The Toda Lattice

This motivates the following definition. Definition 1.2. A conserved quantity in a Hamiltonian system is a smooth function f ∈ C∞(M) such that {f,H} = 0. Remark 1.3. If you’re familiar with the Lagrangian model of classical mechanics, you can derive the above setup from a Lagrangian field theory on the real line R, i.e. from classical Lagrangian mechanics. There’s a classical procedure for doing so known as the Legendre transform. Definition 1.4. A classical Hamiltonian system is completely integrable if there exist a family of m conserved quantities f1, . . . , fm, where dim(M) = 2m, such that