We can regard the as a new set of generalized coordinates, since specifying the is equivalent to specifying the (and, hence, the ). The are usually termed

Let us now try to express , , and the equations of motion in terms of the .

The kinetic
energy can be written

(810) |

(811) |

Hence, the kinetic energy takes the form of a

The potential energy can be written

(813) |

(814) |

Hence, the potential energy also takes the form of a

Writing Lagrange's equations of motion in terms of the normal
coordinates, we obtain [*cf.*, Equation (772)]

(816) |

(817) |

where and are arbitrary constants. Hence, it is clear from Equations (808) and (818) that the most general solution to the perturbed equations of motion is indeed given by Equations (800) and (801).

In conclusion, the equations of motion of a many degree of freedom dynamical system which is slightly perturbed from an equilibrium state take a particularly simple form when expressed in terms of the normal coordinates. Each normal coordinate specifies the instantaneous displacement of an independent mode of oscillation (or secular growth) of the system. Moreover, each normal coordinate oscillates at a characteristic frequency (or grows at a characteristic rate), and is completely unaffected by the other coordinates.