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Manifestly invariant Lagrangians for geophysical fluids
Author(s) -
Zadra Ayrton,
Charron Martin
Publication year - 2015
Publication title -
quarterly journal of the royal meteorological society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.744
H-Index - 143
eISSN - 1477-870X
pISSN - 0035-9009
DOI - 10.1002/qj.2617
Subject(s) - covariant transformation , lagrange multiplier , equations of motion , inviscid flow , lagrangian , invariant (physics) , lagrangian mechanics , classical mechanics , mathematics , conservation of mass , conservation law , mathematical analysis , physics , mathematical physics , analytical mechanics , mechanics , mathematical optimization , quantum mechanics , quantum dynamics , quantum
Three manifestly invariant Lagrangians are presented from which the covariant equations of motion for inviscid classical fluids are derived using the least action principle. Invariance and covariance are here defined with respect to synchronous, but otherwise arbitrary, coordinate transformations, i.e. supposing that time intervals are absolute as required by Newtonian mechanics. In the first Lagrangian, the flow is formulated in terms of fluid particles, but conservation of mass and entropy is assumed apriori . In the second Lagrangian, the flow is described by fields, and conservation of mass and entropy is obtained with Lagrange multipliers. The third Lagrangian is also based on a field formulation, but has no Lagrange multipliers and produces all the desired equations of motion, including conservation of mass and entropy. The differences and similarities between these formulations are discussed. Hydrostatic equations are rederived from an asymptotic expansion of the action using the covariant field formulation.