Introducing the Hamiltonian as a "thermodynamic" potential

A conceptually simple physical interpretation of a conserved Hamiltonian $\mathcal{H}$ for a mechanical system with a time-dependent constraint is given. For the case of a bead on a vertical hoop forced to rotate with constant angular velocity $\omega$, $\mathcal{H}$ is nothing but the total energy of the system plus the external actuator keeping $\omega$ fixed. In an analogy with thermodynamics, the Hamiltonian is introduced as a thermodynamic potential obtained from a Legendre transformation of the energy, in a very instructive way. The ideas can be made extensive to different problems with time-dependent constraints.