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By L D Landau and E.M. Lifshitz (Auth.)
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S). The quantities thus determined are the characteristic frequencies or eigenfrequencies of the system. In particular cases, some of the roots may coincide; the corresponding multiple eigenfrequencies are said to be degenerate. 8) are real and positive. 6), and so of the velocities Xj^, of an exponentially decreasing or increasing factor. Such a factor is inadmissible, since it would lead to a time variation of the total energy Ε = tZ-f-Tof the system, which would therefore not be conserved.
It should be emphasised, however, that this prop erty of the frequency depends on the assumption that t h e oscillations are small, a n d ceases t o hold in higher approximations. Mathematic ally, it depends on the fact that the potential energy is a quadratic function of the coordinate. 10) It is proportional t o the square of the amplitude. 8). The constant A is called the complex amplitude; its modulus is t h e ordinary amplitude, a n d its argument is the initial phase. The use of exponential factors is mathematically simpler than that of trigonometrical ones because they are unchanged in form by differentiation.
F o r such a coordinate we have, by Lagrange's equation, (d/dt) dL/dqj = dL/dqi = 0, so that the corresponding generaUsed momentum = dL/dq^ is an integral of the motion. This leads to a considerable simplification of the problem of integrating the equations of motion when there are cyclic coordinates. 8)), and we return to the known law of conservation of angular m o m e n t u m : Μ = mr^<¡) = constant. 2) 32 Integration of the Equations of Motion §12 This law has a simple geometrical interpretation in the plane motion of a single particle in a central field.