On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere

Producción CientíficaSeparable Hamiltonian systems either in sphero-conical coordinates on an S2 sphere or in elliptic coordinates on a R2 plane are described in a unified way. A back and forth route connecting these Liouville Type I separable systems is unveiled. It is shown how the gnomonic projection and its inverse map allow us to pass from a Liouville Type I separable system with a spherical configuration space to its Liouville Type I partners where the configuration space is a plane and back. Several selected spherical separable systems and their planar cousins are discussed in a classical context

New solutions of Heun general equation

We show that in four particular cases the derivative of the solution of Heun general equation can be expressed in terms of a solution to another Heun equation. Starting from this property, we use the Gauss hypergeometric functions to construct series solutions to Heun equation for the mentioned cases. Each of the hypergeometric functions involved has correct singular behavior at only one of the singular points of the equation; the sum, however, has correct behavior

Geodesics around Weyl-Bach's Ring Solution

We explore some of the gravitational features of a uniform ring both in the Newtonian potential theory and in General Relativity. We use a spacetime associated to a Weyl static solution of the vacuum Einstein's equations with ring like singularity. The Newtonian motion for a test particle in the gravitational field of the ring is studied and compared with the corresponding geodesic motion in the given spacetime. We have found a relativistic peculiar attraction: free falling particle geodesics are lead to the inner rim but never hit the ring.Comment: 8 figures, 14 pages. LaTeX w/ subfigure, graphic

On Virtual Displacement and Virtual Work in Lagrangian Dynamics

The confusion and ambiguity encountered by students, in understanding virtual displacement and virtual work, is discussed in this article. A definition of virtual displacement is presented that allows one to express them explicitly for holonomic (velocity independent), non-holonomic (velocity dependent), scleronomous (time independent) and rheonomous (time dependent) constraints. It is observed that for holonomic, scleronomous constraints, the virtual displacements are the displacements allowed by the constraints. However, this is not so for a general class of constraints. For simple physical systems, it is shown that, the work done by the constraint forces on virtual displacements is zero. This motivates Lagrange's extension of d'Alembert's principle to system of particles in constrained motion. However a similar zero work principle does not hold for the allowed displacements. It is also demonstrated that d'Alembert's principle of zero virtual work is necessary for the solvability of a constrained mechanical problem. We identify this special class of constraints, physically realized and solvable, as {\it the ideal constraints}. The concept of virtual displacement and the principle of zero virtual work by constraint forces are central to both Lagrange's method of undetermined multipliers, and Lagrange's equations in generalized coordinates.Comment: 12 pages, 10 figures. This article is based on an earlier article physics/0410123. It includes new figures, equations and logical conten

Projective dynamics and classical gravitation

Given a real vector space V of finite dimension, together with a particular homogeneous field of bivectors that we call a "field of projective forces", we define a law of dynamics such that the position of the particle is a "ray" i.e. a half-line drawn from the origin of V. The impulsion is a bivector whose support is a 2-plane containing the ray. Throwing the particle with a given initial impulsion defines a projective trajectory. It is a curve in the space of rays S(V), together with an impulsion attached to each ray. In the simplest example where the force is identically zero, the curve is a straight line and the impulsion a constant bivector. A striking feature of projective dynamics appears: the trajectories are not parameterized. Among the projective force fields corresponding to a central force, the one defining the Kepler problem is simpler than those corresponding to other homogeneities. Here the thrown ray describes a quadratic cone whose section by a hyperplane corresponds to a Keplerian conic. An original point of view on the hidden symmetries of the Kepler problem emerges, and clarifies some remarks due to Halphen and Appell. We also get the unexpected conclusion that there exists a notion of divergence-free field of projective forces if and only if dim V=4. No metric is involved in the axioms of projective dynamics.Comment: 20 pages, 4 figure

Sleep Duration is Increased Following Muscle Damaging Exercise in Hot Environmental Conditions

Sleep and recovery measures are typically negatively affected by a muscle-damaging bout of exercise. However, it remains unknown if the additive effects of hot environmental conditions, resulting in increased core temperature and other thermoregulatory responses during the exercise bout, further progress changes in quantity and performance quality of sleep duration. PURPOSE: To investigate the effect of muscle-damaging exercise in the heat, compared to a thermoneutral condition, on sleep and recovery measures. METHODS: Ten healthy males (age: 23 ± 3yr; body mass: 78.7 ± 11.5kg; height: 176.9 ± 5cm; lactate threshold [LT]: 9.7 ± 1.0km.hr-1) performed two protocols in a randomized, counterbalanced order of downhill running (DHR) for 30-minutes at the LT in either a thermoneutral (ambient temperate [Tamb], 20°C; relative humidity [RH], 20%) or hot environmental condition (Tamb, 35°C; RH, 40%) at a -10% gradient. Sleep and recovery measures were collected from a wearable sleep device participants wore the night after the DHR. Differences in sleep and recovery measures following DHR in the heat compared to a thermoneutral condition were analyzed using paired samples T-tests. RESULTS: Sleep hours, restorative sleep hours, rapid eye movement (REM) sleep hours, and slow wave sleep (SWS) hours were all greater following the heat condition (mean ± SD; sleep hours: 6.70 ± 0.74hr, p = 0.040; restorative sleep hours: 3.31 ± 0.90hr, p = 0.012; REM sleep hours: 1.70 ± 0.64hr, p = 0.046; SWS hours: 1.61 ± 0.35hr, p = 0.015) compared to the thermoneutral condition (sleep hours: 5.24 ± 1.75hr; restorative sleep hours: 2.45 ± 1.11hr; REM sleep hours: 1.23 ± 0.68hr; SWS: 1.22 ± 0.53hr). Also, recovery was higher following the heat condition (recovery: 75.88 ± 15.31, p = 0.023) compared to the thermoneutral condition (recovery: 50.75 ± 21.46). Sleep efficiency, sleep disturbance, sleep deprivation, sleep score, %REM, %SWS, light sleep, resting heart rate, and heart rate variability were not different between conditions (ps \u3e 0.05). CONCLUSION: Following muscle-damaging exercise in the heat, sleep and recovery duration measures were increased compared to a thermoneutral condition. These findings suggest that performing muscle-damaging exercises in hot conditions may require a greater amount of sleep for optimal recovery

Spherical functions on the de Sitter group

Matrix elements and spherical functions of irreducible representations of the de Sitter group are studied on the various homogeneous spaces of this group. It is shown that a universal covering of the de Sitter group gives rise to quaternion Euler angles. An explicit form of Casimir and Laplace-Beltrami operators on the homogeneous spaces is given. Different expressions of the matrix elements and spherical functions are given in terms of multiple hypergeometric functions both for finite-dimensional and unitary representations of the principal series of the de Sitter group.Comment: 40 page

Projective dynamics and first integrals

We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms.Comment: 39 pages, 2 figure

On two-dimensional Bessel functions

The general properties of two-dimensional generalized Bessel functions are discussed. Various asymptotic approximations are derived and applied to analyze the basic structure of the two-dimensional Bessel functions as well as their nodal lines.Comment: 25 pages, 17 figure

One pendulum to run them all

The analytical solution for the three-dimensional linear pendulum in a rotating frame of reference is obtained, including Coriolis and centrifugal accelerations, and expressed in terms of initial conditions. This result offers the possibility of treating Foucault and Bravais pendula as trajectories of the same system of equations, each of them with particular initial conditions. We compare them with the common two-dimensional approximations in textbooks. A previously unnoticed pattern in the three-dimensional Foucault pendulum attractor is presented