figure and could release it to wikipedia if you thought it useful. The path goes from Perth, Australia to Bermuda. Bdushaw 21:35, 6 October 2007 (UTC)

## Geodesic =! Great Circle

"in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle." Strictly speaking, this isn't quite right. The Earth is not a perfect sphere (it's an oblate spheroid or ellipsoid - i.e slightly squashed at the poles), which means that the shortest distance between two points (a geodesic) is not quite a great circle. But I'm not sure how best to explain this in the intro without over-complicating it. Wardog (talk) 11:15, 21 February 2008 (UTC)

## Merger proposal

I don't really see why there should be a separate geodesic (general relativity) article. The existence of two articles suggest that they are different concepts, while they are exactly the same concept. (TimothyRias (talk) 15:25, 17 July 2008 (UTC))

Not exactly the same. In GR, there are timelike, spacelike, and null geodesics, and associated questions of causality. Strictly speaking this is separate from the structure of an affine geodesic. Furthermore, the present article focuses on geodesics in Riemannian geometry, which are strong minimizers of the energy functional, rather than just extremals as is the case in GR, and the related notion of affine geodesics. Anyway, I oppose the merge, since much of the material in Geodesic (general relativity) is far too application-specific to be included here (discussion of timelike Killing fields, etc.) siℓℓy rabbit (talk) 15:39, 17 July 2008 (UTC)
There being time and spacelike geodesics is not particular to GR but to the case of a Lorentzian metric, to be comprehensive this article needs to discuss the pseudo-Riemannian case and some applications any way. So, again I really don't see the need for a separate GR article. (TimothyRias (talk) 08:06, 18 July 2008 (UTC))
Since geodesics are relevant to subjects other than relativity---e.g., geodesy---it would seem strange to have this merged with a GR article. —Preceding unsigned comment added by 131.107.0.70 (talk) 19:37, 27 May 2009 (UTC)
I think it's very useful to have a separate article specifically on the use of the term geodesic in general relativity. Having to wade through a mathematical article when one is looking for a physical concept would be a pain. Warren Dew (talk) 05:40, 28 June 2009 (UTC)
I agree with Warren Dew. I was glad that I didn't have to wade through a mathematical article when I was simply trying to define or understand a physical concept (Geodesic). I prefer this article's particular relevance to GR and or other physics phenomena. Specifically this is what I was looking for. Furthermore, looking at the discussion on this talk page, it looks as though this article can be expanded. I would prefer that this article be expanded rather than merged. Therefore, I don't agree with merging this article. Ti-30X (talk) 02:08, 22 September 2009 (UTC)

Well, OK, there clearly is no consensus for such a merger so I'll remove the merge template. I do think this is a shame though. Much of the opposition to the merger seems to be based on the misconception that there are distinct concepts of geodesic in mathematics and physics, while in fact there is only one such concept, the one in geometry (strictly speaking you could say there are two such concepts which coincide for the Levi-Civita connection, but both are relevant to GR). Pretty much everything that is currently said in the geodesic article is relevant to GR in someway, and similarly the geodesic article needs a good applications section discussing the role of geodesics in applications such as GR. But, instead with have some territorial pissings from the mathematics and physics camp, that lead to the propagation of the illusion that there are two distinct concepts here. Such a shame.TimothyRias (talk) 15:26, 5 December 2010 (UTC)

## staggering the gamma symbols

${\displaystyle \Gamma _{~\mu \nu }^{\lambda }}$ should be ${\displaystyle \Gamma _{\mu \nu }^{\lambda }}$ as the Gamma symbols are not tensors. Therefore there is no need to stagger the indices: they can't be lowered anyway. If everyone agrees I will delete the confusing space before mu. Tkuvho (talk) 13:03, 24 February 2010 (UTC)

A better place to raise this issue is at Christoffel symbols, which uses the same, staggered, convention. Personally, I don't stagger the indices on the Christoffel symbols, but some authors do and I don't feel very strongly one way or the other about it. Certainly, if a metric is available then one can lower the index on the Christoffel symbols, and this is usually known as a Christoffel symbol of the first kind: they are more computationally stable than the usual Christoffel symbols. Also, the assertion that the "Gamma symbols are not tensors" is a bit of dogma that essentially depends on what one means by the notation: as the difference between a pair of affine connections, the Gamma symbols certainly do transform tensorially, but this is not usually what is meant by the assertion. Anyway, both of these rejoinders are peripheral to the stylistic issue of whether the indices should be staggered. Just on aesthetic grounds, I would have no objection to changing it to a non-staggered configuration if the change is applied consistently, provided no one makes a compelling case for having staggered indices. Sławomir Biały (talk) 16:46, 24 February 2010 (UTC)
We should first check if this is a math vs physics thing. If physics textbooks tend to stagger these indices there is no point stirring up a tempest in a kettle. Tkuvho (talk) 17:04, 24 February 2010 (UTC)
I tracked down the original discussion four years ago, and copied it at the talk page of Gamma symbols. You can comment there. It seems to me that the reason mentioned there does not apply. Tkuvho (talk) 15:14, 25 February 2010 (UTC)

## Minimizing energy versus length

The section on Riemannian geometry was recently changed rather dramatically, and I'm not certain that I agree with the edit summary there:

• There was a critical error in the "Riem. geom." section. Minimizing the integral of a squared function only provides an upper bound on the sqrd integral, it does not minimize the sqrd integral

For a piecewise ${\displaystyle C^{1}}$ curve (more generally, a ${\displaystyle W^{1,2}}$ curve), the Cauchy-Schwarz inequality gives

${\displaystyle L(\gamma )^{2}\leq 2(b-a)E(\gamma )}$

with equality if and only if ${\displaystyle g(\gamma ',\gamma ')}$ is equal to a constant a.e. It happens that minimizers of ${\displaystyle E(\gamma )}$ also minimize ${\displaystyle L(\gamma )}$, because they turn out to be affinely parameterized, and the inequality is an equality.

The usefulness of this approach is that the problem of seeking minimizers of E is a more robust variational problem. Indeed, E is a "convex function" of ${\displaystyle \gamma }$, so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers within isotopy classes. In contrast, "minimizers" of the functional ${\displaystyle L(\gamma )}$ are generally not very regular, because arbitrary reparameterizations are allowed. (One could probably also argue that the variational problem for ${\displaystyle L(\gamma )}$ is, by itself, rather ill-posed.) 15:28, 23 January 2016 (UTC)

I find the new section much harder to read than the old one. I've reverted to the previous version, but I've also added your explanation above to the article. Hopefully this answers any concerns about correctness. Ozob (talk) 17:53, 23 January 2016 (UTC)
There is a derivation of the geodesic equation from the Riemannian (with positive definite metric) arc length integral in the "Geodesics in general relativity" article. The geodesic equation is a differential geometry topic, so its derivation belongs here, not in the "Geodesics in general relativity" page. Plus the derivation I posted here is much clearer than the one in the general relativity page. I think it's very important to clarify that the only reason the integrand can be squared is because the curve parameter is arc length. With any of the infinite number of possible parameterizations by something other than a constant multiple of arc length, the geodesic equation is not valid. Jrheller1 (talk) 19:56, 23 January 2016 (UTC)
The analysis of Lorentzian manifolds is completely different than in Riemannian signature. There the problem of finding "length minimimizing" paths really is ill-posed as a variational problem. 20:41, 23 January 2016 (UTC)
When using (+,-,-,-) sign convention, the distance between two points is still ${\displaystyle \textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {g_{\alpha \beta }(u^{\alpha })'(u^{\beta })'}}}\operatorname {d} \!\tau }$ (the number under the square root is always non-negative). When using (-,+,+,+) sign convention, the distance is ${\displaystyle \textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {-g_{\alpha \beta }(u^{\alpha })'(u^{\beta })'}}}\operatorname {d} \!\tau }$ (the number under the square root is again always non-negative). So the derivation of the geodesic equation I posted is valid for Lorentzian manifolds. Jrheller1 (talk) 23:18, 23 January 2016 (UTC)
The length functional is very badly non-convex in Lorentzian manifolds. Generally, there are no shortest paths at all in a homotopy class. For example, with the standard Minkowski metric on ${\displaystyle \mathbb {R} ^{4}}$, the standard helix defined by ${\displaystyle (t,\cos t,\sin t,0)}$ is a curve joining the two timelike separated points ${\displaystyle (0,1,0)}$ and ${\displaystyle (2\pi ,1,0)}$, but whose tangent vector is everywhere null (so it has "length" zero). Helices like this are ${\displaystyle C^{0}}$ dense in the space of all timelike curves. 00:18, 24 January 2016 (UTC)

Yes, I've heard that the vast majority of possible spacetimes are pathological. But isn't it true that for any spacetime actually used in general relativity the distance between two points ${\displaystyle \gamma (t_{1})}$ and ${\displaystyle \gamma (t_{2})}$ on a curve ${\displaystyle \gamma }$ is ${\displaystyle \textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}}}\operatorname {d} \!\tau ?}$ Jrheller1 (talk) 01:00, 24 January 2016 (UTC)

That's the length of the curve. But it's not the distance between the points. The distance between the points is the length of the shortest geodesic between them (in Riemannian signature). However, on Lorentzian manifolds, unlike Riemannian manifolds, the geodesic equations are not determined variationally. In physics, one still pretends that they are, and formally computes Euler-Lagrange equations for a functional with no minima. But ultimately, it's the Euler-Lagrange equations themselves that define the geodesics. One can still get a variational characterization of geodesics on Lorentzian manifolds, but it is more involved. The basic fact in this area is that timelike geodesics (in signature (+---)) maximize the proper time among causal curves. 12:52, 24 January 2016 (UTC)
I meant the distance the object has traveled moving between the two points on the curve. The minimum over all possible curves ${\displaystyle \gamma }$ of ${\displaystyle \textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}}}\operatorname {d} \!\tau }$ definitely exists (at least in a reasonable spacetime such as the Schwarzschild metric) and it is exactly the same as the minimum of ${\displaystyle \textstyle {\int _{\tau _{1}}^{\tau _{2}}\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}\operatorname {d} \!\tau .}$ The only reason squaring the integrand is valid is because of parameterization by arc length. If you apply the Euler-Lagrange equations to ${\displaystyle \textstyle {\int _{\tau _{1}}^{\tau _{2}}{\sqrt {\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}}}\operatorname {d} \!\tau ,}$ you get exactly the same result as when you apply it to ${\displaystyle \textstyle {\int _{\tau _{1}}^{\tau _{2}}\pm g_{\alpha \beta }(\gamma ^{\alpha })'(\gamma ^{\beta })'}\operatorname {d} \!\tau .}$ The math is just messier. In Kreyszig's derivation of the geodesic equation, he did actually apply the E-L equations to the functional with the square root integrand and then later switched to arc length parameterization (which eliminates a few terms in the computation). Jrheller1 (talk) 18:55, 24 January 2016 (UTC)
This is more than a case of the math being messier. The least proper time of a causal curve between two timelike separated points is actually equal to zero. (Imagine connecting them by a timelike curve and taking a tight coil of null rays around that curve - see optical black hole.) What you're after is curves that maximize the length, among timelike curves. These may not even exist. For example, consider a space-time that just has one point deleted. Such a manifold is "geodesically incomplete". Even assuming that some version of "completeness" holds (or that we could formulate completeness without knowing in advance what the geodesics are, as in the Hopf-Rinow theorem), existence of timelike geodesics between two timelike separated points seems hard to me. Roughly, one should use a heat flow argument, with the quadratic energy ${\displaystyle E(\gamma )}$, starting with some initial timelike curve and evolving in an isotopy class. It seems like this sort of argument would at least work with compact manifolds with boundary, and then extended to complete manifolds by a suitable truncation argument. But there's a lot in the details, and it's definitely going to be extremely non-trivial. You can find somewhat of a discussion of these issues in Roger Penrose's "Techniques of differential topology in general relativity". What he does there is to take the timelike geodesics as given (via a Hamiltonian approach), and then considers rectifiability using the resulting ambient paths. There is nothing wrong with that approach, but it's somewhat unsatisfactory as a variational characterization of timelike geodesics because one is already assuming what they are, locally. 20:55, 24 January 2016 (UTC)

This article should be primarily oriented towards explaining geodesics on surfaces in 3-dimensional space, because that is what 99.99% of useful geodesics are. The other 0.01% of useful geodesics are geodesics in general relativity, and 99.9% of useful geodesics in GR are geodesics of the Schwarzschild metric. Don't you agree that the derivation of the geodesic equation I posted (which is very similar to Kreyszig's derivation) is valid for the Schwarzschild metric? If there are other useful spacetimes for which this derivation is not valid, they should be dealt with in the "Geodesics in general relativity" page. The derivation of the geodesic equation on the "Geodesics in general relativity" page is just a less clear and concise version of Kreyszig's derivation. It certainly does not address any of the issues you (Slawekb) raised in your previous post. Jrheller1 (talk) 21:39, 24 January 2016 (UTC)

You're the one that brought up Lorentzian space-times. The analysis of geodesics is completely different in Lorentzian space-times. For geodesics on surfaces, it is a theorem that the length functional is dominated by the energy functional, which is more well-behaved analytically. The "space of paths" between two points is a manifold in an appropriate sense, and the energy is a Morse function on the manifold that can be minimized by taking an appropriate flow (see, e.g., [5]). In contrast, the length functional is not a well-behaved Morse function. So, yes, a variational characterization of geodesics even on "simple" things like compact surfaces uses the energy rather than arc length. 22:24, 24 January 2016 (UTC)
When you parameterize by arc length, minimizing arc length of a curve on a surface or manifold via the Euler-Lagrange equation produces exactly the same result as minimizing the "energy" of the curve. So when you parameterize by arc length, both ways are exactly equivalent. When you parameterize a curve by something other than arc length, minimizing the "energy" will give you the wrong answer, minimizing the arc length will give you the right answer. So in the context of geodesics, this concept of "energy" is purely a device for simplifying mathematical calculations when parameterizing by arc length. Jrheller1 (talk) 23:14, 24 January 2016 (UTC)
'When you parameterize a curve by something other than arc length, minimizing the "energy" will give you the wrong answer, minimizing the arc length will give you the right answer.' This is not true. The critical points of the energy functional on the space of continuous, piecewise ${\displaystyle C^{1}}$ paths ${\displaystyle \gamma :[0,1]\to M}$ with fixed endpoints ${\displaystyle \gamma (0)=p}$ and ${\displaystyle \gamma (1)=q}$ are exactly the affinely parameterized geodesics. (These also turn out to be critical points of the functional L, but that fact is not actually useful, because L is very degenerate and it's pretty easy to cook up ridiculous "critical points" of L (e.g., a cuspidal curve, where the variational derivative of L doesn't even exist!)) One still has to say in what sense the resulting geodesic is "minimizing". At worst, that analysis is the same whether length or energy is used. But because the energy is much "smoother" than the length functional, it's a more robust tool. (There are, e.g., homotopy arguments, Morse theory, heat flow arguments, as well as the Hopf-Rinow type arguments.) Indeed, existence of minimizing geodesics was an early triumph of energy methods in the calculus of variations (see, for example, Lyusternik and Fet (1951) "Variational problems on closed manifolds" Dolklady Akad. Nauk SSSR 81:17-18.). 00:39, 25 January 2016 (UTC)
In the simplest possible example of parameterizing a curve on a surface by something other than arc length (a curve ${\displaystyle (x(t)=t,y(t))}$ on the upper hemisphere ${\displaystyle z={\sqrt {1-x^{2}-y^{2}}}}$), minimizing the arc length integral gives the right answer: a straight line ${\displaystyle (t,kt)}$ in the xy-plane for some constant ${\displaystyle k,}$ ${\displaystyle (t,kt,z(t,kt))}$ is a great circle. Minimizing the energy integral gives the wrong answer: it starts out very close to the straight line ${\displaystyle (t,kt)}$ but as the ODE solution curve gets closer to the hemisphere boundary seriously diverges from the straight line.
The coefficients of the first fundamental form of the upper hemisphere ${\displaystyle z={\sqrt {1-x^{2}-y^{2}}}}$ are ${\displaystyle g_{11}=1+x^{2}/z^{2},}$ ${\displaystyle g_{12}=xy/z^{2},}$ and ${\displaystyle g_{22}=1+y^{2}/z^{2}.}$ The coordinates of the curve are ${\displaystyle u^{1}(t)=t}$ and ${\displaystyle u^{2}(t)=y(t)}$ so the derivatives of the coordinates are ${\displaystyle (u^{1})'=1}$ and ${\displaystyle (u^{2})'=y'(t).}$ So ${\displaystyle g_{\alpha \beta }(u^{\alpha })'(u^{\beta })'}$ is ${\displaystyle (1+x^{2}/z^{2})+(2xy/z^{2})y'+(1+y^{2}/z^{2})(y')^{2}.}$ Replacing x with t and ${\displaystyle z^{2}}$ with ${\displaystyle 1-t^{2}-y^{2}}$ yields ${\displaystyle F(t,y,y')=(1+t^{2}/(1-t^{2}-y^{2}))+2ty/(1-t^{2}-y^{2})y'+(1+y^{2}/(1-t^{2}-y^{2})(y')^{2}.}$ This is is the integrand of the "energy" functional and ${\displaystyle {\sqrt {F}}}$ is the integrand of the arc length functional. Applying Euler-Lagrange equation to the "energy" functional yields ${\displaystyle (t^{2}-1)(y^{2}+t^{2}-1)y''-(t^{2}-1)y(y')^{2}+2ty^{2}y'-y(y^{2}-1)=0.}$ Applying E-L to the arc length functional yields ${\displaystyle (y^{2}+t^{2}-1)y''-t(t^{2}-1)(y')^{3}+(3t^{2}-1)y(y')^{2}-t(3y^{2}-1)y'+y(y^{2}-1)=0.}$ It is easy to show that ${\displaystyle y=kt}$ solves the second ODE (for the arc length functional) but not the first (for the "energy" functional).
You seem to think that a curve parameterized by arc length is something special. This is wrong, any piecewise smooth curve can be parameterized by arc length (or infinitely many other parameterizations). Jrheller1 (talk) 07:27, 25 January 2016 (UTC)
Yes, affine parameterization is special. Arc length parameterization is a special case. All you've shown here is that the critical points of the energy functional on the space of curves of a special form are not geodesics (with affine parameter), which isn't very surprising. You need to write down the EL equations for ${\displaystyle (x(t),y(t))}$. That will then give you the geodesic equations. And regardless, you still haven't said exactly what is being "minimized" by this procedure. You've said that this "minimizes the arclength", but haven't specified a domain. Is the domain homotopy classes of C^1 paths with fixed endpoints? Is it the free loop space (for closed geodesics)? Once you have specified the domain, you have to say in what sense a "minimum" is achieved. Is the second variation formula valid for the arclength functional, and the domain you have selected? Can you conclude that you have a global minimum in your class of functions? A local minimum? Is there a compactness result guaranteeing global minima once you have specified the parameters of the problem? None of this is addressed. 12:48, 25 January 2016 (UTC)
I think it's manifestly true that minimizing the energy integral gives the right answer, but to the wrong question. The question you posed is: On the upper hemisphere, among curves of the form ${\displaystyle (t,y(t))}$, which has the least energy? But this isn't a physical setup for arc length minimization! You've constrained the first coordinate not just to move left-to-right, but to do so at a particular rate. Forcing a particle to move at a certain velocity in the x-direction acts like a driving force in the y-direction because of the curvature of the surface. That's sure to change its behavior, and your calculation confirms that. I suspect that if you built a mechanical setup to simulate this situation (say, a ball on a hemisphere with some kind of attachment that was free in one direction and rigid in the other direction, and the rigid direction was forced to obey ${\displaystyle x(t)=t}$), you would observe precisely the path you derived. Ozob (talk) 13:56, 25 January 2016 (UTC)
I would like to disagree with one of your suppositions. Perhaps it's true that 99.99% of all useful closed geodesics are on surfaces embedded in R3. Perhaps it's even true that most of the other useful geodesics are in GR, and that most of those are geodesics of the Schwarzschild metric, though I strongly doubt this. (Geodesics are useful anytime you model something by a manifold, and lots of things are modeled by manifolds.) That does not mean that this article should cater solely to geodesics on surfaces embedded in R3 and geodesics in GR. The article is not titled geodesics on surfaces or geodesics in general relativity. It is not solely about geodesics on well-behaved surfaces or spacetimes. The article is titled geodesic. It's about all geodesics on all manifolds, and its content should reflect that. Ozob (talk) 02:24, 25 January 2016 (UTC)
I don't disagree with anything you say here. I just think that the thing the majority of readers probably need most from this article is a concise but thorough explanation of the differential geometry of curves on smooth surfaces as it relates to geodesics, like that given by Kreyszig. In just a few pages, he explains the exact equivalence of a curve satisfying the geodesic equation and a curve having vanishing geodesic curvature (meaning the curve normal vector is parallel to the surface normal vector at every point where the curvature of the curve is non-zero). Jrheller1 (talk) 07:50, 25 January 2016 (UTC)

A parameterization of a curve in the (x,y) plane of the form (t,y(t)) can represent any possible curve for which there is only one ${\displaystyle y}$ value for a given ${\displaystyle x}$ value. What this means is you can draw any piecewise smooth curve in the xy-plane that has only one ${\displaystyle y}$ value for a given ${\displaystyle x}$ value and find a parameterization for it of the form (t,y(t)) or equivalently (x,y(x)).

There is no need to use the more general form (x(t),y(t)). This will only produce a more complicated ODE (a system of two second order ODEs) with the same result: minimizing the arc length integral will produce the right solution (straight lines through the origin) and minimizing the "energy" integral will produce the wrong solution. You can use an ODE solver with initial conditions y(0)=0 and y'(0)=a for some constant ${\displaystyle a}$ for the "energy" ODE above and see for yourself that it produces a curve that deviates more and more from the right answer as the curve approaches the hemisphere boundary.

A computationally simpler example of the result of applying the Euler-Lagrange equation to both ${\displaystyle \textstyle {\int _{t_{1}}^{t_{2}}F\operatorname {d} \!t}}$ and ${\displaystyle \textstyle {\int _{t_{1}}^{t_{2}}F^{2}\operatorname {d} \!t}}$ is for the minimal surface problem. The minimal surface problem is to find the surface with minimum area for given boundary conditions. To do this it is necessary to minimize the surface area integral ${\displaystyle \textstyle {\int \int _{R}{\sqrt {1+z_{x}^{2}+z_{y}^{2}}}\operatorname {d} \!x\operatorname {d} \!y}.}$ The E-L equation (function of multiple variables version) applied to the surface area integral results in the PDE ${\displaystyle k_{1}+k_{2}=0}$ (in other words, mean curvature is zero everywhere). The E-L equation applied to ${\displaystyle \textstyle {\int \int _{R}(1+z_{x}^{2}+z_{y}^{2})\operatorname {d} \!x\operatorname {d} \!y}}$ produces the Laplace equation ${\displaystyle z_{xx}+z_{yy}=0.}$ This is the wrong answer. The Laplace equation is only an approximation to the minimal surface equation for height field boundaries with relatively slow variations in z. This is just like the solution to the "energy" ODE from my last post. It is a fairly good approximation to the geodesic close to the origin (where z is varying slowly) but gets worse farther away. Jrheller1 (talk) 04:56, 26 January 2016 (UTC)

The reply for minimal surfaces is the same for the reply for your example for geodesics. One seeks critical points for the Dirichlet energy functional of immersions of a domain in R^2 into R^3. Also, you keep referring to the Euler-Lagrange equations as "solving" a minimization problem, and even claim to have "derived" the Euler-Lagrange equations, but nowhere have you actually properly stated the minimization problem that you are claiming to solve, much less prove any kind of uniqueness. Indeed, the area functional and arclength functional both lead to ill-posed variational problems (there is no uniqueness), in part because the space of functions that they are naturally defined on is very wild (W^{1,1}). 10:39, 26 January 2016 (UTC)
Again, it is the right answer to the wrong question. The energy integral you wrote down is physically useful, but not for describing minimal surfaces. Ask yourself the question: Do I expect the graph of an electrostatic potential to be a minimal surface for its boundary conditions? Some thought should convince you that the answer ought to be no. But if you allow yourself three dimensions of freedom, (x(s,t), y(s,t), z(s,t)), then you are not looking at the graph of an electrostatic potential anymore. The physics has changed, and so should your answer. Ozob (talk) 13:42, 26 January 2016 (UTC)

## Change the sentence order in the lead paragraph

Obviously a lot of technically skilled people have edited this article. But that seems to have blinded them to the obvious. For the majority of readers coming to this article all they are interested is great the circle idea of a geodesic. They just need a simple sentence to inform or confirm their notion of what a geodesic is.

The don't need or want a topic sentence like this : "...In the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. If this connection is the Levi-Civita connection induced by a Riemannian metric, then the geodesics are (locally) the shortest path between points in the space....". This sentence contains a raft of references to concepts and words that a general reader will be unfamiliar with. A general reader is likely to either skip the article or start on a trip through the internet trying to figure out what the heck this sentence means.

What the general reader is probably interested in is contained nicely in this paragraph: "The term "geodesic" comes from geodesy, the science of measuring the size and shape of Earth; in the original sense, a geodesic was the shortest route between two points on the Earth's surface, namely, a segment of a great circle. The term has been generalized to include measurements in much more general mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph."

It seems to this reader that the problems of the lead section could be fixed easily by just reversing the order of the two paragraphs. The technical reader can easily skip past the opening section that he probably is well aware of and begin to try to understand the more technical uses of the word in mathematics and physics with the second paragraph of the lead section serving as a summary of the technical information to follow. — Preceding unsigned comment added by Davefoc (talkcontribs) 05:47, 15 April 2016‎ (UTC)

JRSpriggs made the edit suggested above. Thank you. Davefoc (talk) 05:01, 20 September 2016 (UTC)

## Use of t as an affine parameter

Across, much of the article "t" is used as the affine parameter along a geodesic curve. This is a rather unfortunate choice as to many readers it will suggest a relation with time, where no such connection need to exist (or even make sense in the case or Riemannian geometry). I would suggest changing it to something more "neutral" such as λ.TR 11:26, 29 September 2016 (UTC)