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. It was first introduced by him in 1873. If the frictional force on a particle with velocity ${\displaystyle {\vec {v}}}$ can be written as ${\displaystyle {\vec {F}}_{f}=-k{\vec {v}}}$, where ${\displaystyle k}$ is a diagonal matrix, then the Rayleigh dissipation function can be defined for a system of ${\displaystyle N}$ particles as

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

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, ${\displaystyle {\vec {F}}_{f}=-\nabla _{v}R(v)}$, 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

${\displaystyle F_{f,i}=-{\frac {\partial R}{\partial {\dot {q}}_{i}}}}$.

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

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q_{i}}}}}\right)-{\frac {\partial L}{\partial q_{i}}}=-{\frac {\partial R}{\partial {\dot {q}}_{i}}}}$.

Rayleigh writes the Lagrangian ${\displaystyle L}$ as kinetic energy ${\displaystyle T}$ minus potential energy ${\displaystyle V}$, which yields Rayleigh's equation from 1873.

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial T}{\partial {\dot {q_{i}}}}}\right)-{\frac {\partial T}{\partial q_{i}}}+{\frac {\partial R}{\partial {\dot {q}}_{i}}}+{\frac {\partial V}{\partial q_{i}}}=0}$.

Since the 1970s the name Rayleigh dissipation potential for ${\displaystyle R}$ is more common. Moreover, the original theory is generalized from quadratic functions ${\displaystyle q\mapsto R({\dot {q}})={\frac {1}{2}}{\dot {q}}\cdot \mathbb {V} {\dot {q}}}$ to dissipation potentials that are depending on ${\displaystyle q}$ (then called state dependence) and are non-quadratic, which leads to nonlinear friction laws like in Coulomb friction or in plasticity. The main assumption is then, that the mapping ${\displaystyle {\dot {q}}\mapsto R(q,{\dot {q}})}$ is convex and satisfies ${\displaystyle 0=R(q,0)\leq R(q,{\dot {q}})}$.