110
Chapter 2. Geometrical optics
2.7.1 Canonical transformations
We begin with Hamilton’s equations of motion (2.85), which are
given in their most general form as
∂H
∂H
˙
= − ˙
= Q i ,
P i ,
i = 1, . . . , n,
(2.300)
∂P i
∂Q i
where the Q i are the n generalized coordinates, and P i are the n
conjugate canonical momenta. The Hamiltonian is a function of
all coordinates and momenta, where
H = H(Q 1 , . . . , Q n ; P 1 , . . . , P n ; t).
(2.301)
In general, H can have explicit dependence on the time t, which
we regard as a parameter which uniquely specifies a given point
along the trajectory. For brevity, we adopt a vector notation where
H = H(Q, P; t), and
Q = (Q 1 , . . . , Q n )
P = (P 1 , . . . , P n ).
(2.302)
In principle, Hamilton’s equations can be integrated to find the
coordinates Q i (t) and canonical momenta P i (t) as functions of the
time t. This would constitute a formal solution to the general dynamical problem.
It is always possible to transform to a new system of coordinates
and momenta. As a simple example, one could transform from
Cartesian to spherical coordinates. This would simplify a problem with spherical symmetry, such as scattering by a spherically
symmetric Coulomb potential. The components of canonical momentum would also transform to a spherical system.
In this context, we imagine a transformation to a system where all
coordinates and momenta are constants of the motion. If such a
transformation were possible, this would represent an immediate
formal solution to the general dynamical problem, since the coordinates and momenta would simply be equal to their initial values
Chapter 2. Geometrical optics
2.7.1 Canonical transformations
We begin with Hamilton’s equations of motion (2.85), which are
given in their most general form as
∂H
∂H
˙
= − ˙
= Q i ,
P i ,
i = 1, . . . , n,
(2.300)
∂P i
∂Q i
where the Q i are the n generalized coordinates, and P i are the n
conjugate canonical momenta. The Hamiltonian is a function of
all coordinates and momenta, where
H = H(Q 1 , . . . , Q n ; P 1 , . . . , P n ; t).
(2.301)
In general, H can have explicit dependence on the time t, which
we regard as a parameter which uniquely specifies a given point
along the trajectory. For brevity, we adopt a vector notation where
H = H(Q, P; t), and
Q = (Q 1 , . . . , Q n )
P = (P 1 , . . . , P n ).
(2.302)
In principle, Hamilton’s equations can be integrated to find the
coordinates Q i (t) and canonical momenta P i (t) as functions of the
time t. This would constitute a formal solution to the general dynamical problem.
It is always possible to transform to a new system of coordinates
and momenta. As a simple example, one could transform from
Cartesian to spherical coordinates. This would simplify a problem with spherical symmetry, such as scattering by a spherically
symmetric Coulomb potential. The components of canonical momentum would also transform to a spherical system.
In this context, we imagine a transformation to a system where all
coordinates and momenta are constants of the motion. If such a
transformation were possible, this would represent an immediate
formal solution to the general dynamical problem, since the coordinates and momenta would simply be equal to their initial values