Symplectic geometry, phase space and constraints: heuristics

In the last months I read some books on symplectic geometry. Most of them have in common this line of development: definitions & axioms –> main theorems –> “symplectic geometry is useful to physics!”. I do like the axiomatic approach, but I think an heuristic approach is also useful.

During my masters course I read V.I. Arnold’s Mathematical Methods of Classical Mechanics and found Chap. 8 (Symplectic Manifolds) a very good introduction to the subject. Only about 3 months ago I decided to read Arnold’s Chap. 9 (Canonical formalism), its title had not called my attention, so I jumped to others books. But when I started to read it, wow, its approach to symplectic geometry is very interesting and very different from the others approaches! Symplectic geometry is in that chapter ‘more or less’ constructed by physical assumptions. One may even first read Chap. 9 and only after that read the more standard approach of Chap. 8.

One of his main arguments goes like this (these words are mine, more details in Arnold’s book): the phase space is even dimensional (2n) and should not, a priori, have any preferential direction (some physical systems can have a phase space anisotropy, but this is not necessary). We may define an extented phase space which is 2n+1 dimensional, the extra dimension playing the role of the time. In this extended phase space, every physical system is expected to have a unique preferential direction, determining the phase space evolution unambiguously (1). Its easy to show that 2-forms in odd dimensional spaces are always degenerated (2). Let w be a nondegerated 2-form in the phase space (even dimensional), therefore any extension of it to the extended phase space is degenerated and has one, and exactly one, zero-mode (3). We associate this zero-mode v with the direction of the phase space evolution. (!!)

How this zero-mode is determined for each physical system? First lets suppose we are dealing with a very simple physical system: a one-dimensional free particle. We know that the evolution of q and p are given in the phase space by the usual Hamiltonian equations. On the other hand, the zero-mode v is the responsible for the phase space evolution, which means that

\dot q = v1 and \dot p = v2

Therefore, the first component of v is \partial H/ \partial p and the second is – \partial H / \partial q. Which 2-form we in the extended phase espace has this zero-mode v? The solution is we = dL, where L = p dq – p2/(2m) dt (i.e., L is the Lagrangian of the system expressed in first order in the velocities).

More general physical systems have a more general 1-form L. The most general 1-form L (with no explicit dependence on time) is given by

L (\xi) = a\alpha (\xi) d\xi\alpha – H (\xi) dt

In this more general case w_e is still given by dL and w=da\alpha \wedge d \xi\alpha. Is it necessary to w be exact? No, we can generalize this picture a little bit by requiring w to be only closed (not necessarily exact), but here I’ll not go that further, w being exact is sufficiently general to deal with both unconstrained and constrained systems through the usual procedures. Moreover, being w exact it’s easier to prove Liouville theorem (see Arnold’s Chap. 9 for a very elegant demonstration; Liouville theorem is lost if w is not at least closed).

So the most general case we’re going to deal with is we=dL. One should note that we = dL may in general have more than one zero mode, so there is is a problem to specify the evolution of the phase space. Indeed, this happens with constrained theories. One needs first to insert new variables (Lagrange multipliers) to embed the physical manifold (constraint surface) in a R2n+m phase space, then determine the nondegenerated closed 2-form w, which yields unambiguously the phase space evolution.

Nondegenerated closed 2-forms are called symplect structures. A manifold with a symplectic structure is a symplectic manifold. Occasionally degenerated closed 2-forms are called pre-simplectic structures.

=> I’ll stop here. No one can learn symplectic geometry or hamiltonian reduction through this message alone, but I hope I have written some useful remarks.

For more details in the symplectic approach for constrained systems see for instance L. Faddeev, R. Jackiw Hamiltonian reduction of unconstrained and constrained systems, Phys. Rev. Lett. 60 (1988) 1692; M. Henneaux, C. Teitelboim, Quantization of gauge systems, Princeton (1992). There are many references online on symplectic geometry, see A. Cannas homepage for instance (some references are in Portuguese).

(1) Gauge symmetry and constraints will be commented soon.
(2) Let w = 1/2 w_{\alpha \beta} dx^\alpha \wedge dx^\beta, with \alpha, \beta = 1,2,… D. We know that w_{\alpha \beta} = – w_{\beta \alpha}. Hence, det(w_{\alpha \beta}) = (-1)^D det (w_{\alpha \beta}). If D is odd, we have det(w_{\alpha \beta}) = 0.
(3) A zero-mode is an eigen-vector whose eigen-value is null. If w \not= 0 and w_{\alpha \beta} v^\beta = 0 then v is a zero-mode of w.

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