[QUE/CM-01008]

Find potential if the force acting on a particle in one dimension is $$ F(x) =  - m \omega_0^2(x-2bx^3), b>0. $$ Determine the potential energy asuming $V_0=0$ and plot it as a function of \(x\). For what values of energies, is oscillatory motion possible? Show that the period of oscillations as a function of amplitude $a$ is  $$ T =\frac{2}{\omega_0}\int_{-a}^a \frac{dx}{(a^2-x^2)(1- b(a^2+x^2))} $$ and that for small $b$ we have $T = \frac{2\pi}{\omega_0}(1+ \frac{3}{4}b a^2).$