Rayleigh dissipation function explained
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.[1] If the frictional force on a particle with velocity
can be written as
\vec{F}f=-\vec{k} ⋅ \vec{v}
, 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
qi=\left\{q1,q2,\ldotsqn\right\}
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
[2])
.
Rayleigh writes the Lagrangian
as kinetic energy
minus potential energy
, which yields Rayleigh's Eqn. (26) from 1873.
.
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 e.g.
[3] [4] [5] Notes and References
- Rayleigh . Lord . Some general theorems relating to vibrations. . Proc. London Math. Soc. . s1-4 . 1873 . 10.1112/plms/s1-4.1.357 . 357–368. free.
- Book: Goldstein, Herbert . Herbert Goldstein
. Herbert Goldstein . 1980 . . 2nd . Addison-Wesley . Reading, MA . 0-201-02918-9 . 24.
- Moreau . Jean Jacques . Jean Jacques Moreau . Fonctions de résistance et fonctions de dissipation . Travaux du Séminaire d'Analyse Convexe, Montpellier (Exposé no. 6) . 1971 . (See page 6.3 for "fonction de resistance").
- Book: Lebon . Georgy . Jou . David . Casas-Vàzquez . Jos\'e . 2008 . Understanding Non-equilibrium Thermodynamics . Springer-Verlag . (See Chapter 10.2 for dissipation potentials).
- Mielke . Alexander . An introduction to the analysis of gradient systems . 2023 . (See Definition 3.1 on page 25 for dissipation potentials) . math-ph . 2306.05026 .