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:

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

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 generalised force associated with the coordinate $q_{j}$.

Since $Q_{j}q_{j}$ has the dimensions of work, $Q_{j}$ will have the dimensions of force if $q_{j}$ is a distance, and the dimensions of torque if $q_{j}$ is an angle.