Rayleigh dissipation function

{{Short description|Function used in Lagrangian mechanics}}

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.{{cite journal | last=Rayleigh | first=Lord |

title= Some general theorems relating to vibrations. |

journal=Proc. London Math. Soc. | volume=s1-4 | date=1873 |

doi=10.1112/plms/s1-4.1.357 | pages=357–368|doi-access=free}}

If the frictional force on a particle with velocity \vec{v} can be written as \vec{F}_f = -k\vec{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, \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

: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 (see {{cite book |last=Goldstein |first=Herbert |authorlink=Herbert Goldstein |year=1980 |title=Classical Mechanics |edition=2nd |publisher=Addison-Wesley |location=Reading, MA |isbn=0-201-02918-9 |page=24}})

:\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)=\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. {{cite journal | last = Moreau | first = Jean Jacques |authorlink = Jean Jacques Moreau | title= Fonctions de résistance et fonctions de dissipation | journal = Travaux du Séminaire d'Analyse Convexe, Montpellier (Exposé no. 6) | year = 1971 | url=https://hal.science/hal-02309448 | page= (See page 6.3 for "fonction de resistance")}}

{{cite book |last1=Lebon |first1=Georgy | last2=Jou | first2= David

| last3 = Casas-Vàzquez | first3 = Jos\'e

|year=2008 |title = Understanding Non-equilibrium Thermodynamics |publisher = Springer-Verlag

| page = (See Chapter 10.2 for dissipation potentials)}}

{{cite arXiv | last = Mielke | first = Alexander | title = An introduction to the analysis of gradient systems | year =2023 | page = (See Definition 3.1 on page 25 for dissipation potentials) | class = math-ph | eprint = 2306.05026 }}

References