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 v → {\displaystyle {\vec {v}}} can be written as F → f = − k → ⋅ v → {\displaystyle {\vec {F}}_{f}=-{\vec {k}}\cdot {\vec {v}}} , the Rayleigh dissipation function can be defined for a system of N {\displaystyle N} particles as

R ( v ) = 1 2 ∑ i = 1 N ( k x v i , x 2 + k y v i , y 2 + k z v i , z 2 ) . {\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, F → f = − ∇ v R ( v ) {\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 q i = { q 1 , q 2 , … q n } {\displaystyle q_{i}=\left\{q_{1},q_{2},\ldots q_{n}\right\}} as

F → f = − ∂ R ∂ q ˙ i {\displaystyle {\vec {F}}_{f}=-{\frac {\partial R}{\partial {\dot {q}}_{i}}}} .

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

d d t ∂ L ∂ q i ˙ − ∂ L ∂ q i = Q i {\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 )

d d t ( ∂ L ∂ q i ˙ ) − ∂ L ∂ q i = − ∂ R ∂ q ˙ i {\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 L {\displaystyle L} as kinetic energy T {\displaystyle T} minus potential energy V {\displaystyle V} , which yields Rayleigh's Eqn. (26) from 1873.

d d t ( ∂ T ∂ q i ˙ ) − ∂ T ∂ q i + ∂ R ∂ q ˙ i + ∂ V ∂ q i = 0 {\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 R {\displaystyle R} is more common. Moreover, the original theory is generalized from quadratic functions q ↦ R ( q ˙ ) = 1 2 q ˙ ⋅ V q ˙ {\displaystyle q\mapsto R({\dot {q}})={\frac {1}{2}}{\dot {q}}\cdot \mathbb {V} {\dot {q}}} to dissipation potentials that are depending on q {\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 q ˙ ↦ R ( q , q ˙ ) {\displaystyle {\dot {q}}\mapsto R(q,{\dot {q}})} is convex and satisfies 0 = R ( q , 0 ) ≤ R ( q , q ˙ ) {\displaystyle 0=R(q,0)\leq R(q,{\dot {q}})} , see e.g.