Rayleigh dissipation function

In physics, the Rayleigh dissipation function, named for Lord Rayleigh, is a function used to handle the effects of velocity-proportional frictional forces in Lagrangian mechanics. It is defined for a system of ${\displaystyle N}$ particles as

${\displaystyle G(v)={\frac {1}{2}}\sum _{i=1}^{N}(k_{x}v_{i,x}^{2}+k_{y}v_{i,y}^{2}+k_{z}v_{i,z}^{2}).}$

The force of friction is negative the velocity gradient of the dissipation function, ${\displaystyle \mathbf {F} _{f}=-\nabla _{\mathbf {v} }G(v)}$. The function is half the rate at which energy is being dissipated by the system through friction.

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