35
2.3. Conservation laws
a constant of the motion. In this section, we show that other invariant quantities exist, as a direct consequence of the least action
principle. As in the preceding section, the reader is referred to the
book by Sturrock [86] for a detailed discussion.
2.3.1 The Lagrange invariant
In the preceding sections, we derived the necessary conditions for a
single trajectory to represent physically allowable motion. Henceforth we refer to a physically allowable trajectory satisfying (2.43,
2.58) as a ray. In this section, we consider the behavior of rays
which are infinitesimally displaced from one another. This is shown
schematically in Figure 2.2. From (2.57) the variation in optical
path length between two neighboring rays is given by
δW ab = P b · δx b − P a · δx a .
(2.59)
This infinitesimal quantity is nonzero in general, since the endpoints x a and x b of the two rays are assumed in general to be
displaced from one another. It can be shown that δW ab is an
exact differential [72], in which case
P b = v x b W ab ,
P a = −v xa W ab .
(2.60)
Geometrically, this means that the canonical momentum P is normal to surfaces of constant optical path length, W ab = const at
the endpoints, where we note that the endpoints can be chosen to
be anywhere along the ray path.
We now consider a second perturbation, independent from the
first. It follows that (2.59):
d(δW ab ) = dP b · δx b + P b · d(δx b ) − dP a · δx a − P a · d(δx a ). (2.61)
Interchanging the order of perturbations and subtracting, we obtain
dP a · δx a − δP a · dx a = dP b · δx b − δP b · dx b .
(2.62)
2.3. Conservation laws
a constant of the motion. In this section, we show that other invariant quantities exist, as a direct consequence of the least action
principle. As in the preceding section, the reader is referred to the
book by Sturrock [86] for a detailed discussion.
2.3.1 The Lagrange invariant
In the preceding sections, we derived the necessary conditions for a
single trajectory to represent physically allowable motion. Henceforth we refer to a physically allowable trajectory satisfying (2.43,
2.58) as a ray. In this section, we consider the behavior of rays
which are infinitesimally displaced from one another. This is shown
schematically in Figure 2.2. From (2.57) the variation in optical
path length between two neighboring rays is given by
δW ab = P b · δx b − P a · δx a .
(2.59)
This infinitesimal quantity is nonzero in general, since the endpoints x a and x b of the two rays are assumed in general to be
displaced from one another. It can be shown that δW ab is an
exact differential [72], in which case
P b = v x b W ab ,
P a = −v xa W ab .
(2.60)
Geometrically, this means that the canonical momentum P is normal to surfaces of constant optical path length, W ab = const at
the endpoints, where we note that the endpoints can be chosen to
be anywhere along the ray path.
We now consider a second perturbation, independent from the
first. It follows that (2.59):
d(δW ab ) = dP b · δx b + P b · d(δx b ) − dP a · δx a − P a · d(δx a ). (2.61)
Interchanging the order of perturbations and subtracting, we obtain
dP a · δx a − δP a · dx a = dP b · δx b − δP b · dx b .
(2.62)