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In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces Fi, 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 coordinates

• Generalized forces

• Velocity formulation

• D'Alembert's principle

• See also

• References

Generalized forces can be obtained from the computation of the virtual work, δW, of the applied forces.cite-ref-torby1984-1-0[1]

The virtual work of the forces, Fi, acting on the particles Pi, i = 1, ..., n, is given by δ δ W = ∑ ∑ i = 1 n F i ⋅ ⋅ δ δ r i {\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}} where δri is the virtual displacement of the particle Pi.

Let the position vectors of each of the particles, ri, be a function of the generalized coordinates, qj, j = 1, ..., m. Then the virtual displacements δri are given by δ δ r i = ∑ ∑ j = 1 m ∂ ∂ r i ∂ ∂ q j δ δ q j , i = 1 , … … , n , {\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,} where δqj is the virtual displacement of the generalized coordinate qj.

The virtual work for the system of particles becomes δ δ W = F 1 ⋅ ⋅ ∑ ∑ j = 1 m ∂ ∂ r 1 ∂ ∂ q j δ δ q j + ⋯ ⋯ + F n ⋅ ⋅ ∑ ∑ j = 1 m ∂ ∂ r n ∂ ∂ q j δ δ q j . {\displaystyle \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 δqj so that δ δ W = ∑ ∑ i = 1 n F i ⋅ ⋅ ∂ ∂ r i ∂ ∂ q 1 δ δ q 1 + ⋯ ⋯ + ∑ ∑ i = 1 n F i ⋅ ⋅ ∂ ∂ r i ∂ ∂ q m δ δ q m . {\displaystyle \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}.}

The virtual work of a system of particles can be written in the form δ δ W = Q 1 δ δ q 1 + ⋯ ⋯ + Q m δ δ q m , {\displaystyle \delta W=Q_{1}\delta q_{1}+\dots +Q_{m}\delta q_{m},} where Q j = ∑ ∑ i = 1 n F i ⋅ ⋅ ∂ ∂ r i ∂ ∂ q j , j = 1 , … … , m , {\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}},\quad j=1,\ldots ,m,} are called the generalized forces associated with the generalized coordinates qj, j = 1, ..., m.

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 Pi be Vi, then the virtual displacement δri can also be written in the formcite-ref-2[2] δ δ r i = ∑ ∑ j = 1 m ∂ ∂ V i ∂ ∂ q ˙ ˙ j δ δ q j , i = 1 , … … , n . {\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.}

This means that the generalized force, Qj, can also be determined as Q j = ∑ ∑ i = 1 n F i ⋅ ⋅ ∂ ∂ V i ∂ ∂ q ˙ ˙ j , j = 1 , … … , m . {\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.}

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, Pi, of mass mi is F i ∗ ∗ = − − m i A i , i = 1 , … … , n , {\displaystyle \mathbf {F} _{i}^{*}=-m_{i}\mathbf {A} _{i},\quad i=1,\ldots ,n,} where Ai is the acceleration of the particle.

If the configuration of the particle system depends on the generalized coordinates qj, j = 1, ..., m, then the generalized inertia force is given by Q j ∗ ∗ = ∑ ∑ i = 1 n F i ∗ ∗ ⋅ ⋅ ∂ ∂ V i ∂ ∂ q ˙ ˙ j , j = 1 , … … , m . {\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.}

D'Alembert's form of the principle of virtual work yields δ δ W = ( Q 1 + Q 1 ∗ ∗ ) δ δ q 1 + ⋯ ⋯ + ( Q m + Q m ∗ ∗ ) δ δ q m . {\displaystyle \delta W=(Q_{1}+Q_{1}^{*})\delta q_{1}+\dots +(Q_{m}+Q_{m}^{*})\delta q_{m}.}

• Lagrangian mechanics

• Generalized coordinates

• Degrees of freedom (physics and chemistry)

• Virtual work

cite-note-torby1984-11. ↑ citereftorby1984Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0-03-063366-4.

cite-note-22. ↑ T. R. Kane and D. A. Levinson, Dynamics, Theory and Applications, McGraw-Hill, NY, 2005.