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Singular Hamiltonian systems and symplectic capacities

100%

Künzle A. F.

Banach Center Publications

1996

tom 33

nr 1

171-187

The purpose of this paper is to develop the basics of a theory of Hamiltonian systems with non-differentiable Hamilton functions which have become important in symplectic topology. A characteristic differential inclusion is introduced and its equivalence to Hamiltonian inclusions for certain convex Hamiltonians is established. We give two counterexamples showing that basic properties of smooth systems are violated for non-smooth quasiconvex submersions, e.g. even the energy conservation which nevertheless holds for convex submersions. This also implies that the convexity assumption determines, although not symplectically invariant, a limit case for symplectic geometry. Some applications of this theory are reviewed: symplectic capacities for general convex sets, the symplectic product and a product formula for symplectic capacities.

Symplectic Capacities in Manifolds

100%

Künzle A. F.

Banach Center Publications

1997

tom 39

nr 1

77-87

Symplectic capacities coinciding on convex sets in the standard symplectic vector space are extended to any subsets of symplectic manifolds. It is shown that, using embeddings of non-smooth convex sets and a product formula, calculations of some capacities become very simple. Moreover, it is proved that there exist such capacities which are distinct and that there are star-shaped domains diffeomorphic to the ball but not symplectomorphic to any convex set.

Ograniczanie wyników

2 Banach Center Publications

2 Künzle A. F.

1 1997

1 1996

Wyniki wyszukiwania

Singular Hamiltonian systems and symplectic capacities

Symplectic Capacities in Manifolds