# Talk:Equations of motion

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## Equations of circular motion

Changed:

$\phi=\omega_0t+\tfrac12\alpha t^2\,$

to:

$\phi=\phi_0+\omega_0t+\tfrac12\alpha t^2\,$

to correctly account for the initial angular displacement

## Equation 2

I wonder if it's very correct to say that $\frac{\int_{t_0}^{t_1}v\,dt}{t_1-t_0}=\frac{v_1+v_0}{2}$, or (u+v)/2, even if the time difference is assumed to be very small... that's probably why I never have encountered those equations, at school or at the university... Might be useful though for physics programming, for example. —The preceding unsigned comment was added by 82.181.203.54 (talk) 12:01, August 21, 2007 (UTC)

Can anyone remember how those 4 equations are derived (presumably, from Newton's Laws)? -- Tarquin

Hey, maybe It's different in Australia (though I cannot understand why), but I have always been taught that displacement is represented by x, and s represents speed (as distinct from velocity, speed is distance traveled over time, velocity is displacement over time and is a vector) Just a thought.

## Notation

Is anyone else bothered by the fact that at least 3 different notations are used in this article, e.g. d = distance = s; initialocity = u = v0 = vi; etc. etc.

Also, why 'current'? These equations work perfectly well if the 'final velocity' is not current. Ian Cairns 16:48, 13 Oct 2004 (UTC)
To go from (equation 2) average v = s/t to (motion equation 2) s = 1/2 * (u + v) * t, you need to remember that under a constant acceleration, the average velocity is half the final velocity. -- Anonymous Coward

Just adding here that the average velocity is not the half the final velocity, but rather half the initial velocity minus the final velocity. I added this in to the article. - Aldo — Preceding unsigned comment added by 2001:1388:803:9D53:CC95:8D5:857A:FBE1 (talk) 06:16, 26 September 2014 (UTC)

If we're being picky, d represents distance (a scalar) while s represents displacement (a vector). It does bother me, but they're all easily derived using mental logic. ThomasWinwood 19:07, May 11, 2005 (UTC)

Thinking it may be worthwhile changing everything into SUVAT, and adding a note at the start that Vf=V, Vi=U, etc. Basically, mentioning that there's other notations used, but in this article, the notion used will be SUVAT. Anyone agrees? -Aldo — Preceding unsigned comment added by 2001:1388:803:9D53:E0C6:BB2B:E975:4285 (talk) 21:54, 26 September 2014 (UTC)

## Missing equation

What happened to s = vt - ½at²? ThomasWinwood 19:08, May 11, 2005 (UTC)

## Torricelli's Equation

Torricelli's Equation Should this be merged and redirected?Atomiktoaster 00:32, 10 Jun 2005 (UTC)

Hi, I would like to suggest some corrections in this article. There are many examples of equation of motions, but in fact there is only one equation of motion for mechanical systems,

$\sum forces = m \times a$

where, m is the system mass and a is the acceleration. Thank you.

Paulo

## Displacement question

My science teacher told me that they changed the pronumeral for displacement from "s" to "R". Is this OK? The Updater would like to talk to you! 07:48, 22 March 2007 (UTC)

I was taught it was x, and s was for speed (distance over time NOT velicty which is displacement over time). R is resistance isn't it...? don't motion formulea only use lower case? —Preceding unsigned comment added by 202.45.119.135 (talk) 23:02, 19 November 2007 (UTC)

Notation is abused all the time, however, "r" is the radial distance in polar co-ordinates, "x" is one of the components (the others usually "y" and "z") in cartesian co-ordinates and "s" is something high school teachers come up with for displacement as they've already used "d" for distance. "R" is almost always resistance. At any rate it really doesn't matter as long as it's clearly labeled. Durinix (talk) 13:32, 2 April 2008 (UTC)

You can call it what ever you like. Different text books use different symbols. "s" is the most common for displacement in physics, in engineering it could be "d", "s" or whatever random variable you have in your free body diagram for example. —Preceding unsigned comment added by 66.134.78.115 (talk) 18:02, 22 July 2008 (UTC)

## Only one equation?

Why is this article called "Equation of motion" when it is clearly about more than a single equation? Even the opening sentence says "equations of motion". --Dr Greg (talk) 13:15, 4 March 2008 (UTC)

## Correct Equation of motion

The correct equations are given with proper derivation at User:Narendra_Sisodiya/mechanics

Equation give on the page are not correct as

• they do not maintain notion of function in S(t)
• the initial location and time was assumed to be zero,zero , why to give subset of equations
• they do not have vector associated with them, without a vector a velocity becomes speed. —Preceding unsigned comment added by Narendra Sisodiya (talkcontribs) 21:58, 17 July 2008 (UTC)
The reasons above do not make the equations incorrect. They are just less complete than they could be. See my reply to your question on my talk page. --Dr Greg (talk) 16:26, 18 July 2008 (UTC)

## Reverting redirection

If you're going to do a complete redirection of this article to another one, then it would be better to have some discussion on the talk page as well as the edit summary. I propose that this information stay. It is useful, although technically it is covered elsewhere, without providing the simplified classical statememnts of the equations of motion I think that something is lost in removing this article. I would be happy for a redirect if the final pages ended up having this content in them (without the requirement for readers to understand calculus).Angelamaher (talk) 10:20, 19 September 2008 (UTC)

The problem with this article (Equation of motion)is that the title does not reflect the content. The content is about linear motion (kinematics). But the title is "Equation of Motion" which is Dynamics (or kinetics for some authors),and are related to Newton's second law of motion. The equations shown in the current article are already dealt in the article about Kinematics. I would suggest the content of this article be moved to the article about kinematics, so all these equations can be shown there. And again, this article should not exist in my view. _Sanpaz (talk) 15:30, 19 September 2008 (UTC)
These are the equations of motion (See Momentum):
$\sigma_{ij,j} + F_i = \rho \frac{d v_i}{dt}$

or

$\ \sum{\mathbf{F}} = {\mathrm{d}\mathbf{P} \over \mathrm{d}t}= M \frac{\mathrm{d}\mathbf{v}_{cm}}{\mathrm{d}t}=M\mathbf{a}_{cm}$

- Sanpaz (talk) 15:41, 19 September 2008 (UTC)

## Classic version?

I notice an IP's inconsistently replaced "s" with "x" in some of the equations and descriptions. I'm reverting that. My reasoning is:

• "s" is by far the most common form.
• An encyclopaedia is meant to be current, not a historical re-enactment. A note of the historical use of x is sufficient.
• It's inconsistent with other use on this page (the change was not complete).
• It's inconsistent with other use elsewhere in the encyclopaedia (eg the Norse, Dutch, etc versions of this page). —Preceding unsigned comment added by DewiMorgan (talkcontribs) 07:12, 12 March 2010 (UTC)

## Motion of charged particles

Some equations for the motion of charged particles shold be inserted.--86.126.25.12 (talk) 18:33, 29 August 2011 (UTC)

I think the current definition of "Equations of motion": "Equations of motion are equations that describe the behavior of a system as a function of time.", is too broad, too vague. The heat equation describes the behavior of a system in which heat flow occurs as a function of time, but it isn't an equation of motion. I don't have an alternative formulation at this point, just wanted to see if people would agree with me. /Andreas

## Derivations

The derivations should be updated as they are too brief and need to be more intuitive, the derivations on the [Velocity] page (in the equations of motion section) are much better and easier to follow. "Shine on you crazy diamond" (talk) 14:33, 11 December 2011 (UTC)

## Only one body?

This article appears to present the equations of motion of a single body in a uniform gravitational field, or perhaps in no gravitational field at all. Yet the title does not imply such a limitation. I see no equations that describe the motions of two bodies in mutual gravitational attraction, and no discussion of the difficulty (perhaps the impossibility) of constructing closed-form equations for systems containing three or more bodies (the Three-body problem). Something needs changing. David Spector (talk) 20:54, 29 December 2011 (UTC)

## Recent rewrite

The recent rewrite of this article goes too far imho. The previous version concentrated on the classic equations of motion which, for most people, is what they are seeking in this article. If it is felt there is a need for mention of these more esoteric areas then surely this could be achieved better by pointing to other suitable articles rather than swamping this article and potentially confusing the reader simply seeking information on the "equations of motion". Abtract (talk) 22:47, 30 December 2011 (UTC)

In what way does this article go too far? and I have included links to the main articles, using the {{main}} under each heading. Here are the issues I have with the previous version of the article:
• It was excessively padded, and repeated over and over again on different notations of the equations. Some articles on far more important and general equations, such as the Dirac equation and Schrödinger equation, only do this because they are broad explanations of these fundamental formulae. The silly "SUVAT" equations are not even that important - Newton's law is, and the pure definitions of position, velocity and acceleration are.
• These equations are not even "equations of motion" at all - that’s just the title people usually give to them. They are no more than integrating the definitions of velocity and acceleration, and using common sense of average velocity. From there the other equation can obviously obtained by the others. These are very special cases of kinematics, only true in some situations. The article should have a completely general scope as to what equations of motion are, of course SUVAT equations can be mentioned as special cases but should only be in a short section towards the end. The reader will get the wrong idea/not know what are equations of motion.
• As such I re-wrote the article for what equations of motion actually comprise.
I'm telling you - please read any decent physics or maths text book around degree level and you will definitely find that equations of motion refer to the equations I have inserted: Newton’s law, the Euler-Lagrange equations, and Hamilton’s equations. The SUVAT equations were referenced from GCSE/school level, by the look of them. Are they really reliable sources - given that school pupils are taught things, but then progressing onto university have to re-learn stuff? It has to be simplified at school of course, but this is not a school book, it’s a Wikipedia to provide absolute facts.
I know it’s my fault for not proposing such changes first, and not adding more references - it was a very quick edit. I will definitely add more sources. The reason why I didn't write here is because, as a matter of fact, I actually do write on many talk pages first, but normally no one answers so I end up re-writing the article anyway. No one objects to them. In fact this is the first time!
Does this answer your question? I appreciate that if the style of writing is bad and you think it could be better, then you are more than welcome to tell me off, however I would say the scope now set in the article is appropriate. By the way I'm not finished - i'll add more sources, try to simplify the explainations, and make the article a coherent unity. -- 12:18, 31 December 2011 (UTC)

## New re-write

Here is my intended structure. In each case everything will be very briefly stated, and plenty of links will be included.

Also i'll modify the see also section. -- 12:45, 2 January 2012 (UTC)

Well... after all that time and energy, it took an embarresing amount of time to re-write the article and carefully structure it for a wide-as-possible audience. In drafting before saving, at first I ended up writing far too classical mechanics - as though it was a text book on the subject... I'll add more links and referances - honestly I will get round to that =| I keep saying so and it hasn't happened much yet...-- 19:20, 2 January 2012 (UTC)
Its not even near finished yet, I havn't done much of a good job yet - but it will be awesome, please be patient...--- 19:22, 2 January 2012 (UTC)

Great edit, but please note that given [1]-[2] in the section constant linear acceleration:

\begin{align} \bold{v} & = \int \bold{a} {\rm d}t = \bold{a}t+\bold{v}_0 \quad [1] \\ \bold{r} & = \int (\bold{a}t+\bold{v}_0) {\rm d}t = \frac{\bold{a}t^2}{2}+\bold{v}_0t +\bold{r}_0 \quad [2] \\ \end{align}

The equation of average velocity can be derived as:

$\bold{a}t^2 = \left( \bold{v} - \bold{v}_0 \right)t$
$\bold{a}t^2 + 2 \bold{v}_0t = \left( \bold{v} + \bold{v}_0 \right)t$
$\frac{\bold{a}t^2}{2} + \bold{v}_0t = \left( \frac{\bold{v}+\bold{v}_0}{2} \right )t$
$\bold{r} - \bold{r}_0 = \left( \frac{\bold{v}+\bold{v}_0}{2} \right )t \quad [3]$

That said average velocity is redundant in the derivation. Please refer to other text besides the one cited if this is not clear. Also, spatial information is lost if you take the magnitude, since bold r is a vector and r is just a scalar. hoo0 (talk) 02:10, 14 January 2012 (UTC) Actually, please just refer to Kinematics#Kinematics_of_constant_acceleration, the better way to express this. hoo0 (talk) 03:40, 14 January 2012 (UTC)

## A proposed definition

Please consider the general definition below:

The equations of motion describe the behavior of a physical system as a set of mathematical functions. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in special relativity. If the dynamics of a system is known, the equations are the solutions to the differential equations describing the motion of the dynamics.

Note that by this definition, differential equations are not by default Equations of motion, only their solutions describing the motion. This is consistent with the attempts in the article to express motion as functions of time, position, velocity and acceleration. This definition also separates the kinematics nature of the equations from the dynamics of force, momentum and energy. It is also valid in general relativity, as Einstein field equations are differential equations describing gravity with solutions such as the Schwarzschild_metric. hoo0 (talk) 06:22, 14 January 2012 (UTC)

Fine by me - you are obviously very clear in explainations! =) (I did say my edits wern't complete/decent - the point was to re-transform the article into a proper setting from just those SUVAT equations being the only + dominant theme). Sometimes though, it is the case that the differential equation for the system and/or the solution are referred to as the equation/of motion. This is just analogy with (say) Newton's law of motion - a differential equation (of motion), and the solutions are also equations which describe motion. In principle the line should be drawn as you have done it, in practice this isn't always the case. Anyway by all means - add the definition into the article!
One point on a recent edit of yours [1]: you wrote "A = acceleration". I appreciate your dimensional correction to one of my mistakes in the equation,
$- \frac{GmM}{|\bold{r}|^2} \bold{\hat{e}}_r + \bold{R} = m\frac{{\rm d}^2 \bold{r}}{{\rm d} t^2} + 0 \Rightarrow - \frac{GM}{|\bold{r}|^2} \bold{\hat{e}}_r + \bold{A} = \frac{{\rm d}^2 \bold{r}}{{\rm d} t^2} \,\!$
Indeed R/m is an acceleration, but this should be specific: "A = component of acceleration of particle due to air current at position r and time t." "Acceleration" alone could mean the term on the left side, the resultant acceleration, or the gravitational acceleration becuase the projectile is in a gravitational field, or whatever. Just a minor point. Later I'll come back to this article; a bit busy right now...-- 15:08, 14 January 2012 (UTC)
No one has replied - i'll just do it myself.
• The above definition wil be incorperated into the lead,
• the SUVAT derivations might be modified (though I appreicate the point-out - I hardly see the need to derive the average velocity when it can just be stated on the spot: thats an equation constructed immediately, and less lines of maths),
• and will further add links and extend the scope of the article.
-- 21:20, 16 January 2012 (UTC)

## Hamiltonian symmetry or anti-symmetry?

Given Hamilton's equations:

$\dot{p}_i = -\frac{\partial H}{\partial q_i} \quad \dot{q}_i = + \frac{\partial H}{\partial p_i}$

then Maschen correctly wrote that if you take each equation, switch pi for qi, and H for −H, the equations are the same:

$\dot{p}_i = -\frac{\partial H}{\partial q_i} \quad \xrightarrow[]{p_i \rightleftharpoons q_i,\, H\rightarrow -H} \quad \dot{q}_i = -\frac{\partial (-H)}{\partial p_i} \quad\rightarrow\quad \dot{q}_i = +\frac{\partial H}{\partial p_i}$
$\dot{q}_i = +\frac{\partial H}{\partial p_i} \quad\xrightarrow[]{p_i \rightleftharpoons q_i,\, H\rightarrow -H} \quad\dot{p}_i = +\frac{\partial (-H)}{\partial q_i} \quad\rightarrow \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}$

So its not about the antisymmetry of only interchanging pi and qi, he wrote about simaltaneously switching H for −H also. 07:29, 31 March 2012 (UTC)

I will tweak this bit, it reads better to consider the antisymmetry between H for −H since this is just one change, rather than the additional change between p and q. Maschen (talk) 14:16, 15 August 2012 (UTC)
Nevermind - if you change one pair you change the other, seems fine as it is... Maschen (talk) 14:21, 15 August 2012 (UTC)

## SUVAT equations (again)

The SUVAT equations do have limited application; they can only be used in classical mechanics since it would be possible to accelerate beyond the speed of light, and even then they only apply for constant acceleration.

"These formulae have limited application - they can only be used in situations where velocities are much less than the speed of light and accelerations can be neglected."

is correct. Why delete it? And why "relativity is the exception not the norm"? Isn't relativity the extension from classical mechanics to speeds approaching light? Back-reverted. 10:22, 3 April 2012 (UTC)

Actually the SUVAT equations are perfectly valid within relativity, provided all the quantities are measured relative to the same inertial frame. (I don't understand what "and accelerations can be neglected" is supposed to mean; if it were true you would have to put a = 0.) The equations aren't true if t is proper time and/or a is proper acceleration, but for coordinate time and coordinate acceleration they are correct (exactly, not approximately). The restriction that a is constant always applies, whatever the scenario. It is true that you cannot accelerate a particle with mass beyond the speed of light, but the equation works, exactly, whenever the initial and final speeds do not exceed c. If you were maintaining constant coordinate acceleration a, your proper acceleration $a(1-v^2/c^2)^{-3/2}$ (as measured by an accelerometer) would approach infinity as your coordinate speed approached c, thus preventing you from reaching light speed.
So the "limited application" of the equations is that you can't use them for a particle with mass when either initial or final velocity would exceed light, but otherwise they are exactly correct. When applied to a notional trajectory, not associated with a particle, they are always true without exception. Or to put it yet another way, they are always kinematically correct, but the kinematics may be dynamically inachievable in some circumstances. -- 13:38, 3 April 2012 (UTC)
Sorry - you're correct I wasn't thinking. I think when I wrote the statement "accelerations can be neglected" was to refer to weak gravitational fields (i..e gravitational acceleration), but strictly for that section the acceleration is just the gravitational acceleration g, which is clearly non-zero but constant for what matters. This equation switched me off:

$v = u+at$

Thank you - I'll revert and apologies for not thinking clearly and behaving stupidly and rudely. =( 14:04, 3 April 2012 (UTC)

## Motion on spirals

Some equations (kinematic and dynamic) concerning the motions on spirals should be added.--188.26.22.131 (talk) 14:57, 14 August 2012 (UTC)

There already are. Spiral motion is just a combination of radial and angular motion. These equations describe radial and angular motion, and have their special cases summarized in the table following them. There is also the 3d generalization for rotational motion in space. There is also Newton's 2nd law for rotation, which would predict any form of rotational motion (obviously classically). Maschen (talk) 08:00, 15 August 2012 (UTC)

## Plentiful emphasis on position vector and generalized coordinates?

Is this necessary?

The section Equations of motion (Quantitative) first mentions the position and momentum vectors, then gives a detailed description in section Equations of motion (Position vector) of the position vector in various coordinate systems. This should be in the main article Position vector.

Same for the generalized coordinates in the section Equations of motion (Generalized coordinates). Much of this could be moved to Analytical mechanics (D'Alembert's principle, generalized coordinates and constraints), because it’s a brief general description, rather than the main article generalized coordinates since links direct to there.

I intend to make these changes now. Maschen (talk) 10:47, 15 August 2012 (UTC)

Done. Maschen (talk) 13:32, 15 August 2012 (UTC)

## Turn it into an equation, someone?

I added some information regarding the average velocity for constant acceleration: Arithmetic mean of the final velocity minus the initial velocity.

However, I have no idea how to put this in so it looks like an equation. if someone could help out, it'd be greatly appreciated. -Aldo — Preceding unsigned comment added by 2001:1388:803:9d53:cc95:8d5:857a:fbe1 (talk) 06:18, September 26, 2014 (UTC)

All of the SUVAT equations are included, including the average velocity equation. Your edit was in good faith but redundant and reverted (and I agree with the reversion). M∧Ŝc2ħεИτlk 20:41, 26 September 2014 (UTC)

I actually got the equation wrong, it was the initial PLUS the final, divided by 2.

It is included in the SUVAT equations, but there's no pointing out that it's the equation for average velocity. It's just displacement = (vf-v0)/2 * t, which is the displacement, USING the average velocity. However, before that equation, there's no indication that it's the formula for the average velocity. Perhaps this should be added? What are your thoughts on this? -Aldo — Preceding unsigned comment added by 2001:1388:803:9D53:E0C6:BB2B:E975:4285 (talk) 21:48, 26 September 2014 (UTC)

Overall I think the SUVAT equations are covered fine. We don't really need to emphasize the average velocity in one of the equations - anyone with some basic mathematical background would be able to tell what the average velocity is when they see it. If readers do not know where the equations come from, they can click the "derivations" tabs to open up the derivations (for example under Equations of motion#Constant linear acceleration: collinear vectors). M∧Ŝc2ħεИτlk 23:03, 26 September 2014 (UTC)
Hope this and this are OK. M∧Ŝc2ħεИτlk 23:22, 26 September 2014 (UTC)

Works for me; what about the notation though? As someone mentioned above, the article uses both SUVAT and Vf, Vi, etc... Perhaps we should add a section mentioning that there's many notations in use, and saying that the article will only use one (Personally, I'd go for SUVAT). Maybe that would make it clearer?

At the moment, it seems like a collage of equations (correct equations, obviously) but taken from different sources, which makes the article kind of difficult to follow. Thoughts? -Aldo — Preceding unsigned comment added by 2001:1388:803:9D53:E0C6:BB2B:E975:4285 (talk) 01:17, 27 September 2014 (UTC)

Apologies for the late reply and I mean no rudeness at all, but are you reading the article?
The standard notations used by most sources, including the "suvat" symbols, are already there. We don't need to include a plethora of notations for the quantities in the equations, this is an article on what "equations of motion" generally means and should not sidetrack too much on notations for physical quantities. Sometimes different sets of notations for sets of equations need clarification, like for the thermodynamic potentials where possible conflicts may occur, and similiarly for the constant acceleration equations here. But in this article we alrady have the notation which fits in most kinematic equations (t, r, r0, v, v0, a) and for familiarity mention the s, u, v, a, t symbols too. We should not bias one or the other. Other sources will use other (perhaps obscure/non-standard) notations, but it is up to them (authors of the sources) to make their notation clear. Also, why is it difficult to follow if there are different sources citing the sets of equations?
Nevertheless, many people will expect the suvat equations in this article and they have been repeatedly edited in terms of notations and derivations. I'll have a look later on making the entire section clearer in notation, presentation, and sources, but in the mean time it doesn't seem to be an urgent problem. Best, M∧Ŝc2ħεИτlk 21:33, 27 September 2014 (UTC)