Non Linear Oscillations
We have seen that the potential energy of a particle about its position of stable equilibrium can be expressed as,
U ( q ) = ½ C2 q2 + 1/6 C3 q3 + 1/24 C4 q4 + ….
where q is displacement. The restoring force, therefore, is
F ( q ) = –C2 q – ½ C3 q2 – 1/6 C4 q3 – ….
If either all the coefficients C3, C4, etc. (except C2) are zero, or if terms containing q2 and higher powers are neglected, the restoring force is linear in displacement, F ( q ) = –C2 q. The particle then executes simple harmonic motion about position of equilibrium at q = 0, with frequency 
.
On the other hand, if terms containing higher powers of q are present in the expression for force, the force is said to depend non-linearly on displacement. In such cases, the motion of the particle shows two important deviations from simple harmonic motion:
1. In general, the resultant motion of the particle is a superposition of harmonic oscillations of fundamental frequency and higher harmonics containing frequencies 2 ω, 3 ω, etc.
2. In general, the equilibrium point about which oscillations occur gets displaced from q = 0 to some other position q = q0. The particle oscillates symmetrically about point q = q0, and not about q = 0.
Suppose a particle is subjected to a restoring force is given by,
F ( q ) = –C2 q – ½ C3 q2
The force contains a linear term (–C2 q) and a non-linear term [(–C3/2)q2]
The equation of motion of the particle is written as,
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