We construct Poisson structures for Ermakov systems, using the Ermakov invariant as the Hamiltonian. Two classes of Poisson structures are obtained, one of them degenerate, in which case we derive the Casimir functions. In some situations, the existence of Casimir functions can give rise to superintegrable Ermakov systems. Finally, we characterize the cases where linearization of the equations of motion is possible.
We generalize the two dimensional autonomous Hamiltonian Kepler Ermakov dynamical system to three dimensions using the sl(2,R) invariance of Noether symmetries and determine all three dimensional autonomous Hamiltonian Kepler Ermakov dynamical systems which are Liouville integrable via Noether symmetries. Subsequently we generalize the autonomous Kepler Ermakov system in a Riemannian space which admits a gradient homothetic vector by the requirements (a) that it admits a first integral (the Riemannian Ermakov invariant) and (b) it has sl(2,R) invariance. We consider both the non-Hamiltonian an…
Using older and recent results on the integrability of two-dimensional (2d) dynamical systems, we prove that the results obtained in a recent publication concerning the 2d generalized Ermakov system can be obtained as special cases of a more general approach. This approach is geometric and can be used to study efficiently similar dynamical systems.
A 2+1-dimensional anisentropic magnetogasdynamic system with a polytropic gas law is shown to admit an integrable elliptic vortex reduction when $γ=2$ to a nonlinear dynamical subsystem with underlying integrable Hamiltonian-Ermakov structure. Exact solutions of the magnetogasdynamic system are thereby obtained which describe a rotating elliptic plasma cylinder. The semi-axes of the elliptical cross-section, remarkably, satisfy a Ermakov-Ray-Reid system.
We revise recent results on the classification of the generalized three-dimensional Hamiltonian Ermakov system. We show that a statement published recently is incorrect, while the solution for the classification problem was incomplete. We present the correct classification for the three-dimensional system by using results which related the background space with the dynamics. Finally, we extend our results for the generalized $n$% -dimensional Hamiltonian Ermakov system.
We employ the method of nonlocal generalized Sundman transformations to formulate the linearization problem for equations of the generalized Liénard type and show that they may be mapped to equations of the dissipative Ermakov-Milne-Pinney type. We obtain the corresponding new first integrals of these derived equations, this method yields a natural generalization of the construction of Ermakov-Lewis invariant for a time dependent oscillator to (coupled) Liénard and Liénard type equations. We also study the linearization problem for the coupled Liénard equation using nonlocal transformations an…
Reid's m'th-order generalized Ermakov systems of nonlinear coupling constant alpha are equivalent to an integrable Emden-Fowler equation. The standard Ermakov-Lewis invariant is discussed from this perspective, and a closed formula for the invariant is obtained for the higher-order Reid systems (m\geq 3). We also discuss the parametric solutions of these systems of equations through the integration of the Emden-Fowler equation and present an example of a dynamical system for which the invariant is equivalent to the total energy
In this work, we introduce a new realization of exactly-solvable time-dependent Hamiltonians based on the solutions of the fourth Painlevé and the Ermakov equations. The latter is achieved by introducing a shape-invariant condition between an unknown quantum invariant and a set of third-order intertwining operators with time-dependent coefficients. The new quantum invariant is constructed by adding a deformation term to the well-known parametric oscillator invariant. Such a deformation depends explicitly on time through the solutions of the Ermakov equation, which ensures the regularity of the…
The Ermakov-Milne-Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov-Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov-Milne-Pinney equation and associated first integral. The special relativistic extension of the Ray-Reid system and invariant is obtained. General properties of the relativisti…
We consider linear and quadratic integrals of motion for general variable quadratic Hamiltonians. Fundamental relations between the eigenvalue problem for linear dynamical invariants and solutions of the corresponding Cauchy initial value problems for the time-dependent Schroedinger equation are emphasized. An eigenfunction expansionof the solution of the initial value problem is also found. A nonlinear superposition principle for the generalized Ermakov systems is established as a result of decomposition of the general quadratic invariant in terms of the linear ones.
We transform the time-dependent Schroedinger equation for the most general variable quadratic Hamiltonians into a standard autonomous form. As a result, the time-evolution of exact wave functions of generalized harmonic oscillators is determined in terms of solutions of certain Ermakov and Riccati-type systems. In addition, we show that the classical Arnold transformation is naturally connected with Ehrenfest's theorem for generalized harmonic oscillators.
We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov-Milne-Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other …
In the wake of a preceding article \cite{RogUnt06} introducing the Schrödinger-Virasoro group, we study its affine action on a space of $(1+1)$-dimensional Schrödinger operators with time- and space-dependent potential $V$ periodic in time. We focus on the subspace corresponding to potentials that are at most quadratic in the space coordinate, which is in some sense the natural quantization of the space of Hill (Sturm-Liouville) operators on the one-dimensional torus. The orbits in this subspace have finite codimension, and their classification by studying the stabilizers can be obtained by ex…
Based on the Gaussian wave packet solution for the harmonic oscillator and the corresponding creation and annihilation operators, a generalization is presented that also applies for wave packets with time-dependent width as they occur for systems with different initial conditions, time-dependent frequency or in contact with a dissipative environment. In all these cases the corresponding coherent states, position and momentum uncertainties and quantum mechanical energy contributions can be obtained in the same form if the creation and annihilation operators are expressed in terms of a complex v…
We generalise Livens theorem, showing that Hamiltonian equation on the vector bundle $E^\ast\rightarrow M$, dual to a general algebroid $E\rightarrow M$, can be derived by means of a variational principle. The framework can be used to describe Hamiltonian dynamics associated with (both vaconomic and nonholonomic) constraints in the bundle $E$.
We study Hamiltonian analysis of three-dimensional advection flow $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ of incompressible nature $\nabla \cdot {\bf v} ={\bf 0}$ assuming that dynamics is generated by the curl of a vector potential $\mathbf{v} = \nabla \times \mathbf{A}$. More concretely, we elaborate Nambu-Hamiltonian and bi-Hamiltonian characters of such systems under the light of vanishing or non-vanishing of the quantity $\mathbf{A} \cdot \nabla \times \mathbf{A}$. We present an example (satisfying $\mathbf{A} \cdot \nabla \times \mathbf{A} \neq 0$) which can be written as in the form of N…
In \cite{b-i-t}, F. Bagarello, A. Inoue and C. Trapani investigated some operators defined by Riesz bases. These operators connect with ${\it quasi}$-${\it hermitian \; quantum \; mechanics}$, and its relatives. In this paper, we change the frameworks of these operators, and then almost results obtained in \cite{b-i-t} become trivial. Furthermore, we introduce a notion of generalized Riesz bases which is a generalization of Riesz bases and investigate some operators defined by generalized Riesz bases.
In this paper a class of classical Hamiltonian systems is geometrically formulated. This class is such that a Hamiltonian can be written as the sum of a kinetic energy function and a potential energy function. In addition, these energy functions are assumed strictly convex. For this class of Hamiltonian systems Hessian and information geometric formulation is given. With this formulation, a generalized Toda's dual transform is proposed, where his original transform was used in deriving his integrable lattice system. Then a relation between the generalized Toda's dual transform and the Legendre…
We characterize all cases when a certain natural generalization of the Infeld--Rowlands equation admits nontrivial local conservation laws of any order, and give explicit form of these conservation laws modulo trivial ones. Furthermore, we prove that the equation under study admits no nontrivial local Hamiltonian and symplectic structures and no nontrivial local Noether and inverse Noether operators; the method of establishing the said nonexistence results can be readily applied to many other PDEs.
We give an outlook, how to realize the ideas of complex scaling from Grothaus, Vogel and Streit to phase space path integrals in the framework of White Noise Analysis. The idea of this scaling method goes back to Doss. Therefore we extend the concept complex scaling to scaling with suitable bounded operators.
Let the pair of operators, $(H, T)$, satisfy the weak Weyl relation: $Te^{-itH} = e^{-itH}(T + t)$, where $H$ is self-adjoint and $T$ is closed symmetric. Suppose that g is a realvalued Lebesgue measurable function on $\RR$ such that $g \in C^2(R K)$ for some closed subset $K \subset \RR$ with Lebesgue measure zero. Then we can construct a closed symmetric operator $D$ such that $(g(H), D)$ also obeys the weak Weyl relation.
This paper deals with an extension of our previous work [J. Phys. A: Math. Theor. {\bf 40} F817] by considering an alternative construction of canonical and deformed vector coherent states (VCSs) of the Gazeau-Klauder type associated with generalized spin-orbit Hamiltonians. We define an annihilation operator which takes into account the finite dimensional space of states induced by the $k$-photon transition processes of the two-level atom interacting with the single-mode radiation field. The class of nonlinear VCSs (NVCSs) corresponding to the action of the annihilation operator is deduced an…
The Mishchenko-Fomenko theorem on superintegrable Hamiltonian systems is generalized to superintegrable Hamiltonian systems with noncompact invariant submanifolds. It is formulated in the case of globally superintegrable Hamiltonian systems which admit global generalized action-angle coordinates. The well known Kepler system falls into two different globally superintegrable systems with compact and noncompact invariant submanifolds.
In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system on Lie algebroids are given. Here we use the general properties of Lie algebroids to express and prove two geometric version of the Hamilton-Jacobi theorem for Hamiltonian system on Lie algebroids. Then this results are generalized and two types of time-dependent Hamilton-Jacobi theorem of Hamiltonian system on Lie algebroids are obtained.
This paper is a natural continuation of the previous paper \cite{TyuVo13} where generalized oscillator representations for Calogero Hamiltonians with potential $V(x)=α/x^2$, $α\geq-1/4$, were constructed. In this paper, we present generalized oscillator representations for all generalized Calogero Hamiltonians with potential $V(x)=g_{1}/x^2+g_{2}x^2$, $g_{1}\geq-1/4$, $g_{2}>0$. These representations are generally highly nonunique, but there exists an optimum representation for each Hamiltonian, representation that explicitly determines the ground state and the ground-state energy. For general…
We proved a KAM theorem on existence of invariant tori in generalized Hamiltonian systems without action-angle variables. It is a generalization of the result of de la Llave et al. [Llave, 2005] that deals with canonical Hamiltonian system.
On the basis of Liouville theorem the generalization of the Nambu mechanics is considered. Is shown, that Poisson manifolds of n-dimensional multi-symplectic phase space have inducting by (n-1) Hamiltonian k-vector fields, each of which requires of (k)-hamiltonians.
This paper is a natural continuation of the previous paper J.Phys. A: Math.Theor. 44 (2011) 425204, arXiv 0907.1736 [quant-ph] where oscillator representations for nonnegative Calogero Hamiltonians with coupling constant $α\geq-1/4$ were constructed. Here, we present generalized oscillator representations for all Calogero Hamiltonians with $α\geq-1/4$.These representations are generally highly nonunique, but there exists an optimum representation for each Hamiltonian.
We study a generalized scheme of Swanson Hamiltonian from a second-derivative pseudosupersymmetric approach. We discuss plausible choices of the underlying quasi-Hamiltonian and consider the viability of applications to systems like the isotonic oscillator and CPRS potential.
In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub-manifolds.
In this paper, a new generalized $5\times5$ matrix spectral problem of Ablowitz-Kaup-Newell-Segur(AKNS) type associated with the enlarged matrix Lie super algebra is proposed and its corresponding super soliton hierarchy is established. The super variational identities is used to furnish super-Hamiltonian structures for the resulting super soliton hierarchy.
We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of unco…
We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional ωN manifold guarantees the separation of variables. As an application, we construct such chains for the Hénon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.
In this paper we are concerned with completely integrable Hamiltonian systems in the setting of contact geometry. Unlike the symplectic case, contact structures are automatically Hamiltonian. Using the Jacobi brackets defined on contact manifolds, we discuss the commutativity of the first integrals for contact Hamiltonian systems and introduce the generalized contact action-angle variables. We exemplify the general scheme in the case of the five-dimensional toric Sasaki-Einstein spaces $T^{1,1}$ and $Y^{p,q}$.
A generalized non-Hermitian oscillator Hamiltonian is proposed that consists of additional linear terms which break PT-symmetry explicitly. The model is put into an equivalent Hermitian form by means of a similarity transformation and the criterion of N-fold supersymmetry with a position-dependent mass is shown to reside in it.
This text is addressed to mathematicians who are interested in generalized functions and unbounded operators on a Hilbert space. We expose in detail (in a "formal way" - as done by Heisenberg and Pauli - i.e. without mathematical definitions and then, of course, without mathematical rigour) the Heisenberg-Pauli calculations on the simplest model close to physics. The problem for mathematicians is to give a mathematical sense to these calculations, which is possible without any knowledge in physics, since they mimick exactly usual calculations on infinitely differentiable functions and on bound…
This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermite polynomials. This is achieved by exploiting the intrinsic relation between third-order shape-invariant Hamiltonians and the fourth Painlevé equation, such that the generalized Hermite polynomials emerge from the $-1/x$ and $-2x$ hierarchies of rational solutions. Such a relation unequivocally establishes the discrete spectrum structure, which, in general, is composed as the union of a finite- and infinite-dimensional sequence of equidistant eigenvalues separated by …
The generalized Swanson Hamiltonian $H_{GS} = w (\tilde{a}\tilde{a}^†+ 1/2) + α\tilde{a}^2 + β\tilde{a}^{†^2}$ with $\tilde{a} = A(x)d/dx + B(x)$, can be transformed into an equivalent Hermitian Hamiltonian with the help of a similarity transformation. It is shown that the equivalent Hermitian Hamiltonian can be further transformed into the harmonic oscillator Hamiltonian so long as $[\tilde{a},\tilde{a}^†]=$ constant. However, the main objective of this paper is to show that though the commutator of $\tilde{a}$ and $\tilde{a}^†$ is constant, the generalized Swanson Hamiltonian is not necessar…
We present a coordinate-free approach for constructing approximate first integrals of generalized slow-fast Hamiltonian systems, based on the global averaging method on parameter-dependent phase spaces with $\mathbb{S}^1 -$symmetry. Explicit global formulas for approximate second-order first integrals are derived. As examples, we analyze the case quadratic in the fast variables (in particular, the elastic pendulum), and the charged particle in a slowly-varying magnetic field.
In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized Kähler manifolds.
A new description, different by the classical theory of Hamiltonian Mechanics, in the general framework of generalized Lie algebroids is presented. In the particular case of Lie algebroids, new and important results are obtained. We present the \emph{dual mechanical systems} called by use, \emph{dual mechanical}$(ρ,η) $\emph{-systems, Hamilton mechanical}$(ρ,η) $\emph{-systems} or \emph{% Cartan mechanical}$(ρ,η) $\emph{-systems.} and we develop their geometries. We obtain the canonical $(ρ,η) $\emph{-}semi(spray) associated to a dual mechanical $(ρ,η) $-system. The Hamilton mechanical $(ρ,η)$…
We study the hydrodynamic-type system of differential equations modeling isothermal no-slip drift flux. Using the facts that the system is partially coupled and its subsystem reduces to the (1+1)-dimensional Klein--Gordon equation, we exhaustively describe generalized symmetries, cosymmetries and local conservation laws of this system. A generating set of local conservation laws under the action of generalized symmetries is proved to consist of two zeroth-order conservation laws. The subspace of translation-invariant conservation laws is singled out from the entire space of local conservation …
We first point out it is conditional to apply the variational approach to the nonlocal nonlinear Schrödinger equation (NNLSE), that is, the response function must be an even function. Different from the variational approach, the canonical Hamiltonian formulation for the first-order differential system are used to deal with the problems of the nonlinear light-envelope propagations. The Hamiltonian of the system modeled by the NNLSE is obtained, which can be expressed as the sum of the generalized kinetic energy and the generalized potential. The solitons correspond to extreme points of the gene…
We demonstrate that a Poisson structure can always be associated to a general nonautonomous 3D vector field of ODEs by means of a diffeomorphism that preserves both the orientation and the volume of phase-space. The only prerequisite is the existence of one constant of motion.
