From circular paths to elliptic orbits: A geometric approach to Kepler's motion

The hodograph, i.e. the path traced by a body in velocity space, was introduced by Hamilton in 1846 as an alternative for studying certain dynamical problems. The hodograph of the Kepler problem was then investigated and shown to be a circle, it was next used to investigate some other properties of the motion. We here propose a new method for tracing the hodograph and the corresponding configuration space orbit in Kepler's problem starting from the initial conditions given and trying to use no more than the methods of synthetic geometry in a sort of Newtonian approach. All of our geometric constructions require straight edge and compass only.Comment: 9 pages, 4 figure

The effect of wave conditions and surfer ability on performance and the physiological response of recreational surfers.

This study investigated the effects of wave conditions on performance and the physiological responses of surfers. After institutional ethical approval 39 recreational surfers participated in 60 surfing sessions where performance and physiological response were measured using global positioning system (GPS) heart rate monitors. Using GPS, the percentage time spent in surfing activity categories was on average 41.6, 47.0, 8.1, and 3.1% for waiting, paddling, riding, and miscellaneous activities, respectively. Ability level of the surfers, wave size, and wave period are significantly associated with the physiological, ride, and performance parameters during surfing. As the ability level of the surfers increases there is a reduction in the relative exercise intensity (e.g., average heart rate as a percentage of laboratory maximum, rpartial = -0.412, p < 0.01) which is in contrast to increases in performance parameters (e.g., maximum ride speed (0.454, p < 0.01). As the wave size increased there were reductions in physiological demand (e.g., total energy expenditure rpartial = -0.351, p ≤ 0.05) but increases in ride speed and distance measures (e.g., the maximum ride speed, 0.454, p < 0.01). As the wave period increased there were increases in intensity (e.g., average heart rate as a percentage of laboratory maximum, rp = 0.490, p < 0.01) and increases in ride speed and distance measures (e.g., the maximum ride speed, rpartial = 0.371, p < 0.01). This original study is the first to show that wave parameters and surfer ability are significantly associated with the physiological response and performance characteristics of surfing

Hamiltonian Noether theorem for gauge systems and two time physics

The Noether theorem for Hamiltonian constrained systems is revisited. In particular, our review presents a novel method to show that the gauge transformations are generated by the conserved quantities associated with the first class constraints. We apply our results to the relativistic point particle, to the Friedberg et al. model and, with special emphasis, to two time physics.Comment: 20 pages, Latex, references added, the "massless" sense of (87) is clarifie