Showing papers on "Symplectic manifold published in 1981"
TL;DR: In this paper, it was shown that compatible symplectic structures lead in a natural way to hereditary symmetries, and that a hereditary symmetry is an operator-valued function which immediately yields a hierarchy of evolution equations, each having infinitely many commuting symmetry all generated by this hereditary symmetry.
1,651 citations
TL;DR: In this article, a Poisson structure for the Yang-Mills-Vlasov equations was derived by using general methods of symplectic geometry and the main ingredients of the construction were the symplectic structure on the co-adjoint orbits for the group of canonical transformations.
367 citations
TL;DR: In this article, it was shown that the restricted three-body problems are locally like a reversible system, where the fixed set of ƒ is a Lagrangian submanifold L ⊂ M, and there exist cotangent bundle coordinates in a neighborhood of L in M such that ǫ in these coordinates maps a covector into its negative.
40 citations
23 citations
TL;DR: In this paper, a Hamiltonian formulation for classical field theories is presented, in which the Hamilton equation is written by means of the energy-momentum function E and the symplectic 2−form Ω.
Abstract: We present a Hamiltonian formulation for classical field theories. In a general case we write the Hamilton equation by means of the energy–momentum function E and the symplectic 2‐form Ω. We investigate thoroughly an important example, the gravitational field coupled to a matter tensor field. It will be shown that the energy‐momentum differential 3‐form yields a generalization of the Komar energy formula. We prove that the energy–momentum function E, the symplectic 2‐form Ω, the Hamilton equation, and four constraint equations for initial values of canonical variables give rise to the system which is equivalent to the Euler–Lagrange variational equations. We also discuss relations between the Hamilton equation of evolution, the degeneracy of the symplectic 2‐form Ω, and the action of the diffeomorphism group of spacetime in the set of solutions.
15 citations
TL;DR: In this paper, the relation between the local Chevalley cohomologies related to the adjoint representation of the Poisson Lie algebra of a symplectic manifold and the Lie algebras of all globally Hamiltonian vector fields of the manifold is discussed.
Abstract: We describe the relations between the local Chevalley cohomologies related to the adjoint representation of the Poisson Lie algebra of a symplectic manifold and the Lie algebras of all symplectic or globally Hamiltonian vector fields of the manifold. The proofs are based on the computation of the cohomology of the complex (E, ∂), where E is the space of multilinear local maps from a vector bundle of a manifold M into the space of forms on M and ∂L=d ∘ L.
14 citations
13 citations
Journal Article•
6 citations
5 citations
TL;DR: In this article, it was shown that if the Euler class vanishes, then the manifold M admits a nice polarization with dim polarization, for q>2n−k, and for q > 2n −k, then M vanishes.
Abstract: In this note, we prove and discuss the following theorem: if the symplectic or almost symplectic manifoldM admits a nice polarizationF with dim
$$(F \cap \bar F) = k
e 0$$
, we must have Chern
q
M=0 and Pont
q
M=0, forq>2n−k, and, in particular, the Euler class ofM vanishes.
3 citations
01 Mar 1981
TL;DR: In this article, the authors consider a group G on a symplectic manifold P which admits a momentum mapping and prove that if G has a coadjoint equivariant momentum mapping, and if HI(%,: R) = H2(52: R)) 0, then the symplectic action of G had a co-adjoint equilibrium momentum mapping where 1 and 52 are the Lie algebras of G1 and G2 respectively.
Abstract: Ansrmacr. We consider a symplectic action of a group G on a symplectic manifold P, which admits a momentum mapping. Assume that G is a semidirect product of G1 by G2. We prove that if the symplectic action of GI has a coadjoint equivariant momentum mapping, and if HI(%,: R) = H2(52: R) 0, then the symplectic action of G has a coadjoint equivariant momentum mapping, where 1 and 52 are the Lie algebras of G1 and G2 respectively.