Generalized forces

In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces Template:Math, 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, Template:Mvar, of the applied forces.Template:Rp

The virtual work of the forces, Template:Math, acting on the particles Template:Math, is given by $$\delta W=\sum _{i=1}^n{𝐅}_i\cdot \delta {𝐫}_i$$ where Template:Math is the virtual displacement of the particle Template:Mvar.

Generalized coordinates

Let the position vectors of each of the particles, Template:Math, be a function of the generalized coordinates, Template:Math. Then the virtual displacements Template:Math are given by $$\delta {𝐫}_i=\sum _{j=1}^m\frac{\partial {𝐫}_i}{\partial q_j}\delta q_j,i=1,\dots ,n,$$ where Template:Mvar is the virtual displacement of the generalized coordinate Template:Mvar.

The virtual work for the system of particles becomes $$\delta W={𝐅}_1\cdot \sum _{j=1}^m\frac{\partial {𝐫}_1}{\partial q_j}\delta q_j+\dots +{𝐅}_n\cdot \sum _{j=1}^m\frac{\partial {𝐫}_n}{\partial q_j}\delta q_j.$$ Collect the coefficients of Template:Mvar so that $$\delta W=\sum _{i=1}^n{𝐅}_i\cdot \frac{\partial {𝐫}_i}{\partial q_1}\delta q_1+\dots +\sum _{i=1}^n{𝐅}_i\cdot \frac{\partial {𝐫}_i}{\partial q_m}\delta q_m.$$

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 $$Q_j=\sum _{i=1}^n{𝐅}_i\cdot \frac{\partial {𝐫}_i}{\partial q_j},j=1,\dots ,m,$$ are called the generalized forces associated with the generalized coordinates Template:Math.

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 Template:Math, then the virtual displacement Template:Math can also be written in the form $$\delta {𝐫}_i=\sum _{j=1}^m\frac{\partial {𝐕}_i}{\partial {\dot{q}}_j}\delta q_j,i=1,\dots ,n.$$

This means that the generalized force, Template:Mvar, can also be determined as $$Q_j=\sum _{i=1}^n{𝐅}_i\cdot \frac{\partial {𝐕}_i}{\partial {\dot{q}}_j},j=1,\dots ,m.$$

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, Template:Mvar, of mass Template:Mvar is $${𝐅}_i^{\ast }=-m_i{𝐀}_i,i=1,\dots ,n,$$ where Template:Math is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates Template:Math, then the generalized inertia force is given by $$Q_j^{\ast }=\sum _{i=1}^n{𝐅}_i^{\ast }\cdot \frac{\partial {𝐕}_i}{\partial {\dot{q}}_j},j=1,\dots ,m.$$

D'Alembert's form of the principle of virtual work yields $$\delta W=(Q_1+Q_1^{\ast })\delta q_1+\dots +(Q_m+Q_m^{\ast })\delta q_m.$$

See also

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