Lagrangian dynamics by nonlocal constants of motion

Abstract

A simple general theorem is used as a tool that generates nonlocal constants of motion for Lagrangian systems. We review some cases where the constants that we find are useful in the study of the systems: the homogeneous potentials of degree −2, the mechanical systems with viscous fluid resistance and the conservative and dissipative Maxwell-Bloch equations of laser dynamics. We also prove a new result on explosion in the past for mechanical system with hydraulic (quadratic) fluid resistance and bounded potential.

A New Class of Separable Lagrangian Systems Generalizing Sawada–Kotera System

“…Whenever conditions (22) hold, the Euler-Lagrange equations coincide, and E and Ẽ are the first integrals for both. In this case, in terms of the new variables (x, y) = (q 1 + q 2 , q 1 − q 2 ) (just a little simpler than r 1 , r 2 above), the Lagrangian function L becomes…”

“…that have simple solutions. Incidentally, to this system, we can apply the following result ( [22] Section 3): for general Lagrangian functions 1 2 m∥ q∥ 2 − U(q), q ∈ R n , with U homogeneous of degree −2, we have that…”

Nonlocal constants of motion in Lagrangian Dynamics of any order

“…• A particle under a time-independent potential field U(q), q ∈ R n , and hydraulic, i.e. quadratic, fluid resistance, a result taken from Gorni-Zampieri [10], for which the result in dynamics is: if 0 ≤ U ≤ U sup < +∞, all the solutions for which the initial kinetic energy is strictly greater than U sup explode in the past in finite time.…”

“…The paper [10] presents the result on hydraulic fluid resistance and gives a survey on all other applications we mentioned till now. The novelty in the present survey is the introduction of Scomparin's generalization of Theorem 1.1 to higher-order Lagrangian systems, which recently appeared in [18].…”