and
length
below
is made to oscillate at A with the distance OA given by
University Physics Notes: Classical Mechanics – An Example of a Time Dependent Hamiltonian/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
Hence
The Lagrangian is
Where
may
be ignored since it is a function of
only
and ignoring it returns the same Lagrangian equation of motion.
We can now find the Hamiltonian by expressing the Lagrangian in
terms of
and
to
obtain
Then