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 $\overrightarrow{v}$ can be written as ${\overrightarrow{F}}_f=-k\overrightarrow{v}$, where $k$ is a diagonal matrix, then 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, ${\overrightarrow{F}}_f=-{\mathrm{\nabla }}_{v}R(v)$, 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 $q_i=\{q_1,q_2,\dots q_n\}$ as

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

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

$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q_i}}\right)-\frac{\partial L}{\partial q_i}=-\frac{\partial R}{\partial {\dot{q}}_i}$.

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

$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q_i}}\right)-\frac{\partial T}{\partial q_i}+\frac{\partial R}{\partial {\dot{q}}_i}+\frac{\partial V}{\partial q_i}=0$.

Since the 1970s the name Rayleigh dissipation potential for $R$ is more common. Moreover, the original theory is generalized from quadratic functions $q\mapsto R(\dot{q})=\frac{1}{2}\dot{q}\cdot 𝕍\dot{q}$ 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 $0=R(q,0)\le R(q,\dot{q})$.