Generating Functions for Transformations of Coordinates
In order to describe the motion of a system mathematically we need to be able to specify the instantaneous configuration of the system - the position. For example the motion of a projectile can be specified by the horizontal distance of the projectile from the start point and the height of the projectile above the ground while the position of a pendulum can be specified by the angle the pendulum makes with the vertical. In both of these examples, one coordinate is required, so the system is said to have one degree of freedom. In general, ifcoordinates are required the system is said to have n degrees of freedom.
An independent coordinate which describes the configuration of the system is called a generalised coordinate. The generalised coordinate may be a distance in a certain direction, a length along a curve, an angle or some other measure. Even for motion in a straight line along the – axis, the generalised coordinate may not bebut can be any single valued function ofso a unique value of the function corresponds to a unique value of
Corresponding to each generalised coordinate q there is a generalised velocityand the state of the system is uniquely defined by the coordinates(in two dimensions). These coordinates are displayed on a phase space diagram to give the phase curves.
Of course the choice of coordinates is arbitrary and the equations of motion independent of this choice (a basic law of physics) so the form of the equations of motion is the same whatever the choice of coordinates.
are two representations of the same system withThe first gives rise to equation of motion
The second gives rise to equation