An Example of a Time Dependent Hamiltonian and Lagrangian
Deriving the Hamiltonian and Lagrangian for a time dependent system is not much more complicated than for the time independent case. The pendulum of massand lengthbelow is made to oscillate at A with the distance OA given by
The potential energy is given by
The kinetic energy is given by
The Lagrangian is
Wheremay be ignored since it is a function ofonly and ignoring it returns the same Lagrangian equation of motion.
We can now find the Hamiltonian by expressing the Lagrangian in terms ofandto obtain