A nonlinear Hamiltonian structure for the Euler equations

Peter J. Olver

Research output: Contribution to journalArticlepeer-review

67 Scopus citations


The Euler equations for inviscid incompressible fluid flow have a Hamiltonian structure in Eulerian coordinates, the Hamiltonian operator, though, depending on the vorticity. Conservation laws arise from two sources. One parameter symmetry groups, which are completely classified, yield the invariance of energy and linear and angular momenta. Degeneracies of the Hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point-wise conservation of vorticity. It is conjectured that no further conservation laws exist, indicating that the Euler equations are not completely integrable, in particular, do not have soliton-like solutions.

Original languageEnglish (US)
Pages (from-to)233-250
Number of pages18
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - Sep 1982


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