3Hamilton's principle
IB Variational Principles
3 Hamilton’s principle
As mentioned before, Lagrange and Hamilton reformulated Newtonian dynamics
into a much more robust system based on an action principle.
The first important concept is the idea of a configuration space. This
configuration space is a vector space containing generalized coordinates
ξ
(
t
) that
specify the configuration of the system. The idea is to capture all information
about the system in one single vector.
In the simplest case of a single free particle, these generalized coordinates
would simply be the coordinates of the position of the particle. If we have
two particles given by positions
x
(
t
) = (
x
1
, x
2
, x
3
) and
y
(
t
) = (
y
1
, y
2
, y
3
), our
generalized coordinates might be
ξ
(
t
) = (
x
1
, x
2
, x
3
, y
1
, y
2
, y
3
). In general, if we
have N different free particles, the configuration space has 3N dimensions.
The important thing is that the generalized coordinates need not be just the
usual Cartesian coordinates. If we are describing a pendulum in a plane, we do
not need to specify the
x
and
y
coordinates of the mass. Instead, the system
can be described by just the angle
θ
between the mass and the vertical. So the
generalized coordinates is just
ξ
(
t
) =
θ
(
t
). This is much more natural to work
with and avoids the hassle of imposing constraints on x and y.