Introduction to Hamiltonian Systems

Hamiltonian systems are a class of dynamical systems governed by the Hamiltonian function, which represents the total energy of the system—kinetic and potential energy. These systems are a cornerstone of theoretical physics and are essential for understanding mechanics, astrophysics, and quantum mechanics.

Definition of a Hamiltonian System

A Hamiltonian system can be defined on a symplectic manifold, where the state of the system is described by coordinates $(q_1, \ldots, q_n, p_1, \ldots, p_n)$. Here $q_i$ are the generalized coordinates and $p_i$ are the conjugate momenta.

The Hamiltonian Function

The Hamiltonian $H$ is a function, usually representing the total energy of the system:

$$H = T + V$$

where $T$ is the kinetic energy and $V$ is the potential energy.

Properties

• Time Reversal Symmetry: If the system evolves forward in time, then reversing the direction of time will return the system to its initial state.
• Conservation of Energy: The total energy (Hamiltonian) is conserved if $H$ does not explicitly depend on time.

Hamilton's Equations

Hamilton's equations describe the time evolution of the system and are given by:

$\dot{q}_i = \frac{\partial H}{\partial p_i}$ $\dot{p}_i = -\frac{\partial H}{\partial q_i}$

These equations ensure that the flow of the system in phase space is symplectic, preserving the symplectic structure of the manifold.