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Geometric Dynamics on Riemannian Manifolds

Department of Mathematics-Informatics, Faculty of Applied Sciences, University Politehnica of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania
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Mathematics 2020, 8(1), 79; https://doi.org/10.3390/math8010079
Received: 22 October 2019 / Revised: 23 December 2019 / Accepted: 23 December 2019 / Published: 3 January 2020
The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail five significant decomposed dynamics: (i) the motion of the four outer planets relative to the sun fixed by a Hamiltonian, (ii) the motion in a closed Newmann economical system fixed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a flow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-flow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into flows and motions transversal to the flows. View Full-Text
Keywords: dynamical systems; generated ODEs and PDEs; single-time geometric dynamics; multi-time geometric dynamics; decomposable dynamics dynamical systems; generated ODEs and PDEs; single-time geometric dynamics; multi-time geometric dynamics; decomposable dynamics
MDPI and ACS Style

Udriste, C.; Tevy, I. Geometric Dynamics on Riemannian Manifolds. Mathematics 2020, 8, 79.

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