The Lagrangian for a simple pendulum is given by L = 12ml2θ̇2 - mgl(1 - cosθ). If H is the Hamiltonian of the pendulum, what is the value of dHdt?

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: