Abstract
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.
Highlights
The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics; (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics; (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics
The topics of the papers that inspired us in developing the theory of decomposable dynamics can be classified as follows: Applied ODEs and PDEs [11,12,13,14] are evolution equations modeling systems evolving with respect to a “time” parameter
If F is generated by X and g, the ODE (2) represents a single-time geometric dynamics or a geodesic motion in a gyroscopic field of forces [1,2,3,4,5,6,7,8]
Summary
The subject of dynamical systems concerns the evolution of systems in single-time or multi-time (multivariate). The topics of the papers that inspired us in developing the theory of decomposable dynamics can be classified as follows: Applied ODEs and PDEs [11,12,13,14] are evolution equations modeling systems evolving with respect to a “time” parameter. When solving such evolution equations, the appropriate formulation of the problem is usually as an initial value, or Cauchy, problem. The main aim of this paper is to give necessary and sufficient conditions for the decomposition of a general (single-time or multi-time) dynamics into a flow and a transversal movement
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