Friction of particle in double well

jdvlok

The dynamics of mechano-elastic systems such as the Duffing oscillator can be described by the movement of a particle trapped in a double well potential V(x). The friction force is usually expressed as F = δv with δ the damping coefficient and v the particle velocity.

Assuming an actual physical double well y=V(x) is constructed such that gravity acts in the minus y direction and a particle is placed on the surface of this well at (x, V(x)) with an initial velocity, the friction force F acting on the particle should depend on:
1. The velocity v of the particle
2. The normal force N acting on the particle
v and N are varying according to the position of the particle along V(x). How should F be expressed? Using F = δv seems to ignore N, while F = μN (with μ the coefficient of friction) seems to ignore v.