Talk:Second class constraints

The author calls $r^{2}-R^{2}=0$ a secondary constraint, and calls the following ones with different names, thus giving a wrong impression of what is really meant by a secondary class constraint. Besides, I dont think it is clear how those constraints are acquired.

Sticking to this example it is necessary to clarify why we need all 4 of the constraints, and why they are all classified as secondary class constraints.

--Zjappar 22:37, 5 August 2006 (UTC)

The problem you're having is that you (quite understandably) are confusing "secondary constraint" with "second class constraint" which are completely distinct concepts. A secondary constraint is a constraint that one finds by demanding, on the grounds of consistency, that the primary constraints (whatever constraints you started with) have vanishing time derivative. So not only do you demand that your constraint \phi = 0 you also demand that \dot{\phi} = 0. Sometimes this will lead to a new constraint (frequently not). Constraints arrived in this manner are called "secondary constraints". The distinction between primary constraints (whatever constraints you start with in the Hamiltonian formalism) or secondary constraints or tertiary constraints, etc. is largely artificial and unimportant. A dymanical quantity is called first class if its poisson bracket with all constraints vanishes on-shell, and second-class if its poisson bracket does not vanish with all constraints on-shell, then it is called second class. Hence, a second class constraint has nonvanishing poisson bracket with at least one nother constraint. The distinction between primary and secondary constraints is very important. Perhaps the Dirac bracket article I wrote will clarify. Steve Avery 17:11, 4 December 2007 (UTC)

Constraints versus conditions fixing the form of the Hamiltonian off-shell

I approach this material from a slightly different perspective (see Dirac bracket, I would make $r^{2}-R^{2}$ a primary constraint and drop the Lagrange multiplier), but I believe that this article is slightly incorrect in a couple subtle ways:

1) Relating what has been done here to the formalism I am familiar with, ${\dot {\lambda }}$ corresponds to $u_{1}$, that is the arbitrary coefficient of the constraint added to the naive hamiltonian. I agree, then, with the Hamiltonian, and that we have a primary constraint $p_{\lambda }$. The primary constraint, indeed gives a secondary constraint, $r^{2}-R^{2}$. We then should add this secondary constraint into the Hamiltonian with an arbitrary coefficient $u_{2}$. Consistency conditions result in a "tertiary" constraint $r\cdot p$. We need to then add it, too, into the Hamiltonian with an arbitrary coefficient $u_{3}$. Now, the consistency conditions result in no further constraints, but they do fix some of the u's. If I calculate correctly, we find that

$u_{2}={\frac {1}{r^{2}}}\left({\frac {{\vec {p}}^{2}}{2m}}-{\frac {1}{2}}mgz\right)+{\frac {\lambda }{2}}$

and

$u_{3}=-{\frac {{\vec {r}}\cdot {\vec {p}}}{mr^{2}}}.$

Note that u_1 or ${\dot {\lambda }}$ never got fixed. This corresponds to the fact that $p_{\lambda }$ is in fact a first class constraint. The gauge freedom associated with it, is the freedom to pick $\lambda$. So, in fact, this Lagrangian leads to a Hamiltonian system with one first class constraint, and two second class constraints. This distinction between new constraints and consistency conditions that fix the u's is important when one wants to compute Dirac brackets (if one wants to quantize the system).

2) In the literature that I have encountered, people would call secondary, tertiary, and quaternary constraints all "secondary constraints". This is more or less just an issue of semantics.

3) I am not sure what "off-shell constraint" is supposed to mean. If anything, I would call that an "on-shell" constraint since it need only be satisfied when the equations of motion are satisfied. If the constraint had to be satisfied off-shell too, then we should drop it from the Hamiltonian.

If no one responds within a few days, I will begin fixing the above problems. I think the article needs some more drastic changes too, but I will propose more changes here before making any more serious alterations. Steve Avery (talk) 19:03, 6 December 2007 (UTC)

I finally made good on my promises above. I fixed the problems I saw above; however, I did not change the example by eliminating the Lagrange multiplier term from the Hamiltonian procedure and just use r^2-R^2 as a primary constraint. There may be some value in keeping it as it is, but I think the example would be improved by that streamlining.

I added a section heading for the example. I'll start thinking about the more drastic changes. I'll post here some proposed changes before making further major alterations. Steve Avery (talk) 05:52, 13 January 2008 (UTC)

I think the example in the article as it currently stands is incorrect. The constraints are

$p_{\lambda }\approx 0$

$r^{2}-R^{2}\approx 0$

${\vec {p}}\cdot {\vec {r}}\approx 0$

$\lambda R^{2}-mgz+{\frac {p^{2}}{m}}\approx 0$

and the extended Hamiltonian is

$H={\frac {p^{2}}{2m}}+mgz+{\frac {3g}{R^{2}}}p_{\lambda }p_{z}-{\frac {\lambda }{2}}(r^{2}-R^{2})+u_{1}p_{\lambda }+u_{2}\left(r^{2}-R^{2}\right)+u_{3}{\vec {p}}\cdot {\vec {r}}+u_{4}\left(\lambda R^{2}-mgz+{\frac {p^{2}}{m}}\right)$

subject to the relation $u_{i}\approx 0$. This gives rise to the correct time evolution

${\dot {\vec {r}}}\approx {\frac {\vec {p}}{m}}$

${\dot {\lambda }}\approx {\frac {3g}{R^{2}}}p_{z}$

${\dot {\vec {p}}}\approx \lambda {\vec {r}}-mg{\hat {k}}$

${\dot {p_{\lambda }}}\approx 0$.

In addition, $\left\{\phi _{i},H\right\}_{PB}\approx 0$.

The matrix $M_{ij}=\left\{\phi _{i},\phi _{j}\right\}_{PB}$ is

$M\approx {\begin{pmatrix}0&0&0&-R^{2}\\0&0&2R^{2}&0\\0&-2R^{2}&0&mgz+2{\frac {p^{2}}{m}}\\R^{2}&0&-mgz-2{\frac {p^{2}}{m}}&0\end{pmatrix}}$.

The iterated method of getting the constraints is incorrect because at stage i, solving $\left\{H+\sum _{j\leqslant i}u_{j}\phi _{j},\phi _{i}\right\}_{PB}\approx 0$ fails to take into account $u_{k}\phi _{k}$ terms for $k>i$.

Hep thinker (talk) 11:35, 25 March 2009 (UTC)

Just in case anyone is wondering, I obtained these constraints by starting out with the Euler-Lagrange equations of motion, and the momentum definition $p_{i}=\partial L/\partial {\dot {q^{i}}}$. Then, using the Euler-Lagrange equations and the momentum definition again, I calculated the time derivatives ${\dot {q}}$'s and ${\dot {p}}$'s, and only then did I try to find out an extended Hamiltonian which would give rise to these time derivatives (after imposing the constraints). Hep thinker (talk) 13:42, 25 March 2009 (UTC)