How do symplectic manifolds describe kinematics/dynamics ?

  • #1
I understand that symplectic manifolds are phase spaces in classical mechanics, I just don't understand why we would use them. I understand both the mathematics and the physics here, it is the connection between these areas that is cloudy...

What on Earth does the symplectic form have to do with the physics, or the motion, of such a system?

I was reading in Singers "Symmetry in mechanics" and she wrote about a one dimensional motion, such that the cotangent bundle was a two-dimensional symplectic manifold. She did this by showing that there exists an "area form" which mixes position coordinates and momentum coordinates. But whyyyy?!?! The area form, though it may be a symplectic form, tells me nothing about how such a one-dimensional mechanics problem will turn out/time-develop?

Very angry, please help me :)
 

Answers and Replies

  • #2
quasar987
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Simply put, there is a canonical symplectic form w on the cotangent bundle T*Q of the configuration space Q of a physical system, and given the hamiltonian H:T*Q-->R of this physical system, we can use w in a certain way to get a certain vector field on T*Q, written {H, } or X_H and called the symplectic gradient of H (because it is obtained from w exactly the same way as the ordinary gradient is obtained from a Riemannian metric (read "scalar product")). And it turns out that the flow equations dc(t)\dt = X_H(c(t)) for this vector field, when written in coordinates, are precisely the hamiltonian equations of motion. Conclusion: the physical path taken by a system of hamiltonian H in the state (q0,p0) at time t=0 is the unique flow line of the hamiltonian vector field X_H that passes through (q0,p0) at t=0.
 
  • #3
Simply put, there is a canonical symplectic form w on the cotangent bundle T*Q of the configuration space Q of a physical system, and given the hamiltonian H:T*Q-->R of this physical system, we can use w in a certain way to get a certain vector field on T*Q, written {H, } or X_H and called the symplectic gradient of H (because it is obtained from w exactly the same way as the ordinary gradient is obtained from a Riemannian metric (read "scalar product")). And it turns out that the flow equations dc(t)\dt = X_H(c(t)) for this vector field, when written in coordinates, are precisely the hamiltonian equations of motion. Conclusion: the physical path taken by a system of hamiltonian H in the state (q0,p0) at time t=0 is the unique flow line of the hamiltonian vector field X_H that passes through (q0,p0) at t=0.
Aha! Singer does not mention H.

So the total system must have a Hamiltonian also, that is (Manifold, w, H), for us to be able to get the dynamics (= chosen path as function of time) of the system?

If that is the case, then I understand it now. Thank you very much, Sir, for your reply!
 
  • #4
quasar987
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So the total system must have a Hamiltonian also, that is (Manifold, w, H), for us to be able to get the dynamics (= chosen path as function of time) of the system?
Right!
 

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