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# Generalized Forces

The work done on the dynamical system when its Cartesian coordinates change by is simply (B.3)

Here, the are the Cartesian components of the forces acting on the various particles making up the system. Thus, are the components of the force acting on the first particle, the components of the force acting on the second particle, et cetera. Using Equation (B.2), we can also write (B.4)

The previous expression can be rearranged to give (B.5)

where (B.6)

Here, the are termed generalized forces. More explicitly, is termed the force conjugate to the coordinate . Note that a generalized force does not necessarily have the dimensions of force. However, the product must have the dimensions of work. Thus, if a particular is a Cartesian coordinate then the associated is a force. Conversely, if a particular is an angle then the associated is a torque.

Suppose that the dynamical system in question is energy conserving. It follows that (B.7)

for , where is the system's potential energy. Hence, according to Equation (B.6), (B.8)

for .   Next: Lagrange's Equation Up: Classical Mechanics Previous: Generalized Coordinates
Richard Fitzpatrick 2016-01-25