represents
the energy of the system which is the sum of kinetic and potential
energy, labelled
and
respectively.
For a one dimensional system, we may write
University Physics Notes: Classical Mechanics – The Hamiltonian
The Hamiltonianrepresents
the energy of the system which is the sum of kinetic and potential
energy, labelledandrespectively.
For a one dimensional system, we may write
so
where
Note
that
is
a function of
only
and
is
a function of
only.
In generalis
a function of
only
andis
a function of the coordinates, however they are defined.
The value of the Hamiltonian is the total energy of the system. For a closed system, it is the sum of the kinetic and potential energy in the system and is conserved. The Hamiltonian equations give the time evolution of the system. These are
where
and
The time-derivative of the momentumequals
the force acting so the first Hamilton equation means that the force
on the particle equals the rate at which it loses potential energy
with respect to changes in
its
position.
The time-derivative of
here
means the velocity: the second Hamilton equation here means that the
particle’s velocity equals the derivative of its kinetic energy
with respect to its momentum.
We can derive Hamilton's equations by looking at how the total
differential of the Lagrangian depends on time, generalized
positions
and
generalized velocities
(1)
and
to
give
Substitute these into (1):
which
we can rewrite as
and
rearrange to get
The term on the left-hand side is the Hamiltonian so
where the second equality holds because it is equal to
Associating
terms from both sides of the equation above yields Hamilton's
equations
and