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.
Resulting equations for pure wave motion.
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 formula_1 with respect to the invariant wave phase formula_2 then gives rise to a conservation law:
This equation states the "conservation of wave action" – a generalization of the concept of an adiabatic invariant to continuum mechanics – with
being the wave action formula_5 and wave action flux formula_6 respectively. Further formula_7 and formula_8 denote space and time respectively, while formula_9 is the gradient operator. The angular frequency formula_10 and wavenumber formula_11 are defined as
and both are assumed to be slowly varying. Due to this definition, formula_10 and formula_11 have to satisfy the consistency relations:
The first consistency equation is known as the conservation of wave crests, and the second states that the wavenumber field formula_11 is irrotational (i.e. has zero curl).
Method.
The averaged Lagrangian approach applies to wave motion – possibly superposed on a mean motion – that can be described in a Lagrangian formulation. Using an ansatz on the form of the wave part of the motion, the Lagrangian is phase averaged. Since the Lagrangian is associated with the kinetic energy and potential energy of the motion, the oscillations contribute to the Lagrangian, although the mean value of the wave's oscillatory excursion is zero (or very small).
The resulting averaged Lagrangian contains wave characteristics like the wavenumber, angular frequency and amplitude (or equivalently the wave's energy density or wave action). But the wave phase itself is absent due to the phase averaging. Consequently, through Noether's theorem, there is a conservation law called the conservation of wave action.
Originally the averaged Lagrangian method was developed by Whitham for slowly-varying dispersive wave trains. Several extensions have been made, e.g. to interacting wave components, Hamiltonian mechanics, higher-order modulational effects, dissipation effects.
Variational formulation.
The averaged Lagrangian method requires the existence of a Lagrangian describing the wave motion. For instance for a field formula_15, described by a Lagrangian density formula_16 the principle of stationary action is:
with formula_9 the gradient operator and formula_19 the time derivative operator. This action principle results in the Euler–Lagrange equation:
which is the second-order partial differential equation describing the dynamics of formula_21 Higher-order partial differential equations require the inclusion of higher than first-order derivatives in the Lagrangian.
For example, consider a non-dimensional and non-linear Klein–Gordon equation in one space dimension formula_22:
This Euler–Lagrange equation emerges from the Lagrangian density:
The small-amplitude approximation for the Sine–Gordon equation corresponds with the value formula_23 For formula_24 the system is linear and the classical one-dimensional Klein–Gordon equation is obtained.
Slowly-varying waves.
Slowly-varying linear waves.
Whitham developed several approaches to obtain an averaged Lagrangian method. The simplest one is for slowly-varying linear wavetrains, which method will be applied here.
The slowly-varying wavetrain – without mean motion – in a linear dispersive system is described as:
where formula_2 is the real-valued wave phase, formula_29 denotes the absolute value of the complex-valued amplitude formula_30 while formula_31 is its argument and formula_32 denotes its real part. The real-valued amplitude and phase shift are denoted by formula_33 and formula_34 respectively.
Now, "by definition", the angular frequency formula_35 and wavenumber vector formula_36 are expressed as the time derivative and gradient of the wave phase formula_2 as:
As a consequence, formula_10 and formula_11 have to satisfy the consistency relations:
These two consistency relations denote the "conservation of wave crests", and the irrotationality of the wavenumber field.
Because of the assumption of slow variations in the wave train – as well as in a possible inhomogeneous medium and mean motion – the quantities formula_44 formula_45 formula_46 formula_36 and formula_34 all vary slowly in space formula_7 and time formula_8 – but the wave phase formula_51 itself does not vary slowly. Consequently, derivatives of formula_45 formula_46 formula_36 and formula_34 are neglected in the determination of the derivatives of formula_15 for use in the averaged Lagrangian:
Next these assumptions on formula_15 and its derivatives are applied to the Lagrangian density formula_60
Slowly-varying non-linear waves.
Several approaches to slowly-varying non-linear wavetrains are possible. One is by the use of Stokes expansions, used by Whitham to analyse slowly-varying Stokes waves. A Stokes expansion of the field formula_15 can be written as:
where the amplitudes formula_45 formula_64 etc. are slowly varying, as are the phases formula_65 formula_66 etc. As for the linear wave case, in lowest order (as far as modulational effects are concerned) derivatives of amplitudes and phases are neglected, except for derivatives formula_35 and formula_36 of the fast phase formula_69
These approximations are to be applied in the Lagrangian density formula_72, and its phase average formula_73
Averaged Lagrangian for slowly-varying waves.
For pure wave motion the Lagrangian formula_74 is expressed in terms of the field formula_15 and its derivatives. In the averaged Lagrangian method, the above-given assumptions on the field formula_15 – and its derivatives – are applied to calculate the Lagrangian. The Lagrangian is thereafter averaged over the wave phase formula_69
As a last step, this averaging result formula_79 can be expressed as the "averaged Lagrangian" density formula_80 – which is a function of the slowly varying parameters formula_46 formula_36 and formula_33 and independent of the wave phase formula_51 itself.
The averaged Lagrangian density formula_1 is now proposed by Whitham to follow the average variational principle:
From the variations of formula_1 follow the dynamical equations for the slowly-varying wave properties.
Continuing on the example of the nonlinear Klein–Gordon equation, see equations and , and applying the above approximations for formula_87 formula_88 and formula_89 (for this 1D example) in the Lagrangian density, the result after averaging over formula_51 is:
where it has been assumed that, in big-O notation, formula_92 and formula_93. Variation of formula_79 with respect to formula_95 leads to formula_96 So the averaged Lagrangian is:
For linear wave motion the averaged Lagrangian is obtained by setting formula_97 equal to zero.
Set of equations emerging from the averaged Lagrangian.
Applying the averaged Lagrangian principle, variation with respect to the wave phase formula_51 leads to the conservation of wave action:
since formula_100 and formula_101 while the wave phase formula_51 does not appear in the averaged Lagrangian density formula_1 due to the phase averaging. 
Defining the wave action as formula_104 and the wave action flux as formula_105 the result is:
The wave action equation is accompanied with the consistency equations for formula_35 and formula_36 which are:
Variation with respect to the amplitude formula_33 leads to the dispersion relation formula_112
Continuing with the nonlinear Klein–Gordon equation, using the average variational principle on equation , the wave action equation becomes by variation with respect to the wave phase formula_69
and the nonlinear dispersion relation follows from variation with respect to the amplitude formula_115
So the wave action is formula_117 and the wave action flux formula_118 The group velocity formula_119 is formula_120
Conservation of wave action.
The averaged Lagrangian is obtained by integration of the Lagrangian over the wave phase. As a result, the averaged Lagrangian only contains the derivatives of the wave phase formula_51 (these derivatives being, by definition, the angular frequency and wavenumber) and does not depend on the wave phase itself. So the solutions will be independent of the choice of the zero level for the wave phase. Consequently – by Noether's theorem – variation of the averaged Lagrangian formula_122 with respect to the wave phase results in a conservation law:
where
with formula_5 the wave action and formula_6 the wave action flux. Further formula_19 denotes the partial derivative with respect to time, and formula_9 is the gradient operator. By definition, the group velocity formula_129 is given by:
Note that in general the energy of the wave motion does not need to be conserved, since there can be an energy exchange with a mean flow. The total energy – the sum of the energies of the wave motion and the mean flow – is conserved (when there is no work by external forces and no energy dissipation).
Conservation of wave action is also found by applying the generalized Lagrangian mean (GLM) method to the equations of the combined flow of waves and mean motion, using Newtonian mechanics instead of a variational approach.
Connection to the dispersion relation.
Pure wave motion by linear models always leads to an averaged Lagrangian density of the form:
Consequently, the variation with respect to amplitude: formula_132 gives
So this turns out to be the dispersion relation for the linear waves, and the averaged Lagrangian for linear waves is always the dispersion function formula_134 times the amplitude squared.
More generally, for weakly nonlinear and slowly modulated waves propagating in one space dimension and including higher-order dispersion effects – not neglecting the time and space derivatives formula_135 and formula_136 of the amplitude formula_137 when taking derivatives, where formula_138 is a small modulation parameter – the averaged Lagrangian density is of the form:
with the slow variables formula_140 and formula_141
References.
Publications by Whitham on the method.
"An overview can be found in the book:"
"Some publications by Whitham on the method are:"