arXiv subjects

Joanna Gonera

Publications and source records attributed to Joanna Gonera.

At least 19 recordsLinked to original sources

arXiv2025

Coupling constant metamorphosis, Fermat principle and light propagation in Kerr metric

The geodesics of Kerr's metric are described by the four-dimensional Hamiltonian dynamics integrable in the Arnold--Liouville sense. It can be reduced to two-dimensional one by the use of Fermat's principle. The resulting Hamiltonian is, however, rather complicated. We show how one can apply the coupling constant metamorphosis to simplify the Hamiltonian to the one quadratic in momenta and depending on the initial "energy" as parameter. It describes a simple dynamics of two non-linear oscillators and can be integrated directly or evaluated in the framework of perturbation theory by adopting the elegant Lindstedt--Poincaré algorithm. The idea of coupling constant metamorphosis is also applied to the Myers--Perry metric -- a five dimensional generalization of Kerr's metric. The case of single rotation parameter is considered in some detail.

gr-qc↗

arXiv2025

Fermat Principle and weak deflection angle from Lindstedt-Poincare method

The Fermat principle is advocated to be a convenient tool to analyze the light propagation in a curved space time. It is shown that in the weak deflection regime the light ray trajectories can be systematically described by applying the Lindstedt--Poincaré method of solving perturbatively the nonlinear oscillation equations. The expansion in terms of inverse invariant impact parameter for Schwarzschild, Reissner--Nordström and Kerr (equatorial motion) metrics is described. The corresponding deflection angles are computed to the third order. Only algebraic operations are involved in the derivation; no integrations or Fourier expansion of elliptic functions are necessary. It is argued, that contrary to the naive perturbative expansion, the Lindstedt--Poincaré approach correctly represents the main properties of light propagation in asymptotic regime. At each step it preserves the periodicity of the relevant nonlinear oscillations of the inverse radial coordinate which allows to group the trajectories with the same invariant impact parameter into disjoint sets of ones generated by particular oscillations. Moreover, it allows for partial summation of perturbative expansion leading to the uniformly bounded approximations.

gr-qc↗

arXiv2024

Fermat's Principle in General Relativity via Herglotz Variational Formalism

New form of Fermat's principle for light propagation in arbitrary (i.e. in general neither static nor stationary) gravitational field is proposed. It is based on Herglotz extension of canonical formalism and simple relation between the dynamics described by the Lagrangians homogeneous in velocities and the reduced dynamics on lower-dimensional configuration manifold. This approach is more flexible as it allows to extend immediately the Fermat principle to the case of massive particles and to eliminate any space-time coordinate, not only $x^0$.

gr-qc↗

arXiv2024

The families of Hamiltonians sharing common symmetry structure

We describe a general procedure which allows to construct, starting from a given Hamiltonian, the whole family of new ones sharing the same set of unparameterized trajectories in phase space. The symmetry structure of this family can be completely characterized provided the symmetries of initial Hamiltonian are known. Our approach covers numerous models considered in literature as well as it allows to construct novel ones.. It provides a far reaching generalization of Hietarinta et al. coupling-constant metamorphosis method and another proof of Darboux theorem.

math-ph↗

arXiv2023

On the generalization of classical Zernike system

We generalize the results obtained recently (Nonlinearity \underline{36} (2023), 1143) by providing a very simple proof of the superintegrability of the Hamiltonian $H=\vec{p}\,^{2}+F(\vec{q}\cdot\vec{p})$, $\vec{q}, \vec{p}\in\mathbb{R}^{2}$, for any analytic function $F$. The additional integral of motion is constructed explicitly and shown to reduce to a polynomial in canonical variables for polynomial $F$. The generalization to the case $\vec{q}, \vec{p}\in \mathbb{R}^{n}$ is sketched.

math-ph↗

arXiv2022

Kepler problem and chiral effective dynamics

It is shown that by an appropriate canonical transformation Kepler dynamics can be put in the form which allows to exhibit the structure of the symmetry transformations related to the superintegrability. They appear to fit nicely into general scheme of nonlinear realizations. In new coordiantes the Kepler dynamics results from dimensional reduction of that describing low energy mesons with spontaneously broken chiral symmetry.

math-ph↗

arXiv2022

More on superintegrable models on spaces of constant curvature

A known general class of superintegrable systems on 2D spaces of constant curvature can be defined by potentials separating in (geodesic) polar coordinates. The radial parts of these potentials correspond either to an isotropic harmonic oscillator or a generalised Kepler potential. The angular components, on the contrary, are given implicitly by a transcendental, in general, equation. In the present note, devoted to the previously less studied models with the radial potential of the generalised Kepler type, a new two-parameter family of relevant angular potentials is constructed in terms of elementary functions. For an appropriate choice of parameters, the family reduces to an asymmetric spherical Higgs oscillator.

nlin.SI↗

arXiv2021

Classical and quantum speed limits

The new bound on quantum speed limit (in terms of relative purity) is derived by applying the original Mandelstam-Tamm one to the evolution in the space of Hilbert-Schmidt operators acting in the initial space of states. It is shown that it provides the quantum counterpart of the classical speed limit derived in Phys. Rev. Lett. 120 (2018), 070402 and the $\hbar\rightarrow 0$ limit of the former yields the latter. The existence of classical limit is related to the degree of mixing of the quantum state.

quant-ph↗

arXiv2021

Fermat's principle in constant gravitational field

Recently (Int.Journ.Mod.Phys. D27 (2018), 1847025) an interesting property of closed light rings in Kerr black holes has been noticed. We explain its origin and derive a slightly more general result.

gr-qc↗

arXiv2021

Global Symmetries of the Kepler Problem

The global symmetry transformations generated by Runge-Lenz vector of twodimensional Kepler problem are explicitly described. They are given in terms of SU(2) left group multiplication with group elements being suitably parametrized by phase space points. The resulting non-linear action of SU(2) on the phase space is characterized in terms of the theory of nonlinear realizations developed in Phys. Rev. 177 (1969) 2239.

physics.class-ph↗

arXiv2021

Symmetries and integrals of motion of a superintegrable deformed oscillator

The symmetry structure of twodimensional nonlinear isotropic oscillator, introduced in Physica D237 (2008) 505, is discussed. It is shown that it possesses three independent integrals of motion which can be chosen in such a way that they span SU(2), E(2) or SU(1,1) algebras, depending on the value of total energy. They generate the infinitesimal canonical symmetry transformations; integrability of the latter is analyzed. The results are then generalized to the case of arbitrary number of degrees of freedom.

nlin.SI↗

arXiv2019

Conformal symmetry, chiral fermions and semiclassical approximation

The explicit form of conformal generators is found which provides the extension of Poincare symmetry for massless particles of arbitrary helicity. The helicity 1/2 particles are considered as the particular example. The realization of conformal symmetry in the semiclassical regime of Weyl equation is obtained.

hep-th↗

arXiv2019

New superintegrable models on spaces of constant curvature

It is known that the fairly (most?) general class of 2D superintegrable systems defined on 2D spaces of constant curvature and separating in (geodesic) polar coordinates is specified by two types of radial potentials (oscillator or (generalized) Kepler ones) and by corresponding families of angular potentials. Unlike the radial potentials the angular ones are given implicitly (up to a function) by, in general, transcendental equation. In the present paper new two-parameter families of angular potentials are constructed in terms of elementary functions. It is shown that for an appropriate choice of parameters the family corresponding to the oscillator/Kepler type radial potential reduces to Poschl-Teller potential. This allows to consider Hamiltonian systems defined by this family as a generalization of Tremblay-Turbiner-Winternitz (TTW) or Post-Winternitz (PW) models both on plane as well as on curved spaces of constant curvature.

math-ph↗

arXiv2014

Conformal Newton-Hooke symmetry of Pais-Uhlenbeck oscillator

It is demonstrated that the Pais-Uhlenbeck oscillator in arbitrary dimension enjoys the l-conformal Newton-Hooke symmetry provided frequencies of oscillation form the arithmetic sequence omega_k=(2k-1) omega_1, where k=1,...,n, and l is the half-integer (2n-1)/2. The model is shown to be maximally superintegrable. A link to n decoupled isotropic oscillators is discussed and an interplay between the l-conformal Newton-Hooke symmetry and symmetries characterizing each individual isotropic oscillator is analyzed.

hep-th↗

arXiv2013

Nonlinear realizations and the orbit method

Given a symmetry group one can construct the invariant dynamics using the technique of nonlinear realizations or the orbit method. The relationship between these methods is discussed. Few examples are presented.

math-ph↗

arXiv2013

Conformal mechanics

The SL(2,R) invariant Hamiltonian systems are discussed within the frame- work of the orbit method. It is shown that both dynamics and symmetry trans- formations are globally well-defined on phase space. The flexibility in the choice of time variable and Hamiltonian function described in the paper by de Alfaro et al. (Nuovo Cim. 34A (1976),569) is related to the nontrivial global structure of 1 + 0-dimensional space-time. The operational definition of time is discussed.

hep-th↗

arXiv2010

Modified Hamiltonian formalism for higher-derivative theories

The alternative version of Hamiltonian formalism for higher-derivative theories is proposed. As compared with the standard Ostrogradski approach it has the following advantages: (i) the Lagrangian, when expressed in terms of new variables yields proper equations of motion; no additional Lagrange multipliers are necessary (ii) the Legendre transformation can be performed in a straightforward way provided the Lagrangian is nonsingular in Ostrogradski sense. The generalization to singular Lagrangians as well as field theory are presented.

hep-th↗

arXiv2010

Euclidean Path Integral and Higher-Derivative Theories

We consider the Euclidean path integral approach to higher-derivative theories proposed by Hawking and Hertog (Phys. Rev. D65 (2002), 103515). The Pais-Uhlenbeck oscillator is studied in some detail. The operator algebra is reconstructed and the structure of the space of states revealed. It is shown that the quantum theory results from quantizing the classical complex dynamics in which the original dynamics is consistently immersed. The field-theoretical counterpart of Pais-Uhlenbeck oscillator is also considered.

hep-th↗