[Paper Review] From flows and metrics to dynamics

Constantin Udrişte, Aneta Udriste|ArXiv.org|Jul 10, 2000

Advanced Differential Geometry Research5 references3 citations

TL;DR

This paper introduces a geometric dynamics framework that embeds any given flow (via a vector field) into a conservative dynamical system on a semi-Riemannian manifold using a metric-induced connection and a curvature-like tensor. The key result is that trajectories of the original flow become pregeodesics in a modified metric space, solving a long-standing problem of Poincaré by geometrically realizing flows as geodesics via the Lorentz-UdriÈşte world-force law.

ABSTRACT

Recall that a vector field on an n-dimensional differentiable manifold M is a mapping X defined on M with values in the tangent bundle TM that assigns to each point $x\in M$ a vector X(x) in the tangent space $T_x M$. A vector field may be interpreted alternatively as the right-hand side of an autonomous system of first-order ordinary differential equations, i.e., a flow. Now we show that any flow can be enveloped by a conservative dynamics using a semi-Riemann metric g on M. This kind of dynamics was called {\it geometric dynamics} [7]-[9]. The given vector field, the initial semi-Riemann metric, the Levi-Civita connection, and an associated (1,1)-tensor field are used to build a new geometric structure (e.g., semi-Riemann-Jacobi, semi-Riemann-Jacobi-Lagrange, semi-Finsler-Jacobi, etc) on the manifold M ensuring that all the trajectories of a geometric dynamics are pregeodesics (Lorentz-Udrişte world-force law). Implicitly, we solved a problem rised first by Poincaré: find a suitable geometric structure that converts the trajectories of a given vector field into geodesics.

Motivation & Objective

• To embed any given flow (defined by a vector field) into a conservative dynamical system on a semi-Riemannian manifold.
• To solve Poincaré's problem of finding a geometric structure that transforms flow trajectories into geodesics.
• To construct a geometric dynamics framework using the Levi-Civita connection, a metric, and a (1,1)-tensor field F to characterize flow helicity.
• To demonstrate that the resulting dynamics are conservative and governed by a common Hamiltonian, with trajectories being pregeodesics in a conformally rescaled metric.

Proposed method

• Define a flow as a first-order system dx/dt = X(x), where X is a smooth vector field on a manifold M.
• Equip M with a semi-Riemannian metric g to define energy f = ½g(X,X), classifying X as timelike, causal, null, or spacelike.
• Use the Levi-Civita connection ∇ to prolong the flow into a second-order system via ∇X/dt = ∇_X X.
• Introduce the (1,1)-tensor field F = ∇X − g⁻¹⊗g(∇X) to characterize the helicity of the vector field X.
• Modify the prolonged system to ∇²x/dt² = grad f + F(dx/dt), yielding a conservative dynamical system with a common Hamiltonian H = ½g(dx/dt, dx/dt) − f.
• Show that solutions are horizontal pregeodesics in a Riemann-Jacobi-Lagrange manifold (M\E, g_ij, N_j^i), where g_ij = (H₀ + f)δ_ij and N_j^i = −F_j^i.

Experimental results

Research questions

• RQ1 Can any given flow (vector field) be embedded into a conservative dynamical system using geometric structures on a manifold?
• RQ2 Is there a geometric structure that transforms the trajectories of a given vector field into geodesics or pregeodesics?
• RQ3 What role does the (1,1)-tensor field F = ∇X − g⁻¹⊗g(∇X) play in characterizing the dynamics of the flow?
• RQ4 How can the Hamiltonian and Lagrangian formulations be unified for such geometric dynamics?
• RQ5 Under what conditions do the trajectories of the modified system become pregeodesics in a conformally rescaled metric?

Key findings

• The trajectories of the dynamical system ∇²x/dt² = grad f + F(dx/dt) are pregeodesics in the semi-Riemann-Jacobi manifold (M\E, ḡ = (H₀ + f)g), proving the Lorentz-Udrişte world-force law.
• When F = 0, the system reduces to ∇²x/dt² = grad f, and the trajectories are extremals of the Lagrangian L = ½g(dx/dt, dx/dt) + f(x).
• When F ≠ 0, the system ∇²x/dt² = grad f + F(dx/dt) is governed by the Lagrangian L = ½g(dx/dt − X, dx/dt − X), with the same Hamiltonian H = ½g(dx/dt, dx/dt) − f(x) for both cases.
• The ABC flow, a chaotic fluid flow, is shown to generate a geometric dynamics whose solutions are horizontal pregeodesics in the Riemann-Jacobi-Lagrange manifold (R³\E, g_ij, N_j^i) with g_ij = (H + f)δ_ij.
• The solutions of the ABC geometric dynamics are pregeodesics in a conformally rescaled metric, with the equilibrium set E defined by sin x₁ sin x₂ sin x₃ + cos x₁ cos x₂ cos x₃ = 0.
• The system is conservative, with the Hamiltonian H constant along trajectories, and boundary-value problems exist for H < 0, H = 0, and H > 0, even in Riemannian settings.