# Generalized forces

In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces F$_i$, i = 1, …, n, acting on a system that has its configuration defined in terms of generalized coordinates. In the formulation of virtual work, each generalized force is the coefficient of the variation of a generalized coordinate.

## Virtual work

Generalized forces can be obtained from the computation of the virtual work, δW, of the applied forces.$^{:265}$
The virtual work of the forces, F$_i$, acting on the particles P$_i$, i = 1, ..., n, is given by $\delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}$ where δr$_i$ is the virtual displacement of the particle P$_i$.

### Generalized coordinates

Let the position vectors of each of the particles, r$_i$, be a function of the generalized coordinates, q$_j$, j = 1, ..., m. Then the virtual displacements δr$_i$ are given by $\delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j},\quad i=1,\ldots ,n,$ where δq$_j$ is the virtual displacement of the generalized coordinate q$_j$.
The virtual work for the system of particles becomes $\delta W=\mathbf {F} _{1}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{1}}{\partial q_{j}}}\delta q_{j}+\dots +\mathbf {F} _{n}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{n}}{\partial q_{j}}}\delta q_{j}.$ Collect the coefficients of δq$_j$ so that $\delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{1}}}\delta q_{1}+\dots +\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{m}}}\delta q_{m}.$

### Generalized forces

The virtual work of a system of particles can be written in the form $\delta W=Q_{1}\delta q_{1}+\dots +Q_{m}\delta q_{m},$ where $Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}},\quad j=1,\ldots ,m,$ are called the generalized forces associated with the generalized coordinates q$_j$, j = 1, ..., m.

### Velocity formulation

In the application of the principle of virtual work it is often convenient to obtain virtual displacements from the velocities of the system. For the n particle system, let the velocity of each particle P$_i$ be V$_i$, then the virtual displacement δr$_i$ can also be written in the form $\delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}\delta q_{j},\quad i=1,\ldots ,n.$
This means that the generalized force, Q$_j$, can also be determined as $Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.$

## D'Alembert's principle

D'Alembert formulated the dynamics of a particle as the equilibrium of the applied forces with an inertia force (apparent force), called D'Alembert's principle. The inertia force of a particle, P$_i$, of mass m$_i$ is $\mathbf {F} _{i}^{*}=-m_{i}\mathbf {A} _{i},\quad i=1,\ldots ,n,$ where A$_i$ is the acceleration of the particle.
If the configuration of the particle system depends on the generalized coordinates q$_j$, j = 1, ..., m, then the generalized inertia force is given by $Q_{j}^{*}=\sum _{i=1}^{n}\mathbf {F} _{i}^{*}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.$
D'Alembert's form of the principle of virtual work yields $\delta W=(Q_{1}+Q_{1}^{*})\delta q_{1}+\dots +(Q_{m}+Q_{m}^{*})\delta q_{m}.$