Hamiltonian systems

This article briefly explains how a Hamiltonian system is described on a manifold. Throughout we let $M$ be an $n$-dimensional manifold, for our purposes it is sufficient to think of it as an $n$-dimensional surface embedded in some higher dimensional space.

These notes are my own summary of Chapter 1.3 of Notes on Dynamical Systems by Moser and Zehnder Hamiltonian Systems In this section we consider Hamiltonian systems on the phase space $\mathbb{R}^{2n}$.

These notes are my own summary of Chapter 1.5 of Notes on Dynamical Systems by Moser and Zehnder Canonical Transformations Indexed by a Group By now we’ve established that canonical transformations are those change-of-coordinates on $\mathbb{R}^{2n}$ that preserve the Hamiltonian nature of a dynamical system.

Calogero-Moser System In joint work with Nam-Gyu Kang and Nikolai Makarov we discovered that the driving functions of the multiple SLE(0) evolve according to the Calogero-Moser dynamical system. The latter describes the trajectories of $N$ particles on the real line $\mathbb{R}$ that interact via a force determined by an inverse pair potential, namely: