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

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

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, ${\vec {F}}_{f}=-\nabla _{v}R(v)$, 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 $q_{i}=\left\{q_{1},q_{2},\ldots q_{n}\right\}$ as

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

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

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

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

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

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

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

Since the 1970s the name Rayleigh dissipation potential for $R$ is more common. Moreover, the original theory is generalized from quadratic functions $q\mapsto R({\dot {q}})={\frac {1}{2}}{\dot {q}}\cdot \mathbb {V} {\dot {q}}$ to dissipation potentials that are depending on $q$ (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 ${\dot {q}}\mapsto R(q,{\dot {q}})$ is convex and satisfies $0=R(q,0)\leq R(q,{\dot {q}})$, see e.g.