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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
20160125