Singular Poisson tensors

Journal Article · Tue Jul 20 04:00:00 EDT 1982 · AIP Conf. Proc.; (United States)

DOI:https://doi.org/10.1063/1.33647· OSTI ID:6864381

Littlejohn, R G

The Hamiltonian structures discovered by Morrison and Greene for various fluid equations were obtained by guessing a Hamiltonian and a suitable Poisson bracket formula, expressed in terms of noncanonical (but physical) coordinates. In general, such a procedure for obtaining a Hamiltonian system does not produce a Hamiltonian phase space in the usual sense (a symplectic manifold), but rather a family of symplectic manifolds. To state the matter in terms of a system with a finite number of degrees of freedom, the family of symplectic manifolds is parametrized by a set of Casimir functions, which are characterized by having vanishing Poisson brackets with all other functions. The number of independent Casimir functions is the corank of the Poisson tensor J/sup ij/, the components of which are the Poisson brackets of the coordinates among themselves. Thus, these Casimir functions exist only when the Poisson tensor is singular.

• Research Organization: University of California, Los Angeles, California 90024
• OSTI ID: 6864381
• Journal Information: AIP Conf. Proc.; (United States), Journal Name: AIP Conf. Proc.; (United States) Vol. 88:1; ISSN APCPC
• Country of Publication: United States
• Language: English

Related Subjects

640410* -- Fluid Physics-- General Fluid Dynamics
75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
CASIMIR OPERATORS
COMMUTATION RELATIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
FLUID MECHANICS
FUNCTIONS
HAMILTONIAN FUNCTION
MATHEMATICAL OPERATORS
MECHANICS
PARTIAL DIFFERENTIAL EQUATIONS
POISSON EQUATION
SINGULARITY
TENSORS
TRANSFORMATIONS