- Consider a system with one degree of freedom and one generalized co-ordinate q. For small displacement from the equilibrium we can expand potential energy function using taylor series expansion about the equilibrium and we will only consider the lowest order terms. So expanding PE function V(q) we have

where derivativesare evaluated at the equilibrium position q=q_{0}and at equlilbrium (∂V/∂q)_{0}= 0 - V(q
_{0}) is potential energy at equilibrium and can be taken as zero, if the origin of potential energy is shifted to be at minimum equilibrium value.

This implies that

putting second derivative term in bracket equal to k and shifting origin to q_{0}=0 we have

V(q)=kq^{2}/2

and k is the positive parameter at the position of stable equilibrium. - If generalized co-ordinates does not involve time explicitely , the K.E. is then homogeneous quadratic function of generalized velocities, or,.

and equation of motion is

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