Classical phase space flow exact solution

[Liquidxlax]

Homework Statement

if i wanted to obtain an "exact" solution for flow s(t|k) k=(q,p) with a hamiltonian

H(k) = x(ak)

use the fact aJa = 0 where J is the poisson matrix

Homework Equations

The Attempt at a Solution

I hate obscure proofs... i like actual question so I'm lost on what the definition of exact solution for the flow is

do i need to prove that s is sympletic and therefore is canonical?

any sort of idea would help

 

[mark726]

Thank you for your question. An "exact" solution for flow s(t|k) refers to a solution that is derived from the Hamiltonian and satisfies the Hamiltonian equations of motion. In other words, it is a solution that is consistent with the physical laws governing the system.

To obtain such a solution, you can use the fact that aJa = 0, where J is the Poisson matrix. This means that the Hamiltonian equations of motion can be written in terms of the Poisson bracket. In this case, the Poisson bracket is given by {H, F} = aF, where H is the Hamiltonian and F is any function of the phase space variables (q,p).

To prove that s is symplectic and therefore canonical, you can use the definition of symplecticity, which states that the symplectic form ω = dq ∧ dp is preserved under the flow s(t|k). In other words, the Hamiltonian equations of motion preserve the symplectic structure of the phase space.

I hope this helps. Please let me know if you need further clarification.Scientist