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

(594) 
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, etc.
Using Equation (593), we can also write

(595) 
The above expression can be rearranged to give

(596) 
where

(597) 
Here, the are termed generalized forces. 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 conservative. It follows that

(598) 
for , where
is the system's potential energy. Hence, according to Equation (597),

(599) 
for .
Next: Lagrange's Equation
Up: Lagrangian Dynamics
Previous: Generalized Coordinates
Richard Fitzpatrick
20110331