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

$\delta W=\sum _{i=1}^n{\mathbf{𝐅}}_i\cdot \delta {\mathbf{𝐫}}_i$

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