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.^[1] 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

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

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

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

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 (see ^[2])

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

Rayleigh writes the Lagrangian ${\displaystyle L}$ as kinetic energy ${\displaystyle T}$ minus potential energy ${\displaystyle V}$, which yields Rayleigh's Eqn. (26) from 1873.

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

Since the 1970s the name Rayleigh dissipation potential for ${\displaystyle R}$ is more common. Moreover, the original theory is generalized from quadratic functions ${\displaystyle q\mapsto R({\dot {\dot {q))})={\frac {1}{2)){\dot {q))\cdot \mathbb {V} {\dot {q))}$ 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 ${\displaystyle 0=R(q,0)\leq R(q,{\dot {q)))}$, see e.g. ^{[3] [4] [5]}