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

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. :yuck: 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.

The Attempt at a Solution

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