While I learning the Variational calculations peoples wrote a Lagrangian Density Equation from the Gross-pitaveskii (GP) or NLS equation. From that they found the effective Lagrangian and other evolution equations. My question is how can I form that Lagrangian Density equation from GP equation? Can anyone tell the derivation in clear. [NB: In PRA 81, 043636 (2010) Eq. (10) is Lagrangian Density Equation]
Not sure if I understand your question, but I will give it a shot ...
Of course there is no unique density that is the ancestor of a differential equation, since one can add a total derivative to the density without affecting the Euler equations.
I would start by left-multiplying any linear terms in the NL differential equation in the wavefunction by the complex conjugate of the wavefunction.
For the remaining non-linear terms I would attempt a functional integration over the complex conjugate of the wavefunction. This integration would be the inverse of the Euler-variation of the complex conjugate wavefunction. (It would be easier to write this down mathematically than express in words.)
Not sure if I understand your question, but I will give it a shot ...
Of course there is no unique density that is the ancestor of a differential equation, since one can add a total derivative to the density without affecting the Euler equations.
I would start by left-multiplying any linear terms in the NL differential equation in the wavefunction by the complex conjugate of the wavefunction.
For the remaining non-linear terms I would attempt a functional integration over the complex conjugate of the wavefunction. This integration would be the inverse of the Euler-variation of the complex conjugate wavefunction. (It would be easier to write this down mathematically than express in words.)
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