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Generalized Coordinates
Let the
, for
, be a set of coordinates that uniquely
specifies the instantaneous configuration of some classical dynamical system.
Here, it is assumed that each of the
can vary independently.
The
might be Cartesian coordinates, or polar
coordinates, or angles, or some mixture of all three types of coordinate, and are, therefore,
termed generalized coordinates. A dynamical system whose
instantaneous configuration is fully specified by
independent
generalized coordinates is said to have
degrees of freedom.
For instance, the instantaneous position of a particle moving freely
in three dimensions is completely specified by its three Cartesian
coordinates,
,
, and
. Moreover, these coordinates are clearly
independent of one another.
Hence, a dynamical
system consisting of a single particle moving
freely in three dimensions has three degrees of freedom. If there are two freely moving
particles then the system has six degrees of freedom, and so on.
Suppose that we have a dynamical system consisting of
particles
moving freely in three dimensions. This is an
degree of
freedom system whose instantaneous configuration can be specified
by
Cartesian coordinates. Let us denote these coordinates the
, for
. Thus,
are the Cartesian coordinates
of the first particle,
the Cartesian coordinates
of the second particle, et cetera. Suppose that the instantaneous configuration of the system can also be specified by
generalized
coordinates, which we shall denote the
, for
. Thus, the
might be the spherical coordinates of the particles.
In general,
we expect the
to be functions of the
. In other words,

(B.1) 
for
. Here, for the sake of generality, we have included the
possibility that the functional relationship between the
and the
might depend on the time,
, explicitly. This would be the
case if the dynamical system were subject to timevarying constraints. For
instance, a system consisting of a particle constrained to move on a surface that is itself
moving. Finally, by the chain rule, the variation of the
caused by a variation of the
(at constant
) is given by

(B.2) 
for
.
Next: Generalized Forces
Up: Classical Mechanics
Previous: Classical Mechanics
Richard Fitzpatrick
20160125