# Generalized forces

**Generalized forces** find use in Lagrangian mechanics, where they play a role 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.[1]^{:265}

The virtual work of the forces, **F**_{i}, acting on the particles P_{i}, i=1,..., n, is given by

where δ**r**_{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 δ**r**_{i} are given by

where δq_{j} is the virtual displacement of the generalized coordinate q_{j}.

The virtual work for the system of particles becomes

Collect the coefficients of δq_{j} so that

### Generalized forces

The virtual work of a system of particles can be written in the form

where

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 δ**r**_{i} can also be written in the form[2]

This means that the generalized force, Q_{j}, can also be determined as

## 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

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

D'Alembert's form of the principle of virtual work yields

## References

- Torby, Bruce (1984). "Energy Methods".
*Advanced Dynamics for Engineers*. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4. - T. R. Kane and D. A. Levinson, Dynamics, Theory and Applications, McGraw-Hill, NY, 2005.