Jacobi multipliers, non-local symmetries and nonlinear oscillators

Constants of motion, Lagrangians and Hamiltonians admitted by a family of relevant nonlinear oscillators are derived using a geometric formalism. The theory of the Jacobi last multiplier allows us to find Lagrangian descriptions and constants of the motion. An application of the jet bundle formulation of symmetries of differential equations is presented in the second part of the paper. After a short review of the general formalism, the particular case of non-local symmetries is studied in detail by making use of an extended formalism. The theory is related to some results previously obtained by Krasil'shchi, Vinogradov and coworkers. Finally the existence of non-local symmetries for such two nonlinear oscillators is proved.Comment: 20 page

On a generalization of Jacobi's elliptic functions and the Double Sine-Gordon kink chain

A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical application we show that periodic kink solutions (kink chains) of the double sine-Gordon model can be described in a canonical form in terms of generalized Jacobi functions.Comment: 18 pages, 9 figures, 3 table

From Lagrangian to Quantum Mechanics with Symmetries

We present an old and regretfully forgotten method by Jacobi which allows one to find many Lagrangians of simple classical models and also of nonconservative systems. We underline that the knowledge of Lie symmetries generates Jacobi last multipliers and each of the latter yields a Lagrangian. Then it is shown that Noether's theorem can identify among those Lagrangians the physical Lagrangian(s) that will successfully lead to quantization. The preservation of the Noether symmetries as Lie symmetries of the corresponding Schr\"odinger equation is the key that takes classical mechanics into quantum mechanics. Some examples are presented.Comment: To appear in: Proceedings of Symmetries in Science XV, Journal of Physics: Conference Series, (2012

Forcing mechanisms of the terdiurnal tide

Using a nonlinear mechanistic global circulation model we analyze the migrating terdiurnal tide in the middle atmosphere with respect to its possible forcing mechanisms, i.e., the absorption of solar radiation in the water vapor and ozone band, nonlinear tidal interactions, and gravity wave–tide interactions. In comparison to the forcing mechanisms of diurnal and semidiurnal tides, these terdiurnal forcings are less well understood and there are contradictory opinions about their respective relevance. In our simulations we remove the wave number 3 pattern for each forcing individually and analyze the remaining tidal wind and temperature fields. We find that the direct solar forcing is dominant and explains most of the migrating terdiurnal tide's amplitude. Nonlinear interactions due to other tides or gravity waves are most important during local winter. Further analyses show that the nonlinear forcings are locally counteracting the solar forcing due to destructive interferences. Therefore, tidal amplitudes can become even larger for simulations with removed nonlinear forcings.</p

Quasi-doubly periodic solutions to a generalized Lame equation

We consider the algebraic form of a generalized Lame equation with five free parameters. By introducing a generalization of Jacobi's elliptic functions we transform this equation to a 1-dim time-independent Schroedinger equation with (quasi-doubly) periodic potential. We show that only for a finite set of integral values for the five parameters quasi-doubly periodic eigenfunctions expressible in terms of generalized Jacobi functions exist. For this purpose we also establish a relation to the generalized Ince equation.Comment: 15 pages,1 table, accepted for publication in Journal of Physics

On uniformization of Burnside's curve y2=x5−xy^2=x^5-x

Main objects of uniformization of the curve $y^2=x^5-x$ are studied: its Burnside's parametrization, corresponding Schwarz's equation, and accessory parameters. As a result we obtain the first examples of solvable Fuchsian equations on torus and exhibit number-theoretic integer $q$-series for uniformizing functions, relevant modular forms, and analytic series for holomorphic Abelian integrals. A conjecture of Whittaker for hyperelliptic curves and its hypergeometric reducibility are discussed. We also consider the conversion between Burnside's and Whittaker's uniformizations.Comment: Final version. LaTeX, 23 pages, 1 figure. The handbook for elliptic functions has been moved to arXiv:0808.348

Poisson Structures for Aristotelian Model of Three Body Motion

We present explicitly Poisson structures, for both time-dependent and time-independent Hamiltonians, of a dynamical system with three degrees of freedom introduced and studied by Calogero et al [2005]. For the time-independent case, new constant of motion includes all parameters of the system. This extends the result of Calogero et al [2009] for semi-symmetrical motion. We also discuss the case of three bodies two of which are not interacting with each other but are coupled with the interaction of third one

General Kerr-NUT-AdS Metrics in All Dimensions

The Kerr-AdS metric in dimension D has cohomogeneity [D/2]; the metric components depend on the radial coordinate r and [D/2] latitude variables \mu_i that are subject to the constraint \sum_i \mu_i^2=1. We find a coordinate reparameterisation in which the \mu_i variables are replaced by [D/2]-1 unconstrained coordinates y_\alpha, and having the remarkable property that the Kerr-AdS metric becomes diagonal in the coordinate differentials dy_\alpha. The coordinates r and y_\alpha now appear in a very symmetrical way in the metric, leading to an immediate generalisation in which we can introduce [D/2]-1 NUT parameters. We find that (D-5)/2 are non-trivial in odd dimensions, whilst (D-2)/2 are non-trivial in even dimensions. This gives the most general Kerr-NUT-AdS metric in $D$ dimensions. We find that in all dimensions D\ge4 there exist discrete symmetries that involve inverting a rotation parameter through the AdS radius. These symmetries imply that Kerr-NUT-AdS metrics with over-rotating parameters are equivalent to under-rotating metrics. We also consider the BPS limit of the Kerr-NUT-AdS metrics, and thereby obtain, in odd dimensions and after Euclideanisation, new families of Einstein-Sasaki metrics.Comment: Latex, 24 pages, minor typos correcte

Quarterdiurnal signature in sporadic E occurrence rates and comparison with neutral wind shear

The GPS radio occultation (RO) technique is used to study sporadic E (Es) layer plasma irregularities of the Earth's ionosphere on a global scale using GPS signal-to-noise ratio (SNR) profiles from the COSMIC/FORMOSAT-3 satellite. The maximum deviation from the mean SNR can be attributed to the height of the Es layer. Es are generally accepted to be produced by ion convergence due to vertical wind shear in the presence of a horizontal component of the Earth's magnetic field, while the wind shear is provided mainly by the solar tides. Here we present analyses of quarterdiurnal tide (QDT) signatures in Es occurrence rates. From a local comparison with mesosphere/lower thermosphere wind shear obtained with a meteor radar at Collm (51.3∘&thinsp;N, 13.0∘&thinsp;E), we find that the phases of the QDT in Es agree well with those of negative vertical shear of the zonal wind for all seasons except for summer, when the QDT amplitudes are small. We also compare the global QDT Es signal with numerical model results. The global distribution of the Es occurrence rates qualitatively agrees with the modeled zonal wind shears. The results indicate that zonal wind shear is indeed an important driving mechanism for the QDT seen in Es.</p