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.

Virtual work

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 where is the virtual displacement of the particle .

Generalized coordinates

Let the position vectors of each of the particles, , be a function of the generalized coordinates, . Then the virtual displacements where is the virtual displacement of the generalized coordinate .

Collect the coefficients of

Generalized forces

are called the generalized forces associated with the generalized coordinates .

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 Pi be , then the virtual displacement can also be written in the form

This means that the generalized force,

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, , of mass where is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates

See also

• Lagrangian mechanics

• Generalized coordinates

• Degrees of freedom (physics and chemistry)

• Virtual work

References