391 research outputs found
On Elliptical Billiards in the Lobachevsky Space and associated Geodesic Hierarchies
We derive Cayley's type conditions for periodical trajectories for the
billiard within an ellipsoid in the Lobachevsky space. It appears that these
new conditions are of the same form as those obtained before for the Euclidean
case. We explain this coincidence by using theory of geodesically equivalent
metrics and show that Lobachevsky and Euclidean elliptic billiards can be
naturally considered as a part of a hierarchy of integrable elliptical
billiards.Comment: 14 pages, to appear in Journal of Geometry and Physic
On Kapteyn-Kummer Series' Integral Form
In this short research note we obtain double definite integral expressions
for the Kapteyn type series built by Kummer's (or confluent hypergeometric
) functions. These kind of series unify in natural way the similar
fashion results for Neumann-, Schl\"omilch- and Kapteyn-Bessel series recently
established by Pog\'any, S\"uli, Baricz and Jankov Ma\v{s}irevi\'c
An extended Lagrangian formalism
A simple formal procedure makes the main properties of the lagrangian
binomial extendable to functions depending to any kind of order of the
time--derivatives of the lagrangian coordinates. Such a broadly formulated
binomial can provide the lagrangian components, in the classical sense of the
Newton's law, for a quite general class of forces. At the same time, the
generalized equations of motions recover some of the classical alternative
formulations of the Lagrangian equations
Cetaev condition for nonlinear nonholonomic systems and homogeneous constraints
We first present a way to formulate the equations of motion for a
nonholonomic system with nonlinear constraints with respect to the velocities.
The formulation is based on the Cetaev condition which aims to extend the
practical method of virtual displacements from the holonomic case to the
nonlinear nonholonomic one. The condition may appear in a certain sense
artificial and motivated only to coherently generalize that concerning the
holonomic case. In the second part we show that for a specific category of
nonholonomic constraints (homogeneous functions with respect to the generalized
velocities) the Cetaev condition reveals the same physical meaning that emerges
in systems with holonomic constraints. In particular the aspect of the
mechanical energy associable to the system is analysed
A simple approach to nonlinear nonholonomic systems with several examples
The main theme of the article is the study of discrete systems of material
points subjected to constraints not only of a geometric type (holonomic
constraints) but also of a kinematic type (nonholonomic constraints). The
setting up of the equations of motion follows a simple principle which
generalizes the holonomic case. Furthermore, attention is paid to the fact that
the kinematic variables retain their velocity meaning, without resorting to the
pseudo-velocity technique. Particular situations are examined in which the
modeling of the constraints can be carried out in several ways to evaluate
their effective equivalence. Numerous examples, many of which taken from the
most recurring ones in the literature, are provided in order to illustrate the
proposed theory
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