Physics:Generalized forces

Short description: Concept in Lagrangian mechanics

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

[math]\displaystyle{ \delta W = \sum_{i=1}^n \mathbf F_i \cdot \delta \mathbf r_i }[/math]

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

[math]\displaystyle{ \delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j,\quad i=1,\ldots, n, }[/math]

where δq_j is the virtual displacement of the generalized coordinate q_j.

The virtual work for the system of particles becomes

[math]\displaystyle{ \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. }[/math]

Collect the coefficients of δq_j so that

[math]\displaystyle{ \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. }[/math]

Generalized forces

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

[math]\displaystyle{ \delta W = Q_1\delta q_1 + \ldots + Q_m\delta q_m, }[/math]

where

[math]\displaystyle{ Q_j = \sum_{i=1}^n \mathbf F_i \cdot \frac {\partial \mathbf r_i} {\partial q_j},\quad j=1,\ldots, m, }[/math]

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

[math]\displaystyle{ \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. }[/math]

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

[math]\displaystyle{ Q_j = \sum_{i=1}^n \mathbf F_i \cdot \frac {\partial \mathbf V_i} {\partial \dot{q}_j}, \quad j=1,\ldots, m. }[/math]

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

[math]\displaystyle{ \mathbf F_i^*=-m_i\mathbf A_i,\quad i=1,\ldots, n, }[/math]

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

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

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

[math]\displaystyle{ \delta W = (Q_1+Q^*_1)\delta q_1 + \ldots + (Q_m+Q^*_m)\delta q_m. }[/math]

See also

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