Hamilton's Equations

Consider a dynamical system with ${\cal F}$ degrees of freedom that is described by the generalized coordinates $q_i$ , for $i=1,{\cal F}$ . Suppose that neither the kinetic energy, $K$ , nor the potential energy, $U$ , depend explicitly on the time, $t$ . In conventional dynamical systems, the potential energy is generally independent of the $\dot{q}_i$ , whereas the kinetic energy takes the form of a homogeneous quadratic function of the $\dot{q}_i$ . In other words,

$$K = \sum_{i,j = 1,{\cal F}} m_{ij} \dot{q}_i \dot{q}_j,$$ (B.68)

where the $m_{ij}$ depend on the $q_i$ , but not on the $\dot{q}_i$ . It is easily demonstrated from the previous equation that

$$\sum_{i=1,{\cal F}} \dot{q}_i \frac{\partial K}{\partial \dot{q}_i} = 2 K.$$ (B.69)

Recall, from Section B.4, that generalized momentum conjugate to the $i$ th generalized coordinate is defined

$$p_i = \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial K}{\partial \dot{q}_i},$$ (B.70)

where $L=K-U$ is the Lagrangian of the system, and we have made use of the fact that $U$ is independent of the $\dot{q}_i$ . Consider the function

$$H = \sum_{i=1,{\cal F}} \dot{q}_i p_i - L = \sum_{i=1,{\cal F}} \dot{q}_i p_i -K + U.$$ (B.71)

If all of the conditions discussed previously are satisfied then Equations (B.69) and (B.70) yield

$$H = K+ U.$$ (B.72)

In other words, the function $H$ is equal to the total energy of the system.

Consider the variation of the function $H$ . We have

$$\delta H = \sum_{i=1,{\cal F}} \left(\delta\dot{q}_i p_i + \dot{... ...t{q}_i} \delta \dot{q}_i - \frac{\partial L}{\partial q_i} \delta q_i\right).$$ (B.73)

The first and third terms in the bracket cancel, because $p_i= \partial L/\partial \dot{q}_i$ . Furthermore, because Lagrange's equation can be written $\dot{p}_i = \partial L/\partial q_i$ (see Section B.4), we obtain

$$\delta H = \sum_{i=1,{\cal F}} \left(\dot{q}_i \delta p_i - \dot{p}_i \delta q_i\right).$$ (B.74)

Suppose, now, that we can express the total energy of the system, $H$ , solely as a function of the $q_i$ and the $p_i$ , with no explicit dependence on the $\dot{q}_i$ . In other words, suppose that we can write $H=H(q_i,p_i)$ . When the energy is written in this fashion it is generally termed the Hamiltonian of the system. The variation of the Hamiltonian function takes the form

$$\delta H =\sum_{i=1,{\cal F}} \left(\frac{\partial H}{\partial p_i} \delta p_i + \frac{\partial H}{\partial q_i} \delta{q}_i\right).$$ (B.75)

A comparison of the previous two equations yields

$$\dot{q}_i = \frac{\partial H}{\partial p_i},$$ (B.76)

$$\dot{p}_i =-\frac{\partial H}{\partial q_i},$$ (B.77)

for $i=1,{\cal F}$ . These $2{\cal F}$ first-order differential equations are known as Hamilton's equations. Hamilton's equations are often a useful alternative to Lagrange's equations, which take the form of ${\cal F}$ second-order differential equations.