Generalized coordinates

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, and so on. 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,

(7.1) |

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 time-varying 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 due to a variation of the (at constant ) is given by

for .