Generalized force

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.

Generalized forces can be obtained from the computation of the virtual work, δW, of the applied forces.

The virtual work of the forces, F_i, acting on the particles P_i, i=1,..., n, is given by

where δr_i is the virtual displacement of the particle P_i.

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

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

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.

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

This means that the generalized force, Q_j, 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, 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

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