{Q j } = Q 1 , Q 2 , . . . , Q n ,
(2.1)
and n is the number of degrees of freedom needed to completely
specify the system. For example, a three-dimensional Cartesian
coordinate system can be used to completely specify position in
ordinary space, and has three degrees of freedom.
The Q j evolve under the influence of forces, and therefore depend
implicitly on the time t. There exist velocities Q ˙ j given by
˙ ˙
˙
{Q ˙ j } = Q 1 , Q 2 , . . . , Q n ,
(2.2)
where the dot denotes differentiation with respect to time, i.e.,
Q ˙ j ≡
d Q j .
(2.3)
dt
20
Chapter 2. Geometrical optics
2.1 Relativistic classical mechanics
In classical mechanics, a system is described by one or more generalized coordinates Q j , where
This is quite general, since n can take on any positive integral
value. For example, a system of N interacting particles has n = 3N
degrees of freedom.
The central problem of classical mechanics can be stated as follows: given a set of coordinates Q j and velocities Q ˙ j at an initial
time t 0 , calculate the Q j and Q ˙ j at any time t. The result of this
calculation represents a complete specification of the system. In
the present study, we will confine our attention to a single particle of rest mass m and charge q under the influence of electric
and magnetic forces. We therefore define generalized coordinates
x j = (x 1 , x 2 , x 3 ) with corresponding velocities v j = (v 1 , v 2 , v 3 ),
where the six-vector components are functions of time t. In this
case the central problem is to calculate these quantities. The prescription is general with respect to the choice of coordinate systems. For example, one could use Cartesian, cylindrical, spherical,
or other coordinates with three degrees of freedom to specify the
(2.1)
and n is the number of degrees of freedom needed to completely
specify the system. For example, a three-dimensional Cartesian
coordinate system can be used to completely specify position in
ordinary space, and has three degrees of freedom.
The Q j evolve under the influence of forces, and therefore depend
implicitly on the time t. There exist velocities Q ˙ j given by
˙ ˙
˙
{Q ˙ j } = Q 1 , Q 2 , . . . , Q n ,
(2.2)
where the dot denotes differentiation with respect to time, i.e.,
Q ˙ j ≡
d Q j .
(2.3)
dt
20
Chapter 2. Geometrical optics
2.1 Relativistic classical mechanics
In classical mechanics, a system is described by one or more generalized coordinates Q j , where
This is quite general, since n can take on any positive integral
value. For example, a system of N interacting particles has n = 3N
degrees of freedom.
The central problem of classical mechanics can be stated as follows: given a set of coordinates Q j and velocities Q ˙ j at an initial
time t 0 , calculate the Q j and Q ˙ j at any time t. The result of this
calculation represents a complete specification of the system. In
the present study, we will confine our attention to a single particle of rest mass m and charge q under the influence of electric
and magnetic forces. We therefore define generalized coordinates
x j = (x 1 , x 2 , x 3 ) with corresponding velocities v j = (v 1 , v 2 , v 3 ),
where the six-vector components are functions of time t. In this
case the central problem is to calculate these quantities. The prescription is general with respect to the choice of coordinate systems. For example, one could use Cartesian, cylindrical, spherical,
or other coordinates with three degrees of freedom to specify the