Английская Википедия:Generalized forces

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

The virtual work of the forces, Шаблон:Math, acting on the particles Шаблон:Math, is given by

<math>\delta W = \sum_{i=1}^n \mathbf F_i \cdot \delta \mathbf r_i</math>

where Шаблон:Math is the virtual displacement of the particle Шаблон:Mvar.

Generalized coordinates

Let the position vectors of each of the particles, Шаблон:Math, be a function of the generalized coordinates, Шаблон:Math. Then the virtual displacements Шаблон:Math are given by

<math>\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 Шаблон:Mvar is the virtual displacement of the generalized coordinate Шаблон:Mvar.

The virtual work for the system of particles becomes

<math>\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 Шаблон:Mvar so that

<math>\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> \delta W = Q_1\delta q_1 + \ldots + Q_m\delta q_m,</math>

where

<math>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 Шаблон: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 Шаблон:Math, then the virtual displacement Шаблон:Math can also be written in the form

<math>\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, Шаблон:Mvar, can also be determined as

<math>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, Шаблон:Mvar, of mass Шаблон:Mvar is

<math>\mathbf F_i^*=-m_i\mathbf A_i,\quad i=1,\ldots, n,</math>

where Шаблон:Math is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates Шаблон:Math, then the generalized inertia force is given by

<math>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> \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