The action corresponding to the motion of a particle in one dimension is given by S = ∫t1t2 dt [ 12mẋ2 - V(x) + αxẍ + βxẋ ] where m is the mass of the particle, α and β are constants, and V(x) is a potential. The position and velocity are held fixed at the end points of the trajectory. The equation of motion of the particle is:

A diatomic molecule is modeled as two atoms of equal mass m connected by a spring of force constant k. The molecule moves in one dimension under the influence of an external potential: V(x1, x2) = 12mω02(x12 + x22) If one normal mode frequency of the molecule is ω0, then the other normal mode frequency is:

The potential energy between two particles is given by V(r) = -a/r4 + b/r10, where 'a' and 'b' are positive constants. The potential energy at equilibrium separation is: