Averaged Lagrangian

The averaged Lagrangian method requires the existence of a Lagrangian describing the wave motion. For instance for a field ${\displaystyle \varphi ({\boldsymbol {x}},t)}$, described by a Lagrangian density ${\displaystyle L\left(\partial _{t}\varphi ,{\boldsymbol {\nabla }}\varphi ,\varphi \right),}$ the principle of stationary action is:[14]

$${\displaystyle \delta \left(\iiiint \,L\left(\partial _{t}\varphi ({\boldsymbol {x}},t),{\boldsymbol {\nabla }}\varphi ({\boldsymbol {x}},t),\varphi ({\boldsymbol {x}},t)\right)\,{\text{d}}{\boldsymbol {x}}\,{\text{d}}t\right)=0,}$$

with ${\displaystyle {\boldsymbol {\nabla }}}$ the gradient operator and ${\displaystyle \partial _{t}}$ the time derivative operator. This action principle results in the Euler–Lagrange equation:[14]

$${\displaystyle \partial _{t}\left({\frac {\partial L}{\partial \left(\partial _{t}\varphi \right)}}\right)+{\boldsymbol {\nabla }}\cdot \left({\frac {\partial L}{\partial \left({\boldsymbol {\nabla }}\varphi \right)}}\right)-{\frac {\partial L}{\partial \varphi }}=0,}$$

which is the second-order partial differential equation describing the dynamics of ${\displaystyle \varphi .}$ Higher-order partial differential equations require the inclusion of higher than first-order derivatives in the Lagrangian.[14]

For example, consider a non-dimensional and non-linear Klein–Gordon equation in one space dimension ${\displaystyle x}$:[15]

$${\displaystyle {\partial _{t}^{2}\varphi }-{\partial _{x}^{2}\varphi }+\varphi +\sigma \varphi ^{3}=0.}$$

(4)

This Euler–Lagrange equation emerges from the Lagrangian density:[15]

$${\displaystyle L\left(\partial _{t}\varphi ,\partial _{x}\varphi ,\varphi \right)={\tfrac {1}{2}}\left(\partial _{t}\varphi \right)^{2}-{\tfrac {1}{2}}\left(\partial _{x}\varphi \right)^{2}-{\tfrac {1}{2}}\varphi ^{2}-{\tfrac {1}{4}}\sigma \varphi ^{4}.}$$

(5)

The small-amplitude approximation for the Sine–Gordon equation corresponds with the value ${\textstyle \sigma =-{\tfrac {1}{24}}.}$[16] For ${\displaystyle \sigma =0}$ the system is linear and the classical one-dimensional Klein–Gordon equation is obtained.

Whitham developed several approaches to obtain an averaged Lagrangian method.[14][17] The simplest one is for slowly-varying linear wavetrains, which method will be applied here.[14]

The slowly-varying wavetrain –without mean motion– in a linear dispersive system is described as:[18]

$${\displaystyle \varphi \sim \Re \left\{A({\boldsymbol {x}},t)\,e^{i\theta ({\boldsymbol {x}},t)}\right\}=a({\boldsymbol {x}},t)\,\cos \left(\theta ({\boldsymbol {x}},t)+\alpha \right),}$$

with

$${\displaystyle a=\left|A\right|}$$

and

$${\displaystyle \alpha =\arg \left\{A\right\},}$$

where ${\displaystyle \theta ({\boldsymbol {x}},t)}$ is the real-valued wave phase, ${\displaystyle |A|}$ denotes the absolute value of the complex-valued amplitude ${\displaystyle A({\boldsymbol {x}},t),}$ while ${\displaystyle \arg\{A\}}$ is its argument and ${\displaystyle \Re \{A\}}$ denotes its real part. The real-valued amplitude and phase shift are denoted by ${\displaystyle a}$ and ${\displaystyle \alpha }$ respectively.

Now, by definition, the angular frequency ${\displaystyle \omega }$ and wavenumber vector ${\displaystyle {\boldsymbol {k}}}$ are expressed as the time derivative and gradient of the wave phase ${\displaystyle \theta ({\boldsymbol {x}},t)}$ as:[7]

$${\displaystyle \omega \equiv -\partial _{t}\theta }$$

and

$${\displaystyle {\boldsymbol {k}}\equiv +{\boldsymbol {\nabla }}\theta .}$$

As a consequence, ${\displaystyle \omega ({\boldsymbol {x}},t)}$ and ${\displaystyle {\boldsymbol {k}}({\boldsymbol {x}},t)}$ have to satisfy the consistency relations:

$${\displaystyle \partial _{t}{\boldsymbol {k}}+{\boldsymbol {\nabla }}\omega ={\boldsymbol {0}}}$$

and

$${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {k}}={\boldsymbol {0}}.}$$

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 ${\displaystyle A,}$ ${\displaystyle a,}$ ${\displaystyle \omega ,}$ ${\displaystyle {\boldsymbol {k}}}$ and ${\displaystyle \alpha }$ all vary slowly in space ${\displaystyle {\boldsymbol {x}}}$ and time ${\displaystyle t}$ – but the wave phase ${\displaystyle \theta }$ itself does not vary slowly. Consequently, derivatives of ${\displaystyle a,}$ ${\displaystyle \omega ,}$ ${\displaystyle {\boldsymbol {k}}}$ and ${\displaystyle \alpha }$ are neglected in the determination of the derivatives of ${\displaystyle \varphi ({\boldsymbol {x}},t)}$ for use in the averaged Lagrangian:[14]

$${\displaystyle \partial _{t}\varphi \approx +\omega \,a\,\sin(\theta +\alpha )}$$

and

$${\displaystyle {\boldsymbol {\nabla }}\varphi \approx -{\boldsymbol {k}}\,a\,\sin(\theta +\alpha ).}$$

Next these assumptions on ${\displaystyle \varphi ({\boldsymbol {x}},t)}$ and its derivatives are applied to the Lagrangian density ${\displaystyle L\left(\partial _{t}\varphi ,{\boldsymbol {\nabla }}\varphi ,\varphi \right).}$

Several approaches to slowly-varying non-linear wavetrains are possible. One is by the use of Stokes expansions,[19] used by Whitham to analyse slowly-varying Stokes waves.[20] A Stokes expansion of the field ${\displaystyle \varphi ({\boldsymbol {x}},t)}$ can be written as:[19]

$${\displaystyle \varphi =a\,\cos \left(\theta +\alpha \right)+a_{2}\,\cos \left(2\theta +\alpha _{2}\right)+a_{3}\,\cos \left(3\theta +\alpha _{3}\right)+\cdots ,}$$

where the amplitudes ${\displaystyle a,}$ ${\displaystyle a_{2},}$ etc. are slowly varying, as are the phases ${\displaystyle \alpha ,}$ ${\displaystyle \alpha _{2},}$ 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 ${\displaystyle \omega }$ and ${\displaystyle {\boldsymbol {k}}}$ of the fast phase ${\displaystyle \theta }$:

$${\displaystyle \partial _{t}\varphi \approx +\omega a\,\sin \left(\theta +\alpha \right)+2\omega a_{2}\,\sin \left(2\theta +\alpha _{2}\right)+3\omega a_{3}\,\sin \left(3\theta +\alpha _{3}\right)+\cdots ,}$$

and

$${\displaystyle {\boldsymbol {\nabla }}\varphi \approx -{\boldsymbol {k}}a\,\sin \left(\theta +\alpha \right)-2{\boldsymbol {k}}a_{2}\,\sin \left(2\theta +\alpha _{2}\right)-3{\boldsymbol {k}}a_{3}\,\sin \left(3\theta +\alpha _{3}\right)+\cdots .}$$

These approximations are to be applied in the Lagrangian density ${\displaystyle L}$, and its phase average ${\displaystyle {\bar {L}}.}$

For pure wave motion the Lagrangian ${\displaystyle L\left(\partial _{t}\varphi ,{\boldsymbol {\nabla }}\varphi ,\varphi \right)}$ is expressed in terms of the field ${\displaystyle \varphi ({\boldsymbol {x}},t)}$ and its derivatives.[14][17] In the averaged Lagrangian method, the above-given assumptions on the field ${\displaystyle \varphi ({\boldsymbol {x}},t)}$ –and its derivatives– are applied to calculate the Lagrangian. The Lagrangian is thereafter averaged over the wave phase ${\displaystyle \theta }$:[14]

$${\displaystyle {\bar {L}}={\frac {1}{2\pi }}\int _{0}^{2\pi }L\left(\partial _{t}\varphi ,{\boldsymbol {\nabla }}\varphi ,\varphi \right){\text{d}}\theta .}$$

As a last step, this averaging result ${\displaystyle {\bar {L}}}$ can be expressed as the averaged Lagrangian density ${\displaystyle {\mathcal {L}}(\omega ,{\boldsymbol {k}},a)}$ – which is a function of the slowly varying parameters ${\displaystyle \omega ,}$ ${\displaystyle {\boldsymbol {k}}}$ and ${\displaystyle a}$ and independent of the wave phase ${\displaystyle \theta }$ itself.[14]

The averaged Lagrangian density ${\displaystyle {\mathcal {L}}}$ is now proposed by Whitham to follow the average variational principle:[14]

From the variations of ${\displaystyle {\mathcal {L}}}$ follow the dynamical equations for the slowly-varying wave properties.

Continuing on the example of the nonlinear Klein–Gordon equation, see equations 4 and 5, and applying the above approximations for ${\displaystyle \varphi ,}$ ${\displaystyle \partial _{t}\varphi }$ and ${\displaystyle \partial _{x}\varphi }$ (for this 1D example) in the Lagrangian density, the result after averaging over ${\displaystyle \theta }$ is:

$${\displaystyle {\bar {L}}={\tfrac {1}{4}}\left(\omega ^{2}-k^{2}-1\right)a^{2}-{\tfrac {3}{32}}\sigma a^{4}+\left(\omega ^{2}-k^{2}-{\tfrac {1}{4}}\right)a_{2}^{2}+{\mathcal {O}}{\left(a^{6}\right)},}$$

where it has been assumed that, in big-O notation, ${\displaystyle a_{2}={\mathcal {O}}(a^{2})}$ and ${\displaystyle a_{3}={\mathcal {O}}(a^{3})}$. Variation of ${\displaystyle {\bar {L}}}$ with respect to ${\displaystyle a_{2}}$ leads to ${\displaystyle a_{2}=0.}$ So the averaged Lagrangian is:

$${\displaystyle {\mathcal {L}}={\tfrac {1}{4}}\left(\omega ^{2}-k^{2}-1\right)a^{2}-{\tfrac {3}{32}}\sigma a^{4}+{\mathcal {O}}{\left(a^{6}\right)}.}$$

(6)

For linear wave motion the averaged Lagrangian is obtained by setting ${\displaystyle \sigma }$ equal to zero.

Applying the averaged Lagrangian principle, variation with respect to the wave phase ${\displaystyle \theta }$ leads to the conservation of wave action:

$${\displaystyle \partial _{t}\left(+{\frac {\partial {\mathcal {L}}}{\partial \omega }}\right)+{\boldsymbol {\nabla }}\cdot \left(-{\frac {\partial {\mathcal {L}}}{\partial {\boldsymbol {k}}}}\right)=0,}$$

since ${\displaystyle \omega =-\partial _{t}\theta }$ and ${\displaystyle {\boldsymbol {k}}={\boldsymbol {\nabla }}\theta }$ while the wave phase ${\displaystyle \theta }$ does not appear in the averaged Lagrangian density ${\displaystyle {\mathcal {L}}}$ due to the phase averaging. Defining the wave action as ${\displaystyle {\mathcal {A}}\equiv +\partial {\mathcal {L}}/\partial \omega }$ and the wave action flux as ${\displaystyle {\boldsymbol {\mathcal {B}}}\equiv -\partial {\mathcal {L}}/\partial {\boldsymbol {k}}}$ the result is:

$${\displaystyle \partial _{t}{\mathcal {A}}+{\boldsymbol {\nabla }}\cdot {\boldsymbol {\mathcal {B}}}=0.}$$

The wave action equation is accompanied with the consistency equations for ${\displaystyle \omega }$ and ${\displaystyle {\boldsymbol {k}}}$ which are:

$${\displaystyle \partial _{t}{\boldsymbol {k}}+{\boldsymbol {\nabla }}\omega ={\boldsymbol {0}}}$$

and

$${\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {k}}={\boldsymbol {0}}.}$$

Variation with respect to the amplitude ${\displaystyle a}$ leads to the dispersion relation ${\displaystyle \partial {\mathcal {L}}/\partial a=0.}$

Continuing with the nonlinear Klein–Gordon equation, using the average variational principle on equation 6, the wave action equation becomes by variation with respect to the wave phase ${\displaystyle \theta$ :}

$${\displaystyle \partial _{t}\left({\tfrac {1}{2}}\omega a^{2}\right)+\partial _{x}\left({\tfrac {1}{2}}ka^{2}\right)=0,}$$

and the nonlinear dispersion relation follows from variation with respect to the amplitude ${\displaystyle a:}$

$${\displaystyle \omega ^{2}=k^{2}+1+{\tfrac {3}{4}}\sigma a^{2}.}$$

So the wave action is ${\textstyle {\mathcal {A}}={\tfrac {1}{2}}\omega a^{2}}$ and the wave action flux ${\textstyle {\mathcal {B}}={\tfrac {1}{2}}ka^{2}.}$ The group velocity ${\displaystyle v_{g}}$ is ${\displaystyle v_{g}\equiv {\mathcal {B}}/{\mathcal {A}}=k/\omega .}$