Averaged Lagrangian

In continuum mechanics, Whitham's averaged Lagrangian method – or in short Whitham's method – is used to study the Lagrangian dynamics of slowly-varying wave trains in an inhomogeneous (moving) medium. The method is applicable to both linear and non-linear systems. As a direct consequence of the averaging used in the method, wave action is a conserved property of the wave motion. In contrast, the wave energy is not necessarily conserved, due to the exchange of energy with the mean motion. However the total energy, the sum of the energies in the wave motion and the mean motion, will be conserved for a time-invariant Lagrangian. Further, the averaged Lagrangian has a strong relation to the dispersion relation of the system.

The method is due to Gerald Whitham, who developed it in the 1960s. It is for instance used in the modelling of surface gravity waves on fluid interfaces, and in plasma physics.

In case a Lagrangian formulation of a continuum mechanics system is available, the averaged Lagrangian methodology can be used to find approximations for the average dynamics of wave motion – and (eventually) for the interaction between the wave motion and the mean motion – assuming the envelope dynamics of the carrier waves is slowly varying. Phase averaging of the Lagrangian results in an averaged Lagrangian, which is always independent of the wave phase itself (but depends on slowly varying wave quantities like wave amplitude, frequency and wavenumber). By Noether's theorem, variation of the averaged Lagrangian ${\displaystyle {\mathcal {L}}}$ with respect to the invariant wave phase ${\displaystyle \theta ({\boldsymbol {x}},t)}$ then gives rise to a conservation law:

...
Wikipedia