Three Spring-Coupled Masses

These equations can be rewritten

where . Let us search for a normal mode solution of the form

(208) | ||

(209) | ||

(210) |

Equations (205)-(210) can be combined to give the homogeneous matrix equation

where . The normal frequencies are determined by setting the determinant of the matrix to zero: that is,

(212) |

or

(213) |

Thus, the normal frequencies are , , and . According to the first and third rows of Equation (211),

(214) |

provided . According to the second row,

(215) |

when . Incidentally, we can only determine the ratios of , , and , rather than the absolute values of these quantities. In other words, only the direction of the vector is well-defined. [This follows because the most general solution, (219), is undetermined to an arbitrary multiplicative constant. That is, if is a solution to the dynamical equations (205)-(207) then so is , where is an arbitrary constant. This, in turn, follows because the dynamical equations are linear.] Let us arbitrarily set the magnitude of to unity. It follows that the normal mode associated with the normal frequency is

(216) |

Likewise, the normal mode associated with the normal frequency is

(217) |

Finally, the normal mode associated with the normal frequency is

(218) |

The vectors , , and are mutually perpendicular. In other words, they are normal vectors. (Hence, the name ``normal'' mode.)

Let . It follows that the most general solution to the problem is

where

(220) | ||

(221) | ||

(222) |

Here, and are constants. Incidentally, we need to introduce the arbitrary amplitudes to make up for the fact that we set the magnitudes of the vectors to unity. Equation (219) yields

(223) |

The preceding equation can be inverted by noting that , et cetera, because , , and are mutually perpendicular unit vectors. Thus, we obtain

(224) |

This equation determines the three normal coordinates, , , , in terms of the three conventional coordinates, , , . In general, the normal coordinates are undetermined to arbitrary multiplicative constants.