Semi-global symplectic invariants of the spherical pendulum

The Journal of Physical Chemistry, 1996

The classical and quantum mechanics of a spherical pendulum are worked out, including the dynamics of a suspending frame with moment of inertia θ. The presence of two separatrices in the bifurcation diagram of the energy-momentum mapping has its mathematical expression in the hyperelliptic nature of the problem. Nevertheless, numerical computation allows to obtain the action variable representation of energy surfaces and to derive frequencies and winding ratios from there. The quantum mechanics is also best understood in terms of these actions. The limit θ f 0 is of particular interest, both classically and quantum mechanically, as it generates two copies of the frameless standard spherical pendulum. This is suggested as a classical interpretation of spin. † Dedicated to John Ross on the occasion of his 70th birthday.

We survey past work and present new algorithms to numerically integrate the trajectories of Hamiltonian dynamical systems.These algorithms exactly preserve the symplectic 2-form, i.e. they preserve all the Poincar6 invariants. The algorithms have been tested on a variety of examples and results are presented for the Fermi-Pasta-Ulam nonlinear string, the Henon-Heiles system, a four-vortex problem, and the geodesic flow on a manifold of constant negative curvature. In all cases the algorithms possess long-time stability and preserve global geometrical structures in phase space.

Regular and Chaotic Dynamics, 2007

We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolichyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system. We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energy-momentum map. This gives general conditions under which the non-degeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C. Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.

Mathematical Research Letters, 1994

Doklady Mathematics, 2006

arXiv: Classical Physics, 2019

In this paper we revisit the construction by which the $SL(2,\mathbb{R})$ symmetry of the Euler equations allows to obtain the simple pendulum from the rigid body. We begin reviewing the original relation found by Holm and Marsden in which, starting from the two Casimir functions of the extended rigid body with Lie algebra $ISO(2)$ and introducing a proper momentum map, it is possible to obtain both the Hamiltonian and equations of motion of the pendulum. Important in this construction is the fact that both Casimirs have the geometry of an elliptic cylinder. By considering the whole $SL(2,\mathbb{R})$ symmetry group, in this contribution we give all possible combinations of the Casimir functions and the corresponding momentum maps that produce the simple pendulum, showing that this system can also appear when the geometry of one of the Casimirs is given by a hyperbolic cylinder and the another one by an elliptic cylinder. As a result we show that from the extended rigid body with Li...

Journal of Symplectic Geometry, 2001

In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relation between the invariants introduced here and the Seiberg-Witten invariants of a product of a Riemann surface with a two-sphere. Contents

2011

This paper develops a symplectic bifurcation theory for integrable systems in dimension four. We prove that if an integrable system has no hyperbolic singularities and its bifurcation diagram has no vertical tangencies, then the fibers of the induced singular Lagrangian fibration are connected. The image of this singular Lagrangian fibration is, up to smooth deformations, a planar region bounded by the graphs of two continuous functions. The bifurcation diagram consists of the boundary points in this image plus a countable collection of rank zero singularities, which are contained in the interior of the image. Because it recently has become clear to the mathematics and mathematical physics communities that the bifurcation diagram of an integrable system provides the best framework to study symplectic invariants, this paper provides a setting for studying quantization questions, and spectral theory of quantum integrable systems.

Communications in Nonlinear Science and Numerical Simulation, 2021

The search of high-order periodic orbits has been typically restricted to problems with symmetries that help to reduce the dimension of the search space. Well-known examples include reversible maps with symmetry lines. The present work proposes a new method to compute high-order periodic orbits in twist maps without the use of symmetries. The method is a combination of the parameterization method in Fourier space and a Newton-Gauss multiple shooting scheme. The parameterization method has been successfully used in the past to compute quasi-periodic invariant circles. However, this is the first time that this method is used in the context of periodic orbits. Numerical examples are presented showing the accuracy and efficiency of the proposed method. The method is also applied to verify the renormalization prediction of the residues' convergence at criticality (extensively studied in reversible maps) in the relatively unexplored case of maps without symmetries.

Physical Review E, 1996

We develop an analytic technique to study the dynamics in the neighborhood of a periodic trajectory of a Hamiltonian system. The theory begins with Poincaré and Birkhoff; major modern contributions are due to Meyer, Arnol'd, and Deprit. The realization of the method relies on local Fourier-Taylor series expansions with numerically obtained coefficients. The procedure and machinery are presented in detail on the example of the ''perpendicular'' (zϭ0) periodic trajectory of the diamagnetic Kepler problem. This simple one-parameter problem well exhibits the power of our technique. Thus, we obtain a precise analytic description of bifurcations observed by J.-M. Mao and J. B. Delos ͓Phys. Rev. A 45, 1746 ͑1992͔͒ and explain the underlying dynamics and symmetries. ͓S1063-651X͑96͒10407-4͔

arXiv (Cornell University), 2022

Part 1. Introduction and preliminaries Chapter 1. Introduction 1.1. Structure and results of this monograph Chapter 2. A primer on singular symplectic manifolds 2.1. b-Poisson manifolds 2.2. On b m-Symplectic manifolds 2.3. Desingularizing b m-Poisson manifolds Chapter 3. A crash course on KAM theory Part 2. Action-angle coordinates and cotangent models Chapter 4. An action-angle theorem for b m-symplectic manifolds 4.1. Basic definitions 4.2. On b m-integrable systems 4.3. Examples of b m-integrable systems 4.4. Looking for a toric action 4.5. Action-angle coordinates on b m-symplectic manifolds Chapter 5. Reformulating the action-angle coordinate via cotangent lifts 5.1. Cotangent lifts and Arnold-Liouville-Mineur in Symplectic Geometry 5.2. The case of b m-symplectic manifolds Part 3. A KAM theorem for b m-symplectic manifolds Chapter 6. A new KAM theorem 6.1. On the structure of the proof 6.2. Technical results 6.3. A KAM theorem on b m-symplectic manifolds Chapter 7. Desingularization of b m-integrable systems Chapter 8. Desingularization of the KAM theorem on b m-symplectic manifolds Chapter 9. Applications to Celestial mechanics 9.1. The Kepler Problem v vi CONTENTS 9.2. The Problem of Two Fixed Centers 9.3. Double Collision and McGehee coordinates 9.4. The restricted three-body problem 9.5. Escape orbits in Celestial mechanics and Fluid dynamics Bibliography 1 By saying that the diffeomorphism is "ǫ-close to the identity" we mean that, for given H, P and r, there is a constant C such that ψ − Id < Cǫ.

TRIUMF Internal Report TRI-DN-03-13, 2013

We derive general formulae for the phase-space trajectories y(x) of the quadratic pendulum oscillator. The formulae are then developed in the form y(z) in terms of an intermediary function z(x); and in this form singular integrals are eliminated from the expressions for dwell time on a path. The dwell time formulae are then used to evaluate the period of motion for the serpentine paths. The period is found to be a symmetric function of Hamiltonian-value and is minimized on the central trajectory which passes through (x,y)=(0,0). A consideration of neighbour paths leads to the suggestion that small improvements in the output phase space of a particle ensemble might be effected by slightly prolonging acceleration. Finally conditions are given, in terms of the orbital path-length parameters, for a particle beam to be injected on to and extracted from the central trajectory.

European Journal of Physics, 2015

Integration of Hamiltonian systems by reduction to action-angle variables has proven to be a successful approach. However, when the solution depends on elliptic functions the transformation to action-angle variables may need to remain in implicit form. This is exactly the case of the simple pendulum, where in order to make explicit the transformation to action-angle variables one needs to resort to nontrivial expansions of special functions and series reversion. Alternatively, it is shown that the explicit expansion of the transformation to action-angle variables can be constructed directly, and that this direct construction leads naturally to the Lie transforms method, in this way avoiding the intricacies related to the traditional expansion of elliptic functions.

Annals of the New York Academy of Sciences, 1988

Journal of Mathematical Physics, 2010

Revisiting canonical integration of the classical pendulum around its unstable equilibrium, normal hyperbolic canonical coordinates are constructed.