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 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 $$ q \mapsto R(\dot \dot q)=\frac12 \dot q \cdot \mathbb V \dot q $$ 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.