From flows and metrics to dynamics (original) (raw)

On the Space of Trajectories of a Generic Vector Field

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric complex associated with a gradient like vector field and show how differential forms can be integrated on its unstable/stable sets. Integration leads to a morphism between the de Rham complex and the geometric complex.

Vector fields with distributions and invariants of ODEs

Journal of Geometric Mechanics, 2013

We study pairs (X, V) where X is a vector field on a smooth manifold M and V ⊂ T M is a vector distribution, both satisfying certain regularity conditions. We construct basic invariants of such objects and solve the equivalence problem. For a given pair (X, V) we construct a canonical connection on a certain frame bundle. The results are applied to the problem of timescale preserving equivalence of ordinary differential equations. The framework of pairs (X, V) is shown to include sprays, Hamiltonian systems, Veronese webs and other structures.

Toward a Lagrangian vector field topology

2010

Abstract In this paper we present an extended critical point concept which allows us to apply vector field topology in the case of unsteady flow. We propose a measure for unsteadiness which describes the rate of change of the velocities in a fluid element over time. This measure allows us to select particles for which topological properties remain intact inside a finite spatio-temporal neighborhood. One benefit of this approach is that the classification of critical points based on the eigenvalues of the Jacobian remains meaningful.

Fluid flow versus Geometric Dynamics

This paper investigates the way in which geometric dynamics on Riemannian manifolds can be applied to fluid flow. The equations of motion from fluid mechanics (momentum equations), expressed on general curvilinear coordinates, are compared with the equations of motion in a gyroscopic field of forces. Attempts are made to seek conditions under which both describe the same motion, especially in the case when trajectories are not field lines. It is proved that all pathlines and streamlines are geodesics in the least square sense and, in some circumstances, other trajectories included in geometric dynamics have physical support.

A note on Newtonian, Lagrangian and Hamiltonian dynamical systems in Riemannian manifolds

2001

Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure formalizing concepts of length and angle. The interplay of Riemannian metric and its metric connection with mechanical structures produces some features which are absent in the case of general (non-Riemannian) manifolds. The aim of present paper is to discuss these features and develop special language for describing Newtonian, Lagrangian, and Hamiltonian dynamical systems in Riemannian manifolds.

Distinguished trajectories in time dependent vector fields

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2009

We introduce a new definition of distinguished trajectory that generalises the concepts of fixed point and periodic orbit to aperiodic dynamical systems. This new definition is valid for identifying distinguished trajectories with hyperbolic and non-hyperbolic types of stability. The definition is implemented numerically and the procedure consist in determining a path of limit coordinates. It has been successfully applied to known examples of distinguished trajectories. In the context of highly aperiodic realistic flows our definition characterises distinguished trajectories in finite time intervals, and states that outside these intervals trajectories are no longer distinguished.

Differentiability properties of the flow of 2d autonomous vector fields

Journal of Differential Equations, 2021

We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation.

Dynamics on differential one-forms

Mathematical models of dynamics employing exterior calculus are shown to be mathematical representations of the same unifying principle; namely, the description of a dynamic system with a characteristic differential one-form on an odd-dimensional differentiable manifold leads, by analysis with exterior calculus, to a set of characteristic differential equations and a characteristic tangent vector which define transformations of the System' This principle, whose origin is Arnold's use of exterior calculus to describe Hamiltonian mechanics and geometric optics, is applied to irreversible thermodynamics and the dynamics of black holes' ilectromagnetism and strings. It is shown that "exterior calculus" mo{bls apply to systems for which the direction of change is given by a characteristic tangent vectpr and "conventional calculus" models apply to systems whose direction of change is arbitr4ry. The relationship berween the two types of models is shown to imply a technical definition of equilibrium for a dynamic system.

Classification of dynamical systems based on a decomposition of their vector fields

Journal of Differential Equations, 2012

We present a method for the global classification of dynamical systems based on a specific decomposition of their vector fields. Every differentiable vector field on R n can be decomposed uniquely in the sum of 2 systems: one gradient and one that leaves invariant the spheres S n−1. We show that, under some conditions, the topological class of a vector field is determined by the topological classes of its summands. We illustrate this method by applying it to a number of vector fields, among them being some members of the so-called Lorenz family. The advantage of such a classification is that equivalent flows exhibit qualitatively the same dynamical phenomena as their parameters are varied.