In analytical mechanics (particularly Lagrangian mechanics), generalized forces are 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.
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, Fi, acting on the particles Pi, i = 1, ..., n, is given by
δ
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
δ
δ
where δqj is the virtual displacement of the generalized coordinate qj.
The virtual work for the system of particles becomes
δ
δ
δ
Collect the coefficients of δqj so that
δ
δ
δ
The virtual work of a system of particles can be written in the form
δ
δ
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 form[2]
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