I am just trying to wade through literature on classical mechanics and I really don't know where to start, everything is Fibre bundle this or manifold that, and doesn't really ease you in to the topic. I'm sure this is a common question but I would like help with a couple of specific points:

  • What would be the main points to take away from the "symplectic structure of phase space". Specifically what does knowing its symplectic do for us? We quote it all the time when talking about Hamiltonian mechanics/Liouville equation/Poisson brackets... etc ... In other words what would I be saying in differential geometry terms when I say " a canonical transformation preserves the symplectic structure?"

I understand this may have been answered before but so far having seen the previous question/answers I'm still struggling. I'm specifically after answers relating to classical mechanics linear/non-linear alike, but not relativity if it can be avoided.

So to summarise: what are the salient points of a geometrical understanding of classical mechanics and what does this do that a basic understanding doesn't!

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    $\begingroup$ This seems like a very broad question for this forum, but it might attract good answers so I am not voting to close (as I usually would for questions that are "too broad"). I do recommend that you consider narrowing the scope of this question and maybe split into two or three - and clean up a bit (for example, you meant to ask about "the difference between a Riemannian manifold and a non-Riemannian manifold" but you lost your train of thought in the middle...) $\endgroup$ – Floris Sep 4 '14 at 20:32
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    $\begingroup$ Related, possible duplicate: physics.stackexchange.com/q/8256, perhaps you could edit your question to be any remaining questions not addressed in that question? $\endgroup$ – alemi Sep 4 '14 at 21:04
  • $\begingroup$ What are you hoping to get out of classical mechanics? Why do you feel that a study of manifolds would help - especially if you don't want to get into relativity? Is it for statistical thermodynamics, non-linear systems, transition from classical to quantum mechanics, ....? Looking at your last sentence: maybe you and I are asking the same question! $\endgroup$ – akrasia Sep 4 '14 at 21:19
  • $\begingroup$ @akrasia I think we are indeed!It is to push my own understanding beyond what I "need" to know! I am looking at chaos personally and field theories using Lagrange/Hamiltonian prescription. I just get bombarded by technical language when I read books on differential geometry and was wondering if someone could start me off I would be able to do the rest myself. How about yourself does this apply to you too? :) $\endgroup$ – AngusTheMan Sep 4 '14 at 21:28
  • $\begingroup$ @Qmechanic, If I split it up into two questions: 1) symplecticity, 2) Manifolds/tori, would this be okay?I do apologise but it was best to formulate my question when it was in my mind! $\endgroup$ – AngusTheMan Sep 4 '14 at 21:29

The main difference between Hamilton and Lagrange/Newton mechanics is, that it happens directly on the phase space, i.e. any point on your manifold already fully determines the state of your system. Intuitively, you realize this by specifying position and momentum coordinates. On a mathematical level, the world we see is some smooth manifold (a priori not even necessarily Riemann), this is the place where you specify your position coordinates. To specify momentum coordinates, you have to consider points from the cotangent space, therefore the structure where you can fully specify your state is the full cotangent bundle of your original manifold.

However, the cotangent bundle may be regarded as manifold on its own right (with twice the dimension of the original manifold). Furthermore, you can construct a canonical(!) symplectic structure quite easily by using the derivative of the natural projection. The reason, why it makes sense to put the "original manifold" aside and operate only on symplectic manifolds resp. cotangent spaces, is Darboux' Theorem, which basically says that every symplectic manifold is actually locally equivalent to the cotangent space of some manifold. Polemically simplified, one could say "The cotangent bundle is the symplectic manifold".

Lets now take a closer look at the symplectic structure: Locally, it looks like $$ \omega=d\vec{q}\wedge d \vec{p}=dq_i\wedge dp^i $$Using this, you can construct a volume form (up to some constant) as $\omega^n$ where $2n$ is the dimension of your manifold. Especially, you see that this volume is even oriented - locally, you can actually imagine some region as a region in euclidean space. The famous Liouville theorem states, that the Hamiltonian phase flux (i.e. the one-parametric group of the corresponding autonomic system) leaves the symplectic structure intact, therefore it also preserves the volume. This is an important consequence especially in statistical mechanics. However, this is not main the point about the symplectic structure. The important thing is, that the symplectic form together with the Hamiltonian determines the trajectory: Trajectories are defined as integral curves of the Hamiltonian vector field $X_H$, which is uniquely determined by the Hamilton function and the symplectic structure by $$ \omega(\cdot,X_H)=dH $$ Vividly spoken: If you throw some particle onto a symplectic manifold, it will move along the Hamiltonian vector field.

The point now about canonical transformations is, that it has the symplectic form as an integral invariant, or, mathematically speaking, it is a special symplectomorphism. Of course, this yields as consequence, that the local equations governing the trajectories are not affected. You can therefore think of a canonical transformation just as a reparametrization of some region of the symplectic manifold, just like a change from Cartesian to spherical coordinates. In this sense, the major advantage over Lagrange is, that you can simultaneously transform both $p$ and $q$, which of course leads to much easier equations in the end, as you can use canonical transformations to make all coordinates cyclic. This is the basic idea of Hamilton-Jacobi theory.


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