Hamiltonian System - Overview

Overview

Informally, a Hamiltonian system is a mathematical formalism developed by Hamilton to describe the evolution equations of a physical system. The advantage of this description is that it gives important insight about the dynamics, even if the initial value problem can not be solved analytically. One example is the planetary movement of three bodies: even if there is no simple solution to the general problem, Poincaré showed for the first time that it exhibits deterministic chaos.

Formally, an Hamiltonian system is a dynamical system completely described by the scalar function, the Hamiltonian. The state of the system, is described by the generalized coordinates 'momentum' and 'position' where both and are vectors with the same dimension N. So, the system is completely described by the 2N dimensional vector

and the evolution equation is given by the Hamilton's equations:

from which the evolution is given by the solution of an initial value problem.