Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, 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.
The virtual work of the forces, Fi, acting on the particles Pi, i=1,..., n, is given by
where δri is the virtual displacement of the particle Pi.
Let the position vectors of each of the particles, ri, be a function of the generalized coordinates, qj, j=1,...,m. Then the virtual displacements δri are given by
where δqj is the virtual displacement of the generalized coordinate qj.
The virtual work for the system of particles becomes
Collect the coefficients of δqj so that
The virtual work of a system of particles can be written in the form
are called the generalized forces associated with the generalized coordinates qj, j=1,...,m.
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 Pi be Vi, then the virtual displacement δri can also be written in the form2
This means that the generalized force, Qj, 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, Pi, of mass mi is
where Ai is the acceleration of the particle.
If the configuration of the particle system depends on the generalized coordinates qj, j=1,...,m, then the generalized inertia force is given by
D'Alembert's form of the principle of virtual work yields