Hamiltonian function (classical)

The total energy function on phase space that generates time evolution in Hamiltonian mechanics.

Hamiltonian function (classical)

For a classical mechanical system with phase space coordinates $(q, p) = (q_1, \ldots, q_n, p_1, \ldots, p_n)$ , the Hamiltonian is a function

H: \Gamma \to \mathbb{R}, \qquad (q, p) \mapsto H(q, p)

that gives the total energy of the system and determines its time evolution via Hamilton’s equations:

\dot{q}_i = \frac{\partial H}{\partial p_i}, \qquad \dot{p}_i = -\frac{\partial H}{\partial q_i}.

Typical form

For many systems (e.g., particles in a potential), the Hamiltonian splits into kinetic and potential energy:

H(q, p) = T(p) + V(q) = \sum_{i=1}^n \frac{p_i^2}{2m_i} + V(q_1, \ldots, q_n).

In the canonical ensemble , the equilibrium probability density on phase space is

\rho(q, p) = \frac{1}{Z} e^{-\beta H(q, p)},

where $\beta = 1/(k_B T)$ is the inverse temperature and $Z$ is the partition function .