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. ∂ t A + ∇ ⋅ B = 0. {displaystyle partial _{t}{mathcal {A}}+{oldsymbol { abla }}cdot {oldsymbol {mathcal {B}}}=0.}     ( 1 ) ω ≡ − ∂ t θ {displaystyle omega equiv -partial _{t} heta }   and   k ≡ + ∇ θ {displaystyle {oldsymbol {k}}equiv +{oldsymbol { abla }} heta }     ( 2 ) ∂ t k + ∇ ω = 0 {displaystyle partial _{t}{oldsymbol {k}}+{oldsymbol { abla }}omega ={oldsymbol {0}}}   and   ∇ × k = 0 . {displaystyle {oldsymbol { abla }} imes {oldsymbol {k}}={oldsymbol {0}}.}     ( 3 ) ∂ t 2 φ − ∂ x 2 φ + φ + σ φ 3 = 0. {displaystyle {partial _{t}^{2}varphi }-{partial _{x}^{2}varphi }+varphi +sigma varphi ^{3}=0.}     ( 4 ) L ( ∂ t φ , ∂ x φ , φ ) = 1 2 ( ∂ t φ ) 2 − 1 2 ( ∂ x φ ) 2 − 1 2 φ 2 − 1 4 σ φ 4 . {displaystyle Lleft(partial _{t}varphi ,partial _{x}varphi ,varphi ight)={frac {1}{2}}left(partial _{t}varphi ight)^{2}-{frac {1}{2}}left(partial _{x}varphi ight)^{2}-{frac {1}{2}}varphi ^{2}-{frac {1}{4}}sigma varphi ^{4}.}     ( 5 ) δ ∬ L ( ω , k , a ) d x d t = 0. {displaystyle delta iint {mathcal {L}}(omega ,{oldsymbol {k}},a);{ ext{d}}{oldsymbol {x}};{ ext{d}}t=0.} L = 1 4 ( ω 2 − k 2 − 1 ) a 2 − 3 32 σ a 4 + O ( a 6 ) . {displaystyle {mathcal {L}}={ frac {1}{4}}(omega ^{2}-k^{2}-1)a^{2}-{ frac {3}{32}}sigma a^{4}+{mathcal {O}}(a^{6}).}     (6) ∂ t A + ∇ ⋅ B = 0 , {displaystyle partial _{t}{mathcal {A}}+{oldsymbol { abla }}cdot {oldsymbol {mathcal {B}}}=0,} An overview can be found in the book: 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 L {displaystyle {mathcal {L}}} with respect to the invariant wave phase θ ( x , t ) {displaystyle heta ({oldsymbol {x}},t)} 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 A {displaystyle {mathcal {A}}} and wave action flux B {displaystyle {oldsymbol {mathcal {B}}}} respectively. Further x {displaystyle {oldsymbol {x}}} and t {displaystyle t} denote space and time respectively, while ∇ {displaystyle {oldsymbol { abla }}} is the gradient operator. The angular frequency ω ( x , t ) {displaystyle omega ({oldsymbol {x}},t)} and wavenumber k ( x , t ) {displaystyle {oldsymbol {k}}({oldsymbol {x}},t)} are defined as and both are assumed to be slowly varying. Due to this definition, ω ( x , t ) {displaystyle omega ({oldsymbol {x}},t)} and k ( x , t ) {displaystyle {oldsymbol {k}}({oldsymbol {x}},t)} 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 k ( x , t ) {displaystyle {oldsymbol {k}}({oldsymbol {x}},t)} is irrotational (i.e. has zero curl). 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.