analytical mechanics In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related formulations of classical mechanics. Analytical mechanics uses '' scalar'' properties of motion representing the sy ...

(particularly

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the ...

), generalized forces are conjugate to

generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.p. 397 ...

. They are obtained from the applied

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

s , acting on a

system A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...

that has its configuration defined in terms of generalized coordinates. In the formulation of

virtual work In mechanics, virtual work arises in the application of the '' principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different fo ...

, 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

, , of the applied forces. The virtual work of the forces, , acting on the particles , is given by

\delta W = \sum_^n \mathbf F_i \cdot \delta \mathbf r_i

where is the

virtual displacement In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very ...

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 are given by

\delta \mathbf_i = \sum_^m \frac   \delta q_j,\quad i=1,\ldots, n,

where is the virtual displacement of the generalized coordinate . The virtual work for the system of particles becomes

\delta W = \mathbf F_1 \cdot \sum_^m \frac   \delta q_j + \dots + \mathbf F_n \cdot \sum_^m \frac   \delta q_j.

Collect the coefficients of so that

\delta W = \sum_^n \mathbf F_i \cdot \frac   \delta q_1 + \dots + \sum_^n \mathbf F_i \cdot \frac   \delta q_m.

Generalized forces

The virtual work of a system of particles can be written in the form

\delta W = Q_1\delta q_1 + \dots + Q_m\delta q_m,

where

Q_j = \sum_^n \mathbf F_i \cdot \frac  ,\quad j=1,\ldots, m,

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 P_i be , then the virtual displacement can also be written in the formT. R. Kane and D. A. Levinson
Dynamics, Theory and Applications
McGraw-Hill, NY, 2005.

\delta \mathbf r_i = \sum_^m \frac   \delta q_j,\quad i=1,\ldots, n.

This means that the generalized force, , can also be determined as

Q_j = \sum_^n \mathbf F_i \cdot \frac  , \quad j=1,\ldots, m.

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 D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical physics, classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d' ...

. The inertia force of a particle, , of mass is

\mathbf F_i^*=-m_i\mathbf A_i,\quad i=1,\ldots, n,

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

Q^*_j = \sum_^n \mathbf F^*_ \cdot \frac  ,\quad j=1,\ldots, m.

D'Alembert's form of the principle of virtual work yields

\delta W = (Q_1 + Q^*_1)\delta q_1 + \dots + (Q_m + Q^*_m)\delta q_m.

References