36
Chapter 2. Geometrical optics
Figure 2.2: Two rays, infinitesimally displaced from one another.
Since x a and x b can be any two points connected by a ray, it follows
that
dP · δx − δP · dx = const,
(2.63)
where the δ- and d-variations refer to two separate rays, each derived by an independent perturbation from the original ray. This
quantity is known as the Lagrange invariant. To appreciate the
meaning of the Lagrange invariant, we consider a special case for
which dx a = δx b = 0. In this case (2.62) reduces to
−δP a · dx a = dP b · δx b .
(2.64)
This is shown schematically in Figure 2.3 , where the unperturbed
Chapter 2. Geometrical optics
Figure 2.2: Two rays, infinitesimally displaced from one another.
Since x a and x b can be any two points connected by a ray, it follows
that
dP · δx − δP · dx = const,
(2.63)
where the δ- and d-variations refer to two separate rays, each derived by an independent perturbation from the original ray. This
quantity is known as the Lagrange invariant. To appreciate the
meaning of the Lagrange invariant, we consider a special case for
which dx a = δx b = 0. In this case (2.62) reduces to
−δP a · dx a = dP b · δx b .
(2.64)
This is shown schematically in Figure 2.3 , where the unperturbed