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# Rayleigh dissipation function

In physics, the Rayleigh dissipation function, named for Lord Rayleigh, is a function used to handle the effects of velocity-proportional frictional forces in Lagrangian mechanics. It is defined for a system of \( {\displaystyle N}\) particles as

\( {\displaystyle F={\frac {1}{2}}\sum _{i=1}^{N}(k_{x}v_{i,x}^{2}+k_{y}v_{i,y}^{2}+k_{z}v_{i,z}^{2}).} \)

The force of friction is negative the velocity gradient of the dissipation function, \( {\ {\displaystyle {\mathbf {F}}_{f}=-\nabla _{\mathbf {v}}F}\) . The function is half the rate at which energy is being dissipated by the system through friction.

References

Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. p. 24. ISBN 0-201-02918-9.

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