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Generalized forces

The work done on the dynamical system when its Cartesian coordinates change by is simply

 (7.3)

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, and so on. Using Equation (7.2), we can also write

 (7.4)

The preceding expression can be rearranged to give

 (7.5)

where

 (7.6)

Here, the are termed generalized forces. 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 conservative. It follows that

 (7.7)

for , where is the system's potential energy. Hence, according to Equation (7.6),

 (7.8)

for .

Next: Lagrange's equation Up: Lagrangian mechanics Previous: Generalized coordinates
Richard Fitzpatrick 2016-03-31