Rayleigh dissipation function

In physics, the Rayleigh dissipation function, named after Lord Rayleigh, is a function used to handle the effects of velocity-proportional frictional forces in Lagrangian mechanics. If the frictional force on a particle with velocity ${\displaystyle {\vec {v}}}$ can be written as ${\displaystyle {\vec {F}}_{f}=-{\vec {k}}\cdot {\vec {v}}}$, the Rayleigh dissipation function can be defined for a system of ${\displaystyle N}$ particles as

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

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

As friction is not conservative, it is included in the Q_j term of Lagrange's equations.

References[]

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

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