Can a system have the total energy conserved but the hamiltonian not conserved?

[robb_]

Homework Statement

Can a system have the total energy conserved but the hamiltonian not conserved?

Homework Equations

If the partial of the lagrangian w.r.t time is zero, energy is conserved.
The hamiltonian is found by the usual method- get the generalized momentum from the lagrangian then plug each into the equation:let me skip typing it in latex. Compare this to the total energy.

The Attempt at a Solution

I can work the equations and find L and H. It seems, conceptually, that anytime H does not equal the total energy, then energy is not conserved. I wonder if this always is so. Also, I know that H is not equal to E when the generalized coordinates depend on time. I also have worked problems where H does not equal E but H is conserved.

 

[NeoDevin]

You already stated a case where the Hamiltonian is not equal to the energy... Maybe take an example you already know that the energy is conserved, and show in certain coordinate systems, the Hamiltonian may not be.

 

[robb_]

thanks. yes I have worked the case you stated. i will try to think of such a case where E is conserved but H isnt. i guess you are saying that it is possible none the less. moreover, i am trying to conceptualize the hamiltonian. i was thinking that it is the intrinsic energy associated with the motion, i.e. if part of the motion is provided by an external agent then this is not included in the hamiltonian but it is part of the total energy-> of course i am excluding external agents associated with conservative forces, i.e. potentials. does that make any sense? where is my thinking flawed? thanks

 

[NeoDevin]

Because you get to choose the coordinates for the Hamiltonian, you can work backwards from there. Start with any system you can think of where the energy is conserved, then try to find some coordinates such that H is not. Figure out the conservation of energy first, in a simple reference frame, then you can arbitrarily change to another (perhaps time dependent) reference frame, where the Hamiltonian isn't conserved. Since conservation of energy doesn't depend on coordinates, you can change coordinates without changing conservation, but the Hamiltonian is dependent on the coordinates, so when you change coordinates, the Hamiltonian may or may not be conserved. All that's left for you is to choose appropriate coordinates.

After you find some syster in which the energy is conserved, just define coords so that $$\frac{\partial H}{\partial t} \neq 0$$

 

[robb_]

Excellent, so this gets at the heart of my question. So, for a given system the Hamiltonian may or may not be conserved, it depends on the choice of generalized coordinates. right?