Generalized forces

In analytical mechanics (particularly Lagrangian mechanics), generalized forces are 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.

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 $$\delta {\mathbf{r}}_i=\sum _{j=1}^m\frac{\partial {\mathbf{r}}_i}{\partial q_j}\delta q_j,i=1,\dots ,n,$$ where δq_j is the virtual displacement of the generalized coordinate q_j. The virtual work for the system of particles becomes $$\delta W={\mathbf{F}}_1\cdot \sum _{j=1}^m\frac{\partial {\mathbf{r}}_1}{\partial q_j}\delta q_j+\dots +{\mathbf{F}}_n\cdot \sum _{j=1}^m\frac{\partial {\mathbf{r}}_n}{\partial q_j}\delta q_j.$$ Collect the coefficients of δq_j so that $$\delta W=\sum _{i=1}^n{\mathbf{F}}_i\cdot \frac{\partial {\mathbf{r}}_i}{\partial q_1}\delta q_1+\dots +\sum _{i=1}^n{\mathbf{F}}_i\cdot \frac{\partial {\mathbf{r}}_i}{\partial q_m}\delta q_m.$$

Generalized forces

The virtual work of a system of particles can be written in the form $$\delta W=Q_1\delta q_1+\dots +Q_m\delta q_m,$$ where $$Q_j=\sum _{i=1}^n{\mathbf{F}}_i\cdot \frac{\partial {\mathbf{r}}_i}{\partial q_j},j=1,\dots ,m,$$ 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 P_i be V_i, then the virtual displacement δr_i can also be written in the form $$\delta {\mathbf{r}}_i=\sum _{j=1}^m\frac{\partial {\mathbf{V}}_i}{\partial {\dot{q}}_j}\delta q_j,i=1,\dots ,n.$$ This means that the generalized force, Q_j, can also be determined as $$Q_j=\sum _{i=1}^n{\mathbf{F}}_i\cdot \frac{\partial {\mathbf{V}}_i}{\partial {\dot{q}}_j},j=1,\dots ,m.$$

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 $${\mathbf{F}}_i^{*}=-m_i{\mathbf{A}}_i,i=1,\dots ,n,$$ 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 $$Q_j^{*}=\sum _{i=1}^n{\mathbf{F}}_i^{*}\cdot \frac{\partial {\mathbf{V}}_i}{\partial {\dot{q}}_j},j=1,\dots ,m.$$ D'Alembert's form of the principle of virtual work yields $$\delta W=(Q_1+Q_1^{*})\delta q_1+\dots +(Q_m+Q_m^{*})\delta q_m.$$

See also

• Lagrangian mechanics
• Generalized coordinates
• Degrees of freedom (physics and chemistry)
• Virtual work