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 can be written as , where is a diagonal matrix, then the Rayleigh dissipation function can be defined for a system of particles as
:
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, , 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 as
:.
As friction is not conservative, it is included in the term of Lagrange's equations,
:.
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}})
:.
Rayleigh writes the Lagrangian as kinetic energy minus potential energy , which yields Rayleigh's equation from 1873.
:
\frac{\partial R}{\partial\dot{q}_{i}}
+\frac{\partial V}{\partial q_{i}}=0 .
Since the 1970s the name Rayleigh dissipation potential for is more common. Moreover, the original theory is generalized from quadratic functions to
dissipation potentials that are depending on (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 is convex and satisfies , see
{{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)}}