In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces , 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.
Generalized forces can be obtained from the computation of the virtual work, , of the applied forces.
The virtual work of the forces, , acting on the particles , is given by
where is the virtual displacement of the particle .
Let the position vectors of each of the particles, , be a function of the generalized coordinates, . Then the virtual displacements are given by
where is the virtual displacement of the generalized coordinate .
The virtual work for the system of particles becomes
Collect the coefficients of so that
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 .
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<sub>i</sub> be , then the virtual displacement can also be written in the form
This means that the generalized force, , can also be determined as
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, , of mass is
where is the acceleration of the particle.
If the configuration of the particle system depends on the generalized coordinates , then the generalized inertia force is given by
D'Alembert's form of the principle of virtual work yields