Action-angle coordinates

Action-angle coordinates are a set of variables used in Hamiltonian mechanics to represent the state of a dynamical system, particularly in the context of integrable systems. They provide a powerful framework for understanding the long-term behavior of such systems, especially when dealing with periodic or quasi-periodic motion. ### Key Concepts: 1. **Action Variables (J):** Action variables are defined for each degree of freedom in a system, and they are typically calculated as integrals over one complete cycle of motion.