Hamiltonian Systems

In this chapter we explain the origin of symplectic geometry, i.e., classical mechanics. For this, first we review some background about flows of vector fields and Lie derivatives. Then we see that if a symplectic form \(\sigma \) is invariant under the flow of a vector field \(v,\) we obtain the equation \(i_v\sigma =-df.\) Such a vector field v satisfying this equation is actually unique and is called Hamiltonian vector field. In coordinates, this equation turns to a Hamiltonian system. In classical mechanics, to solve an equation of motion, one obtains an Euler–Lagrange equation that can be interpreted as a Hamiltonian system, by the Legendre transform. Hence, we will see how the Legendre transform relates an Euler–Lagrange equation to a Hamiltonian system.