# Generalized force

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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 $delta mathbf\left\{r\right\}$ under the influence of a force $mathbf\left\{F\right\}$ the virtual work done by that force is given by:

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

Translating to generalized coordinates:

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

and by reversing the order of summation we get

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

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

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

where

$Q_j = sum_\left\{i\right\}F_i frac \left\{partial x_i\right\}\left\{partial q_j\right\}$

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

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