# Introducing the Hamiltonian as a "thermodynamic" potential

## TL;DR

This paper presents a new interpretation of the Hamiltonian as a thermodynamic potential, illustrating its physical meaning through a mechanical system with time-dependent constraints, and extends the concept to broader problems.

## Contribution

It introduces the Hamiltonian as a thermodynamic potential via a Legendre transformation, providing a clear physical interpretation for systems with time-dependent constraints.

## Key findings

- Hamiltonian interpreted as a thermodynamic potential.
- Application to a bead on a rotating hoop example.
- Framework extendable to various constrained systems.

## Abstract

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.