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1
Content available remote Invariant properties of the generalized canonical mappings
100%
Janeczko S.
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tom 50
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nr 1
151-161
EN
One of the fundamental objectives of the theory of symplectic singularities is to study the symplectic invariants appearing in various geometrical contexts. In the paper we generalize the symplectic cohomological invariant to the class of generalized canonical mappings. We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and describe the local properties of generic symplectic relations.
2
Content available remote On singularities of Hamiltonian mappings
63%
Fukuda T. , Janeczko S.
Banach Center Publications
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2008
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tom 82
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nr 1
55-69
EN
The notion of an implicit Hamiltonian system-an isotropic mapping H: M → (TM,ω̇) into the tangent bundle endowed with the symplectic structure defined by canonical morphism between tangent and cotangent bundles of M-is studied. The corank one singularities of such systems are classified. Their transversality conditions in the 1-jet space of isotropic mappings are described and the corresponding symplectically invariant algebras of Hamiltonian generating functions are calculated.
3
Content available remote Singularities of implicit differential systems and maximum principle
63%
Janeczko S. , Pelletier F.
Banach Center Publications
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2003
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tom 62
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nr 1
117-132
EN
The integrability condition for the Lagrangian implicit differential systems of (TP,ω̇), introduced in [7], is applied for the specialized control theory systems. The Pontryagin maximum principle was reformulated in the framework of implicit differential systems and the corresponding necessary and sufficient conditions were proved. The beginning of the classification list of normal forms for Lagrangian implicit differential systems according to the symplectic equivalence is provided and the corresponding differential caustics are calculated.
4
Content available remote Symplectic singularities of isotropic mappings
63%
Ishikawa G. , Janeczko S.
Banach Center Publications
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2004
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tom 65
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nr 1
85-106
5
Content available remote Singularities of implicit differential systems and their integrability
63%
Fukuda T. , Janeczko S.
Banach Center Publications
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2004
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tom 65
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nr 1
23-47
6
Content available remote Symplectic classification of parametric complex plane curves
63%
Ishikawa G. , Janeczko S.
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tom 99
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nr 3
263-284
EN
Based on the discovery that the δ-invariant is the symplectic codimension of a parametric plane curve singularity, we classify the simple and uni-modal singularities of parametric plane curves under symplectic equivalence. A new symplectic deformation theory of curve singularities is established, and the corresponding cyclic symplectic moduli spaces are reconstructed as canonical ambient spaces for the diffeomorphism moduli spaces which are no longer Hausdorff spaces.
7
Content available remote Characterization of diffeomorphisms that are symplectomorphisms
63%
Janeczko S. , Jelonek Z.
Fundamenta Mathematicae
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2009
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tom 205
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nr 2
147-160
EN
Let $(X,ω_X)$ and $(Y,ω_Y)$ be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that $Φ*ω_{Y} = cω_{X}$.
8
Content available remote Geometry and representation of the singular symplectic forms
51%
Domitrz W. , Janeczko S. , Pasternak-Winiarski Z.
Banach Center Publications
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2003
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tom 62
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nr 1
57-71
EN
In this paper we show to what extent the closed, singular 2-forms are represented, up to the smooth equivalence, by their restrictions to the corresponding singularity set. In the normalization procedure of the singularity set we find the sufficient conditions for the given closed 2-form to be a pullback of the classical Darboux form. We also find the classification list of simple singularities of the maximal isotropic submanifold-germs in the codimension one Martinet's singular symplectic structures. An example of the exotic singular symplectic structure-germ with no existence of Lagrangian germs is constructed and the singularity theory framework for the pulled back singular symplectic forms is provided.
9
Content available remote Preface, Contents
32%
Janeczko S. , Zajączkowski W. M. , Ziemian B.
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tom 33
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nr 1
1-8
10
Content available remote About the Symposium, Foreword, Contents
32%
Janeczko S. , Zakalyukin V. , Vladimir Arnol'd
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tom 50
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nr 1
1-9
11
Content available remote Preface, Contents
26%
Jakubczyk B. , Janeczko S. , Ziemian B.
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nr 1
1-8
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