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Generalized forces

Generalized forces are defined via coordinate transformation of applied forces, \[ \mathbf{F}_i, \]on a system of n particles, i. The concept finds use in Lagrangian mechanics, where it plays a conjugate role to generalized coordinates.

A convenient equation from which to derive the expression for generalized forces is that of the virtual work, \[ \delta W_a, \] caused by applied forces, as seen in D'Alembert's principle in accelerating systems and the principle of virtual work for applied forces in static systems. The subscript a is used here to indicate that this virtual work only accounts for the applied forces, a distinction which is important in dynamic systems.[1]:265

\[ \delta W_a = \sum_{i=1}^n \mathbf {F}_{i} \cdot \delta \mathbf r_i \]

\[ \delta \mathbf r_i \]is the virtual displacement of the system, which does not have to be consistent with the constraints (in this development)

Substitute the definition for the virtual displacement (differential):[1]:265

\[ \delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j
\delta W_a = \sum_{i=1}^n \mathbf {F}_{i} \cdot \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j \]

Using the distributive property of multiplication over addition and the associative property of addition, we have[1]:265

\[ \delta W_a = \sum_{j=1}^m \sum_{i=1}^n \mathbf {F}_{i} \cdot \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j. \]

By analogy with the way work is defined in classical mechanics, we define the generalized force as:[1]:265

\[ Q_j = \sum_{i=1}^n \mathbf {F}_{i} \cdot \frac {\partial \mathbf {r}_i} {\partial q_j}. \]

Thus, the virtual work due to the applied forces is[1]:265

\[ \delta W_a = \sum_{j=1}^m Q_j \delta q_j. \]


^ a b c d e Torby, 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.

See also

Lagrangian mechanics
Generalized coordinates
Degrees of freedom (physics and chemistry)
Virtual work


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