Uniform motions in central fields

Abstract

We present a theoretical problem of uniform motions, i.e. motions with constant magnitude of the velocity in central fields as a nonholonomic system of one particle with a nonlinear constraint. The concept of the article is in analogy with the recent paper [21]. The problem is analysed from the kinematic and dynamic point of view. The corresponding reduced equation of motion in the Newtonian central gravitational field is solved numerically. Appropriate trajectories for suitable initial conditions are presented. Symmetries and conservation laws are investigated using the concept of constrained Noetherian symmetry [9] and the corresponding constrained Noetherian conservation law. Isotachytonic version of the conservation law of mechanical energy is found as one of the corresponding constraint Noetherian conservation law of this nonholonomic system. 2010 Mathematics Subject Classification. Primary: 70F25, 70B05, 70H03, 70H33; Secondary: 70G45, 58Z05.Key words and phrases. Central field, nonholonomic mechanical systems, isotachytonic constraint, isotachytonic conservation law of angular momentum, reduced and deformed equations of motion, constraint symmetry, isotachytonic version of conservation law of mechanical energy.Both authors appreciate support of their departments.

Higher Order Hamiltonian Systems with Generalized Legendre Transformation

“…Note that in the notation of Equation 11 If we suppose that the matrix (19) has maximal rank, then Writing the Lepagean equivalent ρ (9) and (10) in the generalized Legendre transformation, we get…”

“…This approach is not well adapted to study Lagrangians that are singular in the standard sense. Note that an alternative approach to the study of "degenerated" Lagrangians (singular in the standard sense) is the constraint theory from mechanics (see [19,20]) and in the field theory [21].…”