Generalized force

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 displacement $\delta r$ under the influence of a force F the work done by that force is given by:

$\delta W=F\cdot \delta r=\sum _{i}F_{i}\delta x_{i}$.

Translating to Generalized coordinates:

$\delta W=\sum _{i}(\sum _{k=1}^{n}F_{i}{\frac {\partial x_{i}}{\partial q_{k}}}\delta q_{k})$,

and by reversing the order of summation we get

$\delta W=\sum _{k=1}^{n}(\sum _{i}F_{i}{\frac {\partial x_{i}}{\partial q_{k}}})\delta q_{k}$,