Generalized interpolation and definability

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32849/1/0000225.pd

The generalized non-conservative model of a 1-planet system - revisited

We study the long-term dynamics of a planetary system composed of a star and a planet. Both bodies are considered as extended, non-spherical, rotating objects. There are no assumptions made on the relative angles between the orbital angular momentum and the spin vectors of the bodies. Thus, we analyze full, spatial model of the planetary system. Both objects are assumed to be deformed due to their own rotations, as well as due to the mutual tidal interactions. The general relativity corrections are considered in terms of the post-Newtonian approximation. Besides the conservative contributions to the perturbing forces, there are also taken into account non-conservative effects, i.e., the dissipation of the mechanical energy. This dissipation is a result of the tidal perturbation on the velocity field in the internal zones with non-zero turbulent viscosity (convective zones). Our main goal is to derive the equations of the orbital motion as well as the equations governing time-evolution of the spin vectors (angular velocities). We derive the Lagrangian equations of the second kind for systems which do not conserve the mechanical energy. Next, the equations of motion are averaged out over all fast angles with respect to time-scales characteristic for conservative perturbations. The final equations of motion are then used to study the dynamics of the non-conservative model over time scales of the order of the age of the star. We analyze the final state of the system as a function of the initial conditions. Equilibria states of the averaged system are finally discussed.Comment: 37 pages, 13 figures, accepted to Celestial Mechanics and Dynamical Astronom

Lumbar disk prolapse: response to mechanical physiotherapy in the absence of changes in magnetic resonance imaging. Report of 11 cases

BACKGROUND: Lumbar disk prolapses are among the most common neurological conditions. In this open study, we asked whether repeated end-range spinal movements (McKenzie method) as physiotherapy in patients with lumbar disk prolapse induce early changes in location, size and signal intensity of lumbar disc material detectable by magnetic resonance imaging (MRI). We compared clinical with radiographic changes. The clinical efficacy of mechanical physiotherapy according to the McKenzie method within 5 days was documented. METHODS: Eleven consecutive patients with lumbar disk prolapse were included. Patients were treated with repeated end-range spinal movements and MRI was performed before and after 2-5 treatments. RESULTS: All patients achieved a reduction in symptoms and signs of disk prolapse during and after these procedures but none showed any change in the MRI features of the prolapses. CONCLUSIONS: Beneficial effects of specific mechanical physiotherapy in patients with radicular syndromes from lumbar disk prolapse are not paralleled by changes in the MRI appearance of the prolapses. Alternative explanations for the early clinical responses in some patients with lumbar disc prolapse treated according to the McKenzie method must be sought