Generalized force

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The idea of a Generalized Force is a concept stemming from Lagrangian mechanics. It is a consequence of the application of generalized coordinates to a system undergoing acceleration.

When a particle undergoes a virtual displacement \delta \mathbf{r} under the influence of a force \mathbf{F} the virtual work done by that force is given by:

\delta W = \mathbf{F} \cdot \delta \mathbf{r} = \sum_{i} F_i \delta x_i.

Translating to generalized coordinates:

\delta W = \sum_{i} (\sum_{j=1}^n F_i \frac {\partial x_i}{\partial q_j} \delta q_j),

and by reversing the order of summation we get

\delta W = \sum_{j=1}^n ( \sum_{i}F_i \frac {\partial x_i}{\partial q_j})\delta q_j.

It is from this formulation that the idea of a generalized force stems. The above equation can be written as

\delta W = \sum_{j=1}^n (Q_j)\delta q_j

where

Q_j = \sum_{i}(F_i \frac {\partial x_i}{\partial q_j})

is called the generalized force associated with the coordinate qj.

Since Qjqj has the dimension of work, Qj will have the dimension of force if qj is a distance, and the dimension of torque if qj is an angle.

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