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 [math]\displaystyle{ \vec{v} }[/math] can be written as [math]\displaystyle{ \vec{F}_f = -\vec{k}\cdot\vec{v} }[/math], the Rayleigh dissipation function can be defined for a system of [math]\displaystyle{ N }[/math] particles as

[math]\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 ). }[/math]

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, [math]\displaystyle{ \vec{F}_f = -\nabla_v R(v) }[/math], 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 [math]\displaystyle{ q_{i}=\left\{q_{1},q_{2},\ldots q_{n}\right\} }[/math] as

[math]\displaystyle{ \vec{F}_f = -\frac{\partial R}{\partial\dot{q}_{i}} }[/math].

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

[math]\displaystyle{ \frac{d}{dt}\frac{\partial L}{\partial \dot{q_{i}}}-\frac{\partial L}{\partial q_{i}}=Q_{i} }[/math].

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

[math]\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}} }[/math].

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

[math]\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 }[/math].

Since the 1970s the name Rayleigh dissipation potential for [math]\displaystyle{ R }[/math] is more common. Moreover, the original theory is generalized from quadratic functions [math]\displaystyle{ q \mapsto R(\dot \dot q)=\frac12 \dot q \cdot \mathbb V \dot q }[/math] to dissipation potentials that are depending on [math]\displaystyle{ q }[/math] (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 [math]\displaystyle{ \dot q \mapsto R(q,\dot q) }[/math] is convex and satisfies [math]\displaystyle{ 0 = R(q,0)\leq R(q, \dot q) }[/math], see e.g.