Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, F_{i}, i=1,..., n, acting on a

_{i}, acting on the particles P_{i}, i=1,..., n, is given by
:$\backslash delta\; W\; =\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash delta\; \backslash mathbf\; r\_i$
where δr_{i} is the virtual displacement of the particle P_{i}.

_{i}, be a function of the generalized coordinates, q_{j}, j=1,...,m. Then the virtual displacements δr_{i} are given by
:$\backslash delta\; \backslash mathbf\_i\; =\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j,\backslash quad\; i=1,\backslash ldots,\; n,$
where δq_{j} is the virtual displacement of the generalized coordinate q_{j}.
The virtual work for the system of particles becomes
:$\backslash delta\; W\; =\; \backslash mathbf\; \_\; \backslash cdot\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j\; +\backslash ldots+\; \backslash mathbf\; \_\; \backslash cdot\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j.$
Collect the coefficients of δq_{j} so that
:$\backslash delta\; W\; =\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash frac\; \backslash delta\; q\_1\; +\backslash ldots+\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash frac\; \backslash delta\; q\_m.$

_{j}, j=1,...,m.

_{i} be V_{i}, then the virtual displacement δr_{i} can also be written in the formT. R. Kane and D. A. Levinson

Dynamics, Theory and Applications

McGraw-Hill, NY, 2005. :$\backslash delta\; \backslash mathbf\_i\; =\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j,\backslash quad\; i=1,\backslash ldots,\; n.$ This means that the generalized force, Q_{j}, can also be determined as
:$Q\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash frac\; ,\; \backslash quad\; j=1,\backslash ldots,\; m.$

_{i}, of mass m_{i} is
:$\backslash mathbf\_i^*=-m\_i\backslash mathbf\_i,\backslash quad\; i=1,\backslash ldots,\; n,$
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
:$Q^*\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; ^*\_\; \backslash cdot\; \backslash frac\; ,\backslash quad\; j=1,\backslash ldots,\; m.$
D'Alembert's form of the principle of virtual work yields
:$\backslash delta\; W\; =\; (Q\_1+Q^*\_1)\backslash delta\; q\_1\; +\; \backslash ldots\; +\; (Q\_m+Q^*\_m)\backslash delta\; q\_m.$

system
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and expres ...

that has its configuration defined in terms of generalized coordinates. In the formulation of virtual work
In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and Motion, movement of a mechanical system. The Work (physics), work of a force acting on a particle as it moves along a Displacem ...

, each generalized force is the coefficient of the variation of a generalized coordinate.
Virtual work

Generalized forces can be obtained from the computation of thevirtual work
In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and Motion, movement of a mechanical system. The Work (physics), work of a force acting on a particle as it moves along a Displacem ...

, δW, of the applied forces.
The virtual work of the forces, FGeneralized coordinates

Let the position vectors of each of the particles, rGeneralized forces

The virtual work of a system of particles can be written in the form :$\backslash delta\; W\; =\; Q\_1\backslash delta\; q\_1\; +\; \backslash ldots\; +\; Q\_m\backslash delta\; q\_m,$ where :$Q\_j\; =\; \backslash sum\_^n\; \backslash mathbf\; \_\; \backslash cdot\; \backslash frac\; ,\backslash quad\; j=1,\backslash ldots,\; m,$ are called the generalized forces associated with the generalized coordinates qVelocity 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 PDynamics, Theory and Applications

McGraw-Hill, NY, 2005. :$\backslash delta\; \backslash mathbf\_i\; =\; \backslash sum\_^m\; \backslash frac\; \backslash delta\; q\_j,\backslash quad\; i=1,\backslash ldots,\; n.$ This means that the generalized force, Q

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
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial reference frame, non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's la ...

), called D'Alembert's principle. The inertia force of a particle, PReferences

See also

* Lagrangian mechanics * Generalized coordinates *Degrees of freedom (physics and chemistry)
In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of ...

*Virtual work
In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and Motion, movement of a mechanical system. The Work (physics), work of a force acting on a particle as it moves along a Displacem ...

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Lagrangian mechanics