802 research outputs found
Global isochronous potentials
We present a geometric characterization of the nonlinear smooth functions for which the origin is a global isochronous center for the scalar
equation . We revisit Stillinger and Dorignac isochronous
potentials and show a new simple explicit family. Implicit examples are
easily produced
Revisiting Noether's Theorem on constants of motion
In this paper we revisit Noether's theorem on the constants of motion for
Lagrangian mechanical systems in the ODE case, with some new perspectives on
both the theoretical and the applied side. We make full use of invariance up to
a divergence, or, as we call it here, Bessel-Hagen (BH) invariance. By
recognizing that the Bessel-Hagen (BH) function need not be a total time
derivative, we can easily deduce nonlocal constants of motion. We prove that we
can always trivialize either the time change or the BH-function, so that, in
particular, BH-invariance turns out not to be more general than Noether's
original invariance. We also propose a version of time change that simplifies
some key formulas. Applications include Lane-Emden equation, dissipative
systems, homogeneous potentials and superintegrable systems. Most notably, we
give two derivations of the Laplace-Runge-Lenz vector for Kepler's problem that
require space and time change only, without BH invariance, one with and one
without use of the Lagrange equation
Guido Cavalcanti nella novella del Boccaccio (Decameron VI, 9) e in un sonetto di Dino Compagni.
Sin resume
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