Superintegrable 3-body systems on the line

We consider classical three-body interactions on a Euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momenta first integrals. These systems are superseparable (i.e. multiseparable), superintegrable and equivalent (up to rescalings) to a one-particle system in the three-dimensional Euclidean space. Common features of the dynamics are discussed. We show how to determine the quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics. The conformal superseparability is proved and examples of conformal first integrals are given. The systems considered here in generality include the Calogero, Wolfes, and other three-body interactions widely studied in mathematical physics.Comment: Corrected typos. Some improvement

Geometrical classification of Killing tensors on bidimensional flat manifolds

Valence two Killing tensors in the Euclidean and Minkowski planes are classified under the action of the group which preserves the type of the corresponding Killing web. The classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebraic. The approach allows to classify both characteristic and non characteristic Killing tensors.Comment: 27 pages, 20 figures, pictures format changed to .eps, typos correcte

Solutions of Helmholtz and Schrödinger Equations with Side Condition and Nonregular Separation of Variables

Olver and Rosenau studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of finite-dimensional Hamiltonian systems, namely Hamilton-Jacobi, Helmholtz and time-independent Schrödinger equations with potential on N-dimensional Riemannian and pseudo-Riemannian manifolds, but with a linear side condition, where more structure is available. We show that the requirement of N−1 commuting second-order symmetry operators, modulo a second-order linear side condition corresponds to nonregular separation of variables in an orthogonal coordinate system, characterized by a generalized Stäckel matrix. The coordinates and solutions obtainable through true nonregular separation are distinct from those arising through regular separation of variables. We develop the theory for these systems and provide examples

Phase I-Designing a biofeedback device for quadriceps re-education-bridging the gap in exercise compliance

529-532Knee Osteoarthritis is a painful and disabling condition which causes difficulty in activities of daily living. Such patients are referred for physiotherapy sessions where they were prescribed with exercises as home programs, quadriceps isometrics was the commonly prescribed exercises as it causes the least intraarticular inflammation than other types of exercises. Research reports poor compliance to the exercise program with lack of confidence and doubt of performing correctly were being commonly reported answers to the lack of compliance. With the idea to improve the compliance to the exercise program this device is developed which helps the patients to perform the exercises at home and improve their confidence levels too. The Microcontroller used is Arduino Mega 2560 and Force sensor. Force sensor is used to sense the force produced by the patient and MCP6004 operational amplifier is used as Voltage follower for providing the exact voltage. It also has audio feedback and visual feedback, Audio feed back is given by connecting a voice playback module and a buzzer and visual feedback is given by LED’s of three colours- red, yellow, blue. LCD also displays the time taken by the patients. This device will surely improve the compliance with exercises for the wellbeing of patient

Characteristics of Deterministic and Stochastic Sandpile Models in a Rotational Sandpile Model

Rotational constraint representing a local external bias generally has non-trivial effect on the critical behavior of lattice statistical models in equilibrium critical phenomena. In order to study the effect of rotational bias in a out of equilibrium situation like self-organized criticality, a new two state ``quasi-deterministic'' rotational sandpile model is developed here imposing rotational constraint on the flow of sand grains. An extended set of new critical exponents are found to characterize the avalanche properties at the non-equilibrium steady state of the model. The probability distribution functions are found to obey usual finite size scaling supported by negative time autocorrelation between the toppling waves. The model exhibits characteristics of both deterministic and stochastic sandpile models.Comment: 27 pages, 11 figure

Structure results for higher order symmetry algebras of 2D classical superintegrable systems

Recently the authors and J.M. Kress presented a special function recurrence relation method to prove quantum superintegrability of an integrable 2D system that included explicit constructions of higher order symmetries and the structure relations for the closed algebra generated by these symmetries. We applied the method to 5 families of systems, each depending on a rational parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system. Here we work out the analogs of these constructions for all of the associated classical Hamiltonian systems, as well as for a family including the generic potential on the 2-sphere. We do not have a proof in every case that the generating symmetries are of lowest possible order, but we believe this to be so via an extension of our method.Comment: 23 page