Generalized forces

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.

Virtual work
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
 * $$\delta W = \sum_{i=1}^n \mathbf F_i \cdot \delta \mathbf r_i$$

where $δr_{i}$ is the virtual displacement of the particle $P_{i}$.

Generalized coordinates
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
 * $$\delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j,\quad i=1,\ldots, n,$$

where $δq_{j}$ is the virtual displacement of the generalized coordinate $q_{j}$.

The virtual work for the system of particles becomes
 * $$\delta W = \mathbf F_1 \cdot \sum_{j=1}^m \frac {\partial \mathbf r_1} {\partial q_j} \delta q_j +\ldots+ \mathbf F_n \cdot \sum_{j=1}^m \frac {\partial \mathbf r_n} {\partial q_j} \delta q_j.$$

Collect the coefficients of $δq_{j}$ so that
 * $$\delta W = \sum_{i=1}^n \mathbf F_i \cdot \frac {\partial \mathbf r_i} {\partial q_1} \delta q_1 +\ldots+ \sum_{i=1}^n \mathbf F_i \cdot \frac {\partial \mathbf r_i} {\partial q_m} \delta q_m.$$

Generalized forces
The virtual work of a system of particles can be written in the form
 * $$ \delta W = Q_1\delta q_1 + \ldots + Q_m\delta q_m,$$

where
 * $$Q_j = \sum_{i=1}^n \mathbf F_i \cdot \frac {\partial \mathbf r_i} {\partial q_j},\quad j=1,\ldots, m,$$

are called the generalized forces associated with the generalized coordinates $q_{j}, j = 1, ..., m$.

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 $V_{i}$, then the virtual displacement $δr_{i}$ can also be written in the form
 * $$\delta \mathbf r_i = \sum_{j=1}^m \frac {\partial \mathbf V_i} {\partial \dot q_j} \delta q_j,\quad i=1,\ldots, n.$$

This means that the generalized force, $Q_{j}$, can also be determined as
 * $$Q_j = \sum_{i=1}^n \mathbf F_i \cdot \frac {\partial \mathbf V_i} {\partial \dot{q}_j}, \quad j=1,\ldots, m.$$

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. The inertia force of a particle, $P_{i}$, of mass $m_{i}$ is
 * $$\mathbf F_i^*=-m_i\mathbf A_i,\quad i=1,\ldots, n,$$

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
 * $$Q^*_j = \sum_{i=1}^n \mathbf F^*_{i} \cdot \frac {\partial \mathbf V_i} {\partial \dot q_j},\quad j=1,\ldots, m.$$

D'Alembert's form of the principle of virtual work yields
 * $$ \delta W = (Q_1+Q^*_1)\delta q_1 + \ldots + (Q_m+Q^*_m)\delta q_m.$$