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  • 1.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Fysik, elektroteknik och matematik. Linköpings universitet, Tekniska fakulteten.
    Blaszak, Maciej
    Adam Mickiewicz Univ, Poland.
    Miura maps for Stackel systems2023Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 64, nr 12, artikel-id 122903Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce the concept of Miura maps between parameter-dependent algebraic curves of hyper-elliptic type. These Miura maps induce Miura maps between Stackel systems defined (on the extended phase space) by the considered algebraic curves. This construction yields a new way of generating multi-Hamiltonian representations for Stackel systems.

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  • 2.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz Univ, Poland.
    Szablikowski, Blazej m.
    Adam Mickiewicz Univ, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Fysik, elektroteknik och matematik. Linköpings universitet, Tekniska fakulteten.
    Stäckel representations of stationary Kdv systems2023Ingår i: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 92, nr 3, s. 323-346Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article we study Stackel representations of stationary KdV systems. Using Lax formalism we prove that these systems have two different representations as separable Stackel systems of Benenti type, related with different foliations of the stationary manifold. We do it by constructing an explicit transformation between the jet coordinates of stationary KdV systems and separation variables of the corresponding Benenti systems for arbitrary number of degrees of freedom. Moreover, on the stationary manifold, we present the explicit form of Miura map between both representations of stationary KdV systems, which also yields their bi-Hamiltonian formulation.

    Publikationen är tillgänglig i fulltext från 2025-12-30 15:58
  • 3.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz Univ, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Fysik, elektroteknik och matematik. Linköpings universitet, Tekniska fakulteten.
    Domanski, Ziemowit
    Poznan Univ Tech, Poland.
    Systematic construction of nonautonomous Hamiltonian equations of Painleve type. I. Frobenius integrability2022Ingår i: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 148, nr 3, s. 1208-1250Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This article is the first one in a suite of three articles exploring connections between dynamical systems of Stackel type and of Painleve type. In this article, we present a deformation of autonomous Stackel-type systems to nonautonomous Frobenius integrable systems. First, we consider quasi-Stackel systems with quadratic in momenta Hamiltonians containing separable potentials with time-dependent coefficients, and then, we present a procedure of deforming these equations to nonautonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stackel systems with so-called magnetic separable potentials to nonautonomous Frobenius integrable systems. We also provide a complete list of all two- and three-dimensional Frobenius integrable systems, both with ordinary and with magnetic potentials, which originate in our construction. Further, we prove the equivalence between both classes of systems. Finally, we show how Painleve equations PI-PIV can be derived from our scheme.

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  • 4.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz Univ, Poland; Adam Mickiewicz Univ, Poland.
    Domanski, Ziemowit
    Poznan Univ Tech, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Fysik, elektroteknik och matematik. Linköpings universitet, Tekniska fakulteten. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Systematic construction of nonautonomous Hamiltonian equations of Painleve-type. II. Isomonodromic Lax representation2022Ingår i: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 149, nr 2, s. 364-415Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This is the second article in a suite of articles investigating relations between Stackel-type systems and Painleve-type systems. In this paper, we construct isomonodromic Lax representations for Painleve-type systems found in the previous paper by Frobenius integrable deformations of Stackel-type systems. We first construct isomonodromic Lax representations for Painleve-type systems in the so-called magnetic representation and then, using a multitime-dependent canonical transformation, we also construct isomonodromic Lax representations for Painleve-type systems in the nonmagnetic representation. Thus, we prove that the Frobenius integrable systems constructed in Part I are indeed of Painleve-type. We also present isomonodromic Lax representations for all one-, two-, and three-dimensional Painleve-type systems originating in our scheme. Based on these results we propose complete hierarchies of that follow from our construction.

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  • 5.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz Univ, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Fysik, elektroteknik och matematik. Linköpings universitet, Tekniska fakulteten. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Systematic construction of non-autonomous Hamiltonian equations of Painleve-type. III. Quantization2022Ingår i: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 149, nr 2, s. 416-440Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    This is the third article in our series of articles exploring connections between dynamical systems of Stackel-type and of Painleve-type. In this article, we present a method of deforming of minimally quantized quasi-Stackel Hamiltonians, considered in Part I to self-adjoint operators satisfying the quantum Frobenius condition, thus guaranteeing that the corresponding Schrodinger equations possess common, multitime solutions. As in the classical case, we obtain here both magnetic and nonmagnetic families of systems. We also show the existence of multitime-dependent quantum canonical maps between both classes of quantum systems.

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  • 6.
    Maniraguha, Jean de Dieu
    et al.
    University of Rwanda, Kigali, Rwanda.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Fysik, elektroteknik och matematik. Linköpings universitet, Tekniska fakulteten.
    Kurujyibwami, Célestin
    University of Rwanda, Kigali, Rwanda.
    Transforming Stäckel Hamiltonians of Benenti type to polynomial form2022Ingår i: Advances in Theoretical and Mathematical Physics, ISSN 1095-0761, E-ISSN 1095-0753, Vol. 26, nr 3, s. 711-734Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we discuss two canonical transformations that turn Stäckel separable Hamiltonians of Benenti type into polynomial form: transformation to Viète coordinates and transformation to Newton coordinates. Transformation to Newton coordinates has been applied to these systems only very recently and in this paper we present a new proof that this transformation indeed leads to polynomial form of Stäckel Hamiltonians of Benenti type. Moreover we present all geometric ingredients of these Hamiltonians in both Viète and Newton coordinates.

  • 7.
    Blaszak, Maciej
    et al.
    Department of Mathematical Physics and Computer Modeling, A. Mickiewicz University, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Fysik, elektroteknik och matematik. Linköpings universitet, Tekniska fakulteten.
    Sergyeyev, Artur
    Mathematical Institute, Silesian University in Opava, Czech Republic.
    Deforming Lie algebras to Frobenius integrable nonautonomous Hamiltonian systems2021Ingår i: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 87, nr 2, s. 249-263Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Motivated by the theory of Painlevé equations and associated hierarchies, we study nonautonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed into a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stäckel systems from [14].

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  • 8.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz Univ, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Fysik, elektroteknik och matematik. Linköpings universitet, Tekniska fakulteten.
    Stackel transform of Lax equations2020Ingår i: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 145, nr 2, s. 179-196Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We construct a map between Lax equations for pairs of Liouville integrable Hamiltonian systems related by a multiparameter Stackel transform. Using this map, we construct Lax representation for a wide class of separable systems by applying the multiparameter Stackel transform to Lax equations of suitably chosen systems from a seed class. For a given separable system, we obtain in this way a set of nonequivalent Lax equations parameterized by an arbitrary function of the spectral parameter, as it is in the case of a related seed system.

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  • 9.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz University, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska fakulteten.
    Classical and Quantum Superintegrability of Stackel Systems2017Ingår i: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 13, artikel-id 008Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we discuss maximal superintegrability of both classical and quantum Stackel systems. We prove a sufficient condition for a flat or constant curvature Stackel system to be maximally superintegrable. Further, we prove a sufficient condition for a Stackel transform to preserve maximal superintegrability and we apply this condition to our class of Stackel systems, which yields new maximally superintegrable systems as conformal deformations of the original systems. Further, we demonstrate how to perform the procedure of minimal quantization to considered systems in order to produce quantum superintegrable and quantum separable systems.

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  • 10.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska fakulteten.
    Blaszak, Maciej
    Adam Mickiewicz University, Poland.
    Non-Homogeneous Hydrodynamic Systems and Quasi-Stackel Hamiltonians2017Ingår i: SIGMA. Symmetry, Integrability and Geometry, ISSN 1815-0659, E-ISSN 1815-0659, Vol. 13, artikel-id 077Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we present a novel construction of non-homogeneous hydrodynamic equations from what we call quasi-Stackel systems, that is non-commutatively integrable systems constructed from appropriate maximally superintegrable Stackel systems. We describe the relations between Poisson algebras generated by quasi-Stackel Hamiltonians and the corresponding Lie algebras of vector fields of non-homogeneous hydrodynamic systems. We also apply Stackel transform to obtain new non-homogeneous equations of considered type.

  • 11.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz University, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska fakulteten.
    Domanski, Ziemowit
    Polish Academic Science, Poland.
    Separable quantizations of Stackel systems2016Ingår i: Annals of Physics, ISSN 0003-4916, E-ISSN 1096-035X, Vol. 371, s. 460-477Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article we prove that many Hamiltonian systems that cannot be separably quantized in the classical approach of Robertson and Eisenhart can be separably quantized if we extend the class of admissible quantizations through a suitable choice of Riemann space adapted to the Poisson geometry of the system. Actually, in this article we prove that for every quadratic in momenta Stackel system (defined on 2n dimensional Poisson manifold) for which Stackel matrix consists of monomials in position coordinates there exist infinitely many quantizations - parametrized by n arbitrary functions - that turn this system into a quantum separable Stackel system. (C) 2016 Elsevier Inc. All rights reserved.

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  • 12.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska fakulteten.
    Blaszak, Maciej
    Adam Mickiewicz University, Poland.
    Flat coordinates of flat Stackel systems2015Ingår i: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 268, s. 706-716Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this article we explicitly construct Stackel separable systems in separation coordinates with the help of separation curve as introduced by Sklyanin. Further, we construct explicit transformation between separable and flat coordinates for flat Stackel systems. We also exploit the geometric structure of these systems in the obtained flat coordinates. These coordinates generalize the wellknown generalized elliptic and generalized parabolic coordinates introduced by Jacobi. (C) 2015 Elsevier Inc. All rights reserved.

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  • 13.
    Blaszak, M
    et al.
    External - unknown .
    Marciniak, Krzysztof
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska högskolan.
    Invertible Coupled KdV and Coupled Harry Dym Hierarchies2013Ingår i: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 131, nr 3, s. 211-228Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper, we discuss the conditions under which the coupled KdV and coupled Harry Dym hierarchies possess inverse (negative) parts. We further investigate the structure of nonlocal parts of tensor invariants of these hierarchies, in particular, the nonlocal terms of vector fields, conserved one-forms, recursion operators, Poisson and symplectic operators. We show that the invertible coupled KdV hierarchies possess Poisson structures that are at most weakly nonlocal while coupled Harry Dym hierarchies have Poisson structures with nonlocalities of the third order.

  • 14.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska högskolan.
    Blaszak, Maciej
    Department of Physics, Adam Mickiewicz University, Poznan, Poland.
    On Reciprocal Equivalence of Stäckel Systems2012Ingår i: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 129, nr 1, s. 26-50Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we ivestigate Stäckel transforms between different classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.

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  • 15.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, Maciej
    Adam Mickiewicz University Poznan.
    Construction of coupled Harry Dym hierarchy and its solutions from Stackel systems2010Ingår i: NONLINEAR ANALYSIS-THEORY METHODS and APPLICATIONS, ISSN 0362-546X, Vol. 73, nr 9, s. 3004-3017Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical Stackel separable systems. Both nonlocal and purely differential parts of hierarchies are obtained. We also construct various classes of solutions of cHD hierarchy from solutions of corresponding Stackel systems.

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  • 16.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, Maciej
    Insitution för fysik Adam Mickiewicz universitet, Poznan, Poland.
    Non-Hamiltonian systems separable by Hamilton-Jacobi method2008Ingår i: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 58, nr 5, s. 557-575Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that with every separable classical Stäckel system of Benenti type on a Riemannian space one can associate, by a proper deformation of the metric tensor, a multi-parameter family of non-Hamiltonian systems on the same space, sharing the same trajectories and related to the seed system by appropriate reciprocal transformations. These systems are known as bi-cofactor systems and are integrable in quadratures as the seed Hamiltonian system is. We show that with each class of bi-cofactor systems a pair of separation curves can be related. We also investigate the conditions under which a given flat bi-cofactor system can be deformed to a family of geodesically equivalent flat bi-cofactor systems. © 2007 Elsevier Ltd. All rights reserved.

  • 17.
    Blaszak, Maciej
    et al.
    Adam Mickiewicz Univ Poznan, Dept Phys, PL-61614 Poznan, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Stackel systems generating coupled KdV hierarchies and their finite-gap and rational solutions2008Ingår i: JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, ISSN 1751-8113, Vol. 41, nr 48Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show how to generate coupled KdV hierarchies from Stackel separable systems of Benenti type. We further show that the solutions of these Stackel systems generate a large class of finite-gap and rational solutions for cKdV hierarchies. Most of these solutions are new.

  • 18.
    Marciniak, Krzysztof
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Geodesically equivalent flat bi-cofactor systems2007Ingår i: Symmetry and Perturbation Theory SPT,2007, Proceedings of the International Conference on SPT 2007: World Scientific , 2007, s. 261-Konferensbidrag (Refereegranskat)
    Abstract [en]

    We find a sufficient condition for a J-tensor J to generate from any given flat bicofactor system a multiparameter family of geodesically equivalent flat bi-cofactor systems 

  • 19.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Rauch, Stefan
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Separable systems of coordinates for triangular Newton equations q¨i = Mi(q1,..., qi)2007Ingår i: Studies in applied mathematics (Cambridge), ISSN 0022-2526, E-ISSN 1467-9590, Vol. 118, nr 1, s. 45-84Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Triangular form of Newton equations is a strong property. Together with the existence of a single quadratic with respect to velocities integral of motion, it usally implies existence of further n - 1 integrals that are also quadratic. These integrals make the triangular system separable in new type of coordinates. The separation coordinates are built of quadric surfaces that are nonorthogonal and noconfocal and can intersect along lower dimensional singular manifolds. We present here separability theory for n-dimensional triangular systems and analyze the structure of separation coordinates in two and three dimensions. © 2007 by the Massachusetts Institute of Technology.

  • 20.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, Maciej
    Institution för fysik Adam Mickiewicz universitet, Poznan, Polen.
    From Stäckel systems to integrable hierarchies of PDE's: Beneti class of separation relations.2006Ingår i: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 47, s. 0329041-03290426Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We propose a general scheme of constructing of soliton hierarchies from finite dimensional Stäckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e., certain Stäckel systems with quadratic in momenta integrals of motion.

  • 21.
    Rauch, Stefan
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Matematiska institutionen, Tillämpad matematik.
    Marciniak, Krzysztof
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Separation of variables for differential equations2006Ingår i: Encyclopedia of Mathematical Physics / [ed] Jean-Pierre Françoise, Gregory L. Naber, Tsou Sheung Tsun, Oxford: Elsevier , 2006, 1, s. 526-535Kapitel i bok, del av antologi (Övrigt vetenskapligt)
    Abstract [en]

      The Encyclopedia of Mathematical Physics provides a complete resource for researchers, students and lecturers with an interest in mathematical physics. It enables readers to access basic information on topics peripheral to their own areas, to provide a repository of the core information in the area that can be used to refresh the researcher's own memory banks, and aid teachers in directing students to entries relevant to their course-work. The Encyclopedia does contain information that has been distilled, organised and presented as a complete reference tool to the user and a landmark to the body of knowledge that has accumulated in this domain. It also is a stimulus for new researchers working in mathematical physics or in areas using the methods originating from work in mathematical physics by providing them with focused high quality background information.Editorial Board: Jean-Pierre Françoise, Université Pierre et Marie Curie, Paris, France Gregory L. Naber, Drexel University, Philadelphia, PA, USA Tsou Sheung Tsun, University of Oxford, UKAlso available online via ScienceDirect (2006) - featuring extensive browsing, searching, and internal cross-referencing between articles in the work, plus dynamic linking to journal articles and abstract databases, making navigation flexible and easy. For more information, pricing options and availability visit www.info.sciencedirect.com. * First comprehensive interdisciplinary coverage * Mathematical Physics explained to stimulate new developments and foster new applications of its methods to other fields * Written by an international group of experts * Contains several undergraduate-level introductory articles to facilitate acquisition of new expertise * Thematic index and extensive cross-referencing to provide easy access and quick search functionality * Also available online with active linking.

  • 22. Blaszak, M
    et al.
    Marciniak, Krzysztof
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    From separation relations to soliton hierarchies2005Ingår i: Czechoslovak Journal of Physics, ISSN 0011-4626, E-ISSN 1572-9486, Vol. 55, nr 11, s. 1343-1350Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We propose a general scheme of constructing soliton hierarchies from finite-dimensional Stackel systems and related separation relations. In particular, we concentrate on Benenti class of separable systems. © Institute of Physics, Academy of Sciences of Czech Republic 2005.

  • 23.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, Maciej
    Institute of Physics A. Mickiewicz University.
    From Stäckel systems to soliton hierarchies: Benenti class of separation relations2005Rapport (Övrigt vetenskapligt)
    Abstract [en]

       

  • 24.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, M.
    Blaszak, M., Institute of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznán, Poland.
    Geometric reduction of Hamiltonian systems2005Ingår i: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, Vol. 55, nr 3, s. 325-339Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Given a foliation S of a manifold M, a distribution Z in M transversal to S and a Poisson bivector ? on M, we present a geometric method of reducing this operator on the foliation S along the distribution Z. It encompasses the classical ideas of Dirac (Dirac reduction) and more modern theory of J. Marsden and T. Ratiu, but our method leads to formulae that allow for an explicit calculation of the reduced Poisson bracket. Moreover, we analyse the reduction of Hamiltonian systems corresponding to the bivector ?.

  • 25.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, Maciej
    Institute of Physics A. Mickiewicz University.
    Geometric reduction of Poisson operators2005Ingår i: Proceedings of the SPT 2004 Symmetry and Perturbation Theory Conference,2004, Singapore: World Scientific Ltd , 2005, s. 193-Konferensbidrag (Refereegranskat)
    Abstract [en]

       

  • 26.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Rauch-Wojciechowski, S
    Separable systems of coordinates for triangular Newton equations2005Rapport (Övrigt vetenskapligt)
  • 27.
    Blaszak, Maciej
    et al.
    Institute of Physics, A. Mickiewicz University.
    Marciniak, Krzysztof
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Dirac reduction of dual poisson-presymplectic pairs2004Ingår i: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 37, nr 19, s. 5173-5187Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    A new notion of a dual Poisson-presymplectic pair is introduced and its properties are examined. The procedure of Dirac reduction of Poisson operators onto submanifolds proposed by Dirac is in this paper embedded in a geometric procedure of reduction of dual Poisson-presymplectic pairs. The method presented generalizes those introduced by J Marsden and T Ratiu for reductions of Poisson manifolds. Two examples are given.

  • 28.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, Maciej
    A. Mickiewicz University, Poland.
    Geometric reduction of Hamiltonian systems2004Rapport (Övrigt vetenskapligt)
  • 29.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, M
    Dirac reduction revisited2003Ingår i: Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, E-ISSN 1776-0852, Vol. 10, nr 4, s. 451-463Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The procedure of Dirac reduction of Poisson operators on submanifolds is discussed within a particularly useful special realization of the general Marsden-Ratiu reduction procedure. The Dirac classification of constraints on 'first-class' constraints and 'second-class' constraints is reexamined.

  • 30.
    Blaszak, M.
    et al.
    Blaszak, M., Institute of Physics, A Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland.
    Marciniak, Krzysztof
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Separability preserving Dirac reductions of Poisson pencils on Riemannian manifolds2003Ingår i: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 36, nr 5, s. 1337-1356Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Dirac deformation of Poisson operators of arbitrary rank is considered. The question when Dirac reduction does not destroy linear Poisson pencils is studied. A class of separability preserving Dirac reductions in the corresponding quasi-bi-Hamiltonian systems of Benenti type is discussed. Two examples of such reductions are given.

  • 31.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Blaszak, M.
    Blaszak, M., Department of Physics, A Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland.
    Separation of variables in quasi-potential systems of bi-cofactor form2002Ingår i: Journal of Physics A: Mathematical and General, ISSN 0305-4470, E-ISSN 1361-6447, Vol. 35, nr 12, s. 2947-2964Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We perform variable separation in the quasi-potential systems of equations of the form q¨ = -A-1?k = -à -1?k~, where A and à are Killing tensors, by embedding these systems into a bi-Hamiltonian chain and by calculating the corresponding Darboux-Nijenhuis coordinates on the symplectic leaves of one of the Hamiltonian structures of the system. We also present examples of the corresponding separation coordinates in two and three dimensions.

  • 32.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Tekniska högskolan. Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem.
    Rauch-Wojciechowski, S.
    Integrable perturbations of the harmonic oscillator and Poisson pencils2001Ingår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 17, nr 2, s. 191-209Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Integrable perturbations of the two-dimensional harmonic oscillator are studied with the use of the recently developed theory of quasi-Lagrangian equations (equations of the form q¨ = A-1(q)?k(q) where A(q) is a Killing matrix) and with the use of Poisson pencils. A quite general class of integrable perturbations depending on an arbitrary solution of a certain second-order linear PDE is found in the case of harmonic oscillator with equal frequencies. For the case of nonequal frequencies all quadratic perturbations admitting two integrals of motion which are quadratic in velocities are found. A non-potential generalization of the Korteveg-de Vries integrable case of the Hénon-Heiles system is obtained. In the case when the perturbation is of a driven type (i.e. when one of the equations is autonomous) a method of solution of these systems by separation of variables and quadratures is presented.

  • 33.
    Marciniak, Krzysztof
    et al.
    Linköpings universitet, Institutionen för teknik och naturvetenskap, Kommunikations- och transportsystem. Linköpings universitet, Tekniska högskolan.
    Rauch-Wojciechowski, Stefan
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    On integrable perturbations of harmonic oscillator2001Ingår i: Reports on mathematical physics, ISSN 0034-4877, E-ISSN 1879-0674, ISSN 0034-4877, Vol. 48, s. 139-147Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Integrable perturbations of the two-dimensional harmonic oscillator are studied with the use of the recently developed theory of quasi-Lagrangian equations. For the case of nonequal frequencies all quadratic perturbations admitting two integrals of motion which are quadratic in velocities are found. A non-potential generalization of the KdV integrable case of the Hénon—Heiles system is obtained.

  • 34.
    Marciniak, Krzysztof
    Linköpings universitet, Matematiska institutionen, Tillämpad matematik. Linköpings universitet, Tekniska högskolan.
    Soliton Methods in the Theory of Integrable Mechanical Systems1998Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This dissertation is devoted to the study of integrable sets of mechanical equations with the use of soliton techniques. A Lax representation of the coupled Kortevegde Vries (cKdV) hierarchy with sow-ces is derived. It is shown that all stationary and restricted flows of the cKdV and the Harry Dym (HD) hierarchies admit a parametrization in the form of Newton equations. New infinite families of Newton equations are found and their integrable structures (such as Lax representation and bi-Hamiltonian formulation) are derived from the underlying soliton hierarchy. Next, an application of the r-matrix formalism to the theory of lattice integrable systems is presented. New families of lattice integrable systems are derived and several Miuralike maps between them are established. One of the offsprings of the obtained results is a completely new theory - theory of quasi-Lagrangian Newton (qLN) equations with two quadratic integrals of motion which contains as its special realization the classical separability theory for natural two-dimensional Hamiltonian systems. The theory of qLN systems goes however far beyond the separability theory - all qLN equations with two quadratic integrals of motion are characterized by a second order linear partial differential equation with polynomial coefficients of third order which generalizes the Bertrand-Darboux equation describing separable potentials in the plane. Hamiltonian formulation of the obtained qLN systems is presented and their complete integrability is proved. Finally, two families of Poisson structures are investigated. It is shown how these families naturally lead to separable systems or to quasi-Lagrangian Newton systems.

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