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Generalized forces find use in
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
, where they play a role conjugate to
generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
. 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 analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
. 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, Fi, acting on the particles Pi, i=1,..., n, is given by :\delta W = \sum_^n \mathbf _ \cdot \delta \mathbf r_i where δri is the virtual displacement of the particle Pi.


Generalized coordinates

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 :\delta \mathbf_i = \sum_^m \frac \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 :\delta W = \mathbf _ \cdot \sum_^m \frac \delta q_j +\ldots+ \mathbf _ \cdot \sum_^m \frac \delta q_j. Collect the coefficients of δqj so that :\delta W = \sum_^n \mathbf _ \cdot \frac \delta q_1 +\ldots+ \sum_^n \mathbf _ \cdot \frac \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 + \ldots + Q_m\delta q_m, where :Q_j = \sum_^n \mathbf _ \cdot \frac ,\quad j=1,\ldots, m, are called the generalized forces associated with the generalized coordinates qj, 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 Pi be Vi, then the virtual displacement δri can also be written in the formT. R. Kane and D. A. Levinson
Dynamics, Theory and Applications
McGraw-Hill, NY, 2005.
:\delta \mathbf_i = \sum_^m \frac \delta q_j,\quad i=1,\ldots, n. This means that the generalized force, Qj, can also be determined as :Q_j = \sum_^n \mathbf _ \cdot \frac , \quad j=1,\ldots, 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 A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's second law of motion, which trea ...
), called
D'Alembert's principle D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alembert ...
. The inertia force of a particle, Pi, of mass mi is :\mathbf_i^*=-m_i\mathbf_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 = \sum_^n \mathbf ^*_ \cdot \frac ,\quad j=1,\ldots, m. D'Alembert's form of the principle of virtual work yields : \delta W = (Q_1+Q^*_1)\delta q_1 + \ldots + (Q_m+Q^*_m)\delta q_m.


References


See also

*
Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
*
Generalized coordinates In analytical mechanics, generalized coordinates are a set of parameters used to represent the state of a system in a configuration space. These parameters must uniquely define the configuration of the system relative to a reference state.,p. 39 ...
*
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 freedo ...
* Virtual work {{DEFAULTSORT:Generalized Forces Mechanics Classical mechanics Lagrangian mechanics