Lagrangian formulation for perfect fluid equations with the ℓ -conformal Galilei symmetry

Lagrangian formulation for perfect fluid equations which hold invariant under the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mo>ℓ</a:mo></a:math>-conformal Galilei group with half-integer <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo>ℓ</c:mo></c:math> is proposed. It is based on a Clebsch-type parametrization and reproduces Lagrangian description of the Euler fluid equations for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo>ℓ</e:mo><e:mo>=</e:mo><e:mfrac><e:mn>1</e:mn><e:mn>2</e:mn></e:mfrac></e:math>. The transition from the Lagrangian formulation to the Hamiltonian one is analyzed in detail.