Demonstrating something is a constant of motion

[romeo6]

I am given a Hamiltonian and am asked to show that the Hamiltonain is a constant of motion for orbits defined by the corresponding Hamiltonian equations.

can someone decrypt this for me please...




i.e How do I define orbits for a given hamiltonian and then how do I show that the Hamiltonian is a constant of motion?

Thanks in advance.

 

[TMFKAN64]

You don't have to define the orbits... the Hamiltonian does that along with the corresponding Hamiltonian equations. (These are qk' = partial H / partial pk and -pk' = partial H / partial qk. I apologize for not using latex here...)

Anyway, the Hamiltonian is specified as H(qk, pk, t), where qk are the generalized coordinates and pk are the generalized momenta. If this is constant over time, dH/dt = 0, right? So, calculate dH/dt in terms of partials with respect to the arguments, and simplify it using the Hamiltonian equations to show that it's zero.

 

[dextercioby]

The PB of H and H is zero and iff the Hamiltonian is time-independent, then it is a constant of motion.

Daniel.