Linear Oscillations

Abstract

In Chaps. 3 and 4, we have seen that the Lagrangian for a holonomic, scleronomic system with f degrees of freedom has the form (with q =q ₁,..., q _f )

$$ L = L\left( {q,\dot{q}} \right) = \frac{1}{2}\sum\limits_{{i,j = 1}}^{f} {{{g}_{{ij}}}} \left( q \right){{\dot{q}}_{i}}{{\dot{q}}_{j}} - V\left( q \right) $$

in an inertial frame of reference. The equations of motion, i.e. Lagrange’s equations derived from this Lagrangian, are, in general, very complicated non-linear differential equations, which can only be solved numerically.