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 $Q\_jq\_j$ 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.

ee also

*Lagrangian mechanics
*Generalized coordinates
*Degrees of freedom (physics and chemistry)
*Virtual work