Hamilton's Equations of Motion

Hamilton's equation of motion depend on forming the Hamilton, which can be formed from the Lagrangian by

$$H = \sum_j p_j \dot{q}_j - L.$$ (4.9)

$p_j$ is called the conjugate momentum to the coordinate $q_j$. The conjugate momentums can be calculated from the Lagrangian by

$$p_j = \frac{\partial L}{\partial \dot{q}_j}$$ (4.10)

$$\dot{p}_j = \frac{\partial L}{\partial q_j}$$ (4.11)

Hamilton's equations of motion are then

$$\dot{q}_k = \frac{\partial H}{\partial p_k}$$ (4.12)

$$-\dot{p}_k = \frac{\partial H}{\partial q_k}$$ (4.13)

Examining a particle with

$$T = \frac{1}{2} m \dot{x}^2_i U = U(x_i).$$ (4.14)

gives

$$p_i = \frac{\partial L}{\partial \dot{x}_i} = m \dot{x}_i.$$ (4.15)

The Hamiltonian becomes

$$H = \sum_j p_j \dot{q}_j - L = \sum_j m \dot{x}_j \dot{x}_... ...ot{x}^2_j + U = \sum_j \frac{1}{2} m\dot{x}^2_j + U = T + U.$$ (4.16)

It can be shown that if $H$ has no explicit time dependence (e.g. $\frac{\partial H}{\partial t} = 0$) then energy is conserved.