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

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

This function represents half of the rate of energy dissipation of the system through friction. The force of friction is negative the velocity gradient of the dissipation function, ${\displaystyle {\vec {F}}_{f}=-\nabla _{v}G(v)}$, analogous to a force being equal to the negative position gradient of a potential. This relationship is represented in terms of the set of generalized coordinates ${\displaystyle q_{i}=\left\{q_{1},q_{2},\ldots q_{n}\right\}}$ as

${\displaystyle {\vec {F}}_{f}=-{\frac {\partial G}{\partial {\dot {q}}_{i}}}}$.

As friction is not conservative, it is included in the ${\displaystyle Q_{i}}$ term of Lagrange's equations,

${\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q_{i}}}}}={\frac {\partial L}{\partial q_{i}}}+Q_{i}}$.

Applying of the value of the frictional force described by generalized coordinates into the Euler-Lagrange equations gives

${\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q_{i}}}}}={\frac {\partial L}{\partial q_{i}}}-{\frac {\partial G}{\partial {\dot {q}}_{i}}}}$.