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 $\vec$ can be written as $\vec_f = -k\vec$, where $k$ is a diagonal matrix, then the Rayleigh dissipation function can be defined for a system of $N$ particles as

:$R(v) = \frac \sum_^N ( k_x v_^2 + k_y v_^2 + k_z v_^2 ).$

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, $\vec_f = -\nabla_v R(v)$, 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_=\left\$ as

:$F_ = -\frac$.

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

:$\frac\frac-\frac=Q_$.
Applying of the value of the frictional force described by generalized coordinates into the Euler-Lagrange equations gives

:$\frac\left(\frac\right)-\frac=-\frac$.

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

:$\frac\left(\frac\right) - \frac+ \frac +\frac=0$.

Since the 1970s the name Rayleigh dissipation potential for $R$ is more common. Moreover, the original theory is generalized from quadratic functions $q \mapsto R(\dot q)=\frac12 \dot q \cdot \mathbb V \dot q$ 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 $0 = R(q,0)\leq R(q, \dot q)$.

References

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Functions and mappings
Lagrangian mechanics