A particle under double-exponential potential

Question

A particle of mass m is constrained to move along the x-axis. It's potential energy is given by:

U(x) = U0 [ e−2β(x−x0) − 2e−β(x−x0) ]

where U₀, β, and x₀, are known parameters.

a) Assume that the particle is subject to a weak frictional force. What would be the position of the particle after a very long time?

b) Now ignore the existence of the friction. Find an expression for the natural frequency (in Radians/second) of the particle around its equilibrium position

Answer 1

For a) note that since friction exists, the particle will eventually end up at a global minimum. This means, that the first derivative of the potential must equal zero and the second derivative be positive.

For b) write down the Lagrangian, and equations of motion from there. You would end up with natural frequency.