How do symplectic manifolds describe kinematics/dynamics ?

[camel_jockey]

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 :)

 

[quasar987]

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.

 

[camel_jockey]

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!

 

[quasar987]

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!

 

[blue_raver22]

Symplectic manifolds play a crucial role in describing the kinematics and dynamics of classical mechanical systems. In classical mechanics, the state of a system is described by its position and momentum coordinates, also known as the phase space. A symplectic manifold is a mathematical structure that captures the geometry of this phase space.

The use of symplectic manifolds in classical mechanics stems from the fundamental principle of Hamiltonian mechanics, which states that the evolution of a mechanical system can be described by a Hamiltonian function. This function is a scalar quantity that is defined on the phase space and is related to the energy of the system. The Hamiltonian function generates a set of equations known as the Hamiltonian equations, which describe the time evolution of the system's position and momentum coordinates.

Symplectic manifolds provide a natural framework for studying Hamiltonian mechanics. The symplectic structure on the manifold encodes the Poisson bracket, which is a fundamental operation in Hamiltonian mechanics that describes the evolution of quantities in the phase space. The symplectic structure also allows for the definition of conserved quantities, which are important in understanding the dynamics of a system.

In the case of a one-dimensional motion, the cotangent bundle is a symplectic manifold with a two-dimensional phase space. The area form mentioned in Singer's book is the symplectic form on this manifold, which encodes the Poisson bracket and the Hamiltonian equations. This symplectic form, along with the Hamiltonian function, completely describes the kinematics and dynamics of the one-dimensional mechanical system.

In summary, symplectic manifolds provide a powerful mathematical framework for understanding the kinematics and dynamics of classical mechanical systems. They allow us to study the evolution of the system's state and the conserved quantities that govern its behavior. Therefore, symplectic manifolds are an essential tool for scientists studying classical mechanics and its applications in various fields.