Hamiltonian

Can be proved by expressions derived in 1. Generalized Momentum and when it’s conserved

Hamiltonian Function #

$$ \begin{align} \frac{dh}{dt}=\frac{dH}{dt}=-\frac{\partial L}{\partial t}\\\ where\\ h=\sum_k \frac{\partial L}{\partial q_k’}q_k’-L\\ where\\ H=\sum_k p_k\\ q_k’-L \end{align} $$

h, H are constant if L doesn’t contain time explicitly.

h=constant, a first integral of motion called the “Jacobi’s integral”.

$H=T+V$, if system is conservative and T is a homogeneous quadratic function.

Hamilton’s canonical equations of motion #

$$ \begin{align} q_k’=\frac{\partial H}{\partial p_k}\\ p_k’=-\frac{\partial H}{\partial q_k} \end{align} $$