Hamilton's Equations

(B.68) |

where the depend on the , but not on the . It is easily demonstrated from the previous equation that

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

where is the Lagrangian of the system, and we have made use of the fact that is independent of the . Consider the function

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

(B.72) |

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

Consider the variation of the function . We have

(B.73) |

The first and third terms in the bracket cancel, because . Furthermore, because Lagrange's equation can be written (see Section B.4), we obtain

(B.74) |

Suppose, now, that we can express the total energy of the system, , solely as a function of the and the , with no explicit dependence on the . In other words, suppose that we can write . When the energy is written in this fashion it is generally termed the

(B.75) |

A comparison of the previous two equations yields

for . These first-order differential equations are known as