Mostrando recursos 1 - 14 de 14

  1. On Auto and Hetero Bäcklund Transformations for the Hénon-Heiles Systems

    Tsiganov, Andrey; Sozonov, Alexey
    We consider a canonical transformation of parabolic coordinates on the plain associated with integrable Hénon-Heiles systems and suppose that this transformation together with some additional relations may be considered as a counterpart of the auto and hetero Bäcklund transformations.

  2. Deformation Quantization of Kähler Manifolds and Their Twisted Fock Representation

    Sako, Akifumi; Umetsu, Hiroshi
    We introduce the notion of twisted Fock representations of noncommutative Kähler manifolds and give their explicit expressions. The so-called twisted Fock representation is a representation of the Heisenberg like algebra whose states are constructed by acting creation operators on a vacuum state. "Twisted" means that creation operators are not Hermitian conjugate of annihilation operators. In deformation quantization of Kähler manifolds with separation of variables formulated by Karabegov, local complex coordinates and partial derivatives of the Kähler potential with respect to coordinates satisfy the commutation relations between the creation and annihilation operators. Based on these relations, the twisted Fock representation of noncommutative Kähler manifolds is constructed.

  3. On Geodesic Bifurcations

    Ryparová, Lenka; Mikeš, Josef
    In this paper we study fundamental equations of geodesics on surfaces of revolution. We obtain examples of existence of geodesic bifurcation.

  4. Trajectories of the Plate-Ball Problem

    Pulov, Vladimir; Hadzhilazova, Mariana; Mladenov, Ivaïlo
    The plate-ball problem concerns the shortest trajectories traced by a rolling sphere on a horizontal plane between the prescribed initial and final states meaning the positions and orientations of the sphere. Here we present an explicit parametric representation of these trajectories in terms of the Jacobian elliptic functions and elliptic integrals.

  5. Semi-Discrete Constant Mean Curvature Surfaces of Revolution in Minkowski Space

    Müller, Christian; Yasumoto, Masashi
    In this paper we describe semi-discrete isothermic constant mean curvature surfaces of revolution with smooth profile curves in Minkowski three-space. Unlike the case of semi-discrete constant mean curvature surfaces in Euclidean three-space, they might have certain types of singularities in a sense defined by the second author in a previous work. We analyze the singularities of such surfaces.

  6. Geodesic Mappings Onto Riemannian Manifolds and Differentiability

    Hinterleitner, Irena; Mikeš, Josef
    In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto (pseudo-) Riemannian manifolds. We proved that if a manifold with affine (or projective) connection of differentiability class $C^r (r\geq2)$ admits a geodesic mapping onto a \hbox{(pseudo-)} Riemannian manifold of class $C^1$, then this manifold belongs to the differentiability class $C^{r+1}$. From this result follows if an Einstein spaces admits non-trivial geodesic mappings onto (pseudo-) Riemannian manifolds of class $C^1$ then this manifold is an Einstein space, and there exists a common coordinate system in which the components of the metric of these Einstein manifolds are real analytic functions.

  7. Cayley Map and Higher Dimensional Representations of Rotations

    Donchev, Veliko; Mladenova, Clementina; Mladenov, Ivaïlo
    The embeddings of the $\frak{so}(3)$ Lie algebra and the Lie group ${\rm SO}(3)\)$ in higher dimensions is an important construction from both mathematical and physical viewpoint. Here we present results based on a program package for building the generating matrices of the irreducible embeddings of the $\frak{so}(3)$ Lie algebra within $\frak{so}(n)$ in arbitrary dimension $n \geq 3, n \neq 4k+2, k \in \mathbb{N}$ relying on the algorithm developed recently by Campoamor-Strursberg \cite{campoamor}. For the remaining cases $n = 4k+2$ embeddings of $\frak{so}(3)$ into $\frak{so}(n)$ are also constructed. Besides, we investigate the characteristic polynomials of these $\frak{so}(n)$ elements. We show that the Cayley map applied to $\cal{C} \in \frak{so}(n)$ is well...

  8. Local and Non-Local Conservation Laws for Constrained Lagrangians and Applications to Cosmology

    Dimakis, Nikolaos
    We study the existence of conservation laws in constrained systems described by quadratic Lagrangians; the type of which is encountered in mini-superspace cosmology. As is well known, variational symmetries lead to conserved quantities that can be used in the classical and quantum integration of a system. Additionally - and due to the parametrization invariance of such Lagrangians - conditional symmetries defined on phase space can lead to non-local integrals of motion. The latter may be of importance in various cosmological configurations. As an example we present the case of scalar field cosmology with an arbitrary potential.

  9. Rotary Diffeomorphism onto Manifolds with Affine Connection

    Chudá, Hana; Mikeš, Josef; Sochor, Martin
    In this paper we will introduce a newly found knowledge above the existence and the uniqueness of isoperimetric extremals of rotation on two-dimensional (pseudo-) Riemannian manifolds and on surfaces on Euclidean space. We will obtain the fundamental equations of rotary diffeomorphisms from (pseudo-) Riemannian manifolds for twice-differentiable metric tensors onto manifolds with affine connections.

  10. Complex and Real Hypersurfaces of Locally Conformal Kähler Manifolds

    Cherevko, Yevhen; Chepurna, Olena
    We studied complex and real hypersurfaces immersed in locally conformal Kähler manifolds. We have obtained conditions for the immersions, if the manifolds admit existence of such complex hypersurfaces that are orthogonal to both Lee and anti-Lee vector fields. Also we explore real hypersurfaces immersed in LCK-manifolds.

  11. Generalized Euler Angles Viewed as Spherical Coordinates

    Brezov, Danail; Mladenova, Clementina; Mladenov, Ivaïlo
    Here we develop a specific factorization technique for rotations in $\mathbb{R}^3$ into five factors about two or three fixed axes. Although not always providing the most efficient solution, the method allows for avoiding gimbal lock singularities and decouples the dependence on the invariant axis ${\bf n}$ and the angle $\phi$ of the compound rotation. In particular, the solutions in the classical Euler setting are given directly by the angle of rotation $\phi$ and the coordinates of the unit vector $\bf{n}$ without additional calculations. The immediate implementations in rigid body kinematics are also discussed and some generalizations and potential applications in other branches of science and technology...

  12. Geodesic Mappings of Spaces with Affine Connection Onto Symmetric Manifolds

    Berezovskii, Volodymyr; Mikeš, Josef; Peška, Patrik
    In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto symmetric manifolds. We obtain fundamental equations of this problem. At the end of our paper we demonstrate example of studied mappings.

  13. Symmetry, Phases and Quantisation

    Wu, Siye
    We explain rich geometric structures that appear in the quantisation of linear bosonic and fermionic systems. By contrasting with the quantisation of general curved phase spaces, we focus on results that shed light on one of the most basic problems in quantisation: the dependence of the quantum Hilbert space on auxiliary data such as the choice of polarisations that is necessary to define a quantum Hilbert space.

  14. Symmetry, Geometry and Quantization with Hypercomplex Numbers

    Kisil, Vladimir
    These notes describe some links between the group $\(SL_2(\mathbb{R})\)$, the Heisenberg group and hypercomplex numbers - complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this framework. In particular, classical mechanics can be obtained as a theory with $\emph{noncommutative}$ observables and a $\emph{non-zero}$ Planck constant if we replace complex numbers in quantum mechanics by dual numbers. Our consideration is based on induced representations which are build from complex$-/\-$dual$-\-/$double-valued characters. Dynamic equations, rules of additions of probabilities, ladder operators and uncertainty relations are also discussed. Finally, we prove a Calderón--Vaillancourt-type norm estimation for relative convolutions.

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