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 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 $q_{j}$ .

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

See also