Integrability and non-integrability in Hamiltonian mechanics

Abstract

International audienceIn 1834, Hamilton expressed the differential equations of classical mechanics, the Lagrange equations$$\frac{d}{dt} \frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q}$$ with $L:\mathbb{R}^n\times \mathbb R ^n \to \mathbb R$ in the "canonical form": $$\dot{q}=\frac{\partial H}{\partial p},\quad \dot{p}=- \frac{\partial H}{\partial q}$$ Here $p = \partial L/\partial \dot{q} \in \mathbb R^n$ is the generalized momentum and the Hamiltonian function $H= p\dot{q}- L\big|_{p,q}$ is the "total energy" of the mechanical system