3Hamilton's principle

IB Variational Principles



3.2 The Hamiltonian
In 1833, Hamilton took Lagrangian mechanics further and formulated Hamilto-
nian mechanics. The idea is to abandon
˙
x
and use the conjugate momentum
p
=
L
˙
x
instead. Of course, this involves taking the Legendre transform of the
Lagrangian to obtain the Hamiltonian.
Definition
(Hamiltonian)
.
The Hamiltonian of a system is the Legendre trans-
form of the Lagrangian:
H(x, p) = p ·
˙
x L(x,
˙
x),
where
˙
x is a function of p that is the solution to p =
L
˙
x
.
p
is the conjugate momentum of
x
. The space containing the variables
x, p
is known as the phase space.
Since the Legendre transform is its self-inverse, the Lagrangian is the Legendre
transform of the Hamiltonian with respect to p. So
L = p ·
˙
x H(x, p)
with
˙
x =
H
p
.
Hence we can write the action using the Hamiltonian as
S[x, p] =
Z
(p ·
˙
x H(x, p)) dt.
This is the phase-space form of the action. The Euler-Lagrange equations for
these are
˙
x =
H
p
,
˙
p =
H
x
Using the Hamiltonian, the Euler-Lagrange equations put
x
and
p
on a much
more equal footing, and the equations are more symmetric. There are also many
useful concepts arising from the Hamiltonian, which are explored much in-depth
in the II Classical Dynamics course.
So what does the Hamiltonian look like? Consider the case of a single particle.
The Lagrangian is given by
L =
1
2
m|
˙
x|
2
V (x, t).
Then the conjugate momentum is
p =
L
˙
x
= m
˙
x,
which happens to coincide with the usual definition of the momentum. How-
ever, the conjugate momentum is often something more interesting when we
use generalized coordinates. For example, in polar coordinates, the conjugate
momentum of the angle is the angular momentum.
Substituting this into the Hamiltonian, we obtain
H(x, p) = p ·
p
m
1
2
m
p
m
2
+ V (x, t)
=
1
2m
|p|
2
+ V.
So H is the total energy, but expressed in terms of x, p, not x,
˙
x.