[Paper Review] Richness or Semi-Hamiltonicity of quasi-linear systems which are not in evolution form

Michael Michael, Bialy|arXiv (Cornell University)|Jan 31, 2011

Control and Stability of Dynamical Systems6 references3 citations

TL;DR

This paper generalizes the concepts of Richness and Semi-Hamiltonicity to quasi-linear systems not in evolution form by introducing a rotationally invariant condition on characteristic angles. It proves that under this condition, Riemann invariants satisfy Riccati equations along characteristics, enabling blow-up analysis even when both coefficient matrices degenerate. The key contribution is a generalized framework for analyzing hyperbolic systems in geometric contexts, such as polynomial integrals of geodesic flows on the 2-torus.

ABSTRACT

The aim of this paper is to consider quasi-linear systems which are not in the form of evolution equations. We propose new condition of Richness or Semi-Hamiltonicity for such a system and prove that the blow up analysis along characteristic curves can be performed for it in an analogous manner. This opens a possibility to use this ansatz also for geometric problems. We apply the results to the problem of Polynomial integral for geodesic flows on the 2-torus.

Motivation & Objective

• To extend the theory of Rich and Semi-Hamiltonian systems beyond evolution form, where one of the coefficient matrices is non-degenerate.
• To define a rotationally invariant condition for Richness that applies even when both matrices $ A(u) $ and $ B(u) $ degenerate.
• To establish that under this generalized condition, the derivatives of Riemann invariants satisfy Riccati equations along characteristic curves.
• To apply the framework to the problem of polynomial integrals for geodesic flows on the 2-torus, particularly in conformal coordinates.
• To show that the system can be written in conservation law form if and only if the generalized Richness condition holds.

Proposed method

• Introduce a new definition of Richness based on the characteristic angles $ heta_i $ of the system, requiring the condition $ rac{ heta_{j,i}}{ an( heta_i - heta_j)} $ to satisfy a symmetric derivative identity.
• Define Riemann invariants $ r_i $ via a regular change of variables such that $ L_{v_i} r_i = 0 $, where $ v_i $ are unit vector fields along characteristics.
• Derive a Riccati equation for the orthogonal derivative $ w_i = L_{v_i^ot} r_i $, involving a potential function $ G_i $ defined by $ rac{ heta_{j,i}}{ an( heta_i - heta_j)} $.
• Use the commutator identity $ [v_i, v_i^ot] $ to derive the evolution of $ w_i $, leading to the Riccati structure.
• Apply the generalized framework to the geodesic flow on the 2-torus in conformal coordinates, showing that the resulting system satisfies the generalized Richness condition.
• Prove that the system is Rich in the new sense if and only if it can be written as $ n $ conservation laws $ (g_i)_x + (h_i)_y = 0 $, via a change of variables and invertible matrix multiplication.

Experimental results

Research questions

• RQ1 Can the concept of Richness or Semi-Hamiltonicity be extended to quasi-linear systems not in evolution form, where both coefficient matrices may degenerate?
• RQ2 Does a rotationally invariant condition on characteristic angles allow for blow-up analysis along characteristics even in the presence of matrix degeneracy?
• RQ3 Is the generalized Richness condition equivalent to the system being expressible in conservation law form?
• RQ4 How can this framework be applied to geometric problems such as finding polynomial integrals of geodesic flows on the 2-torus?
• RQ5 What is the behavior of the Riccati equation for the orthogonal derivative of Riemann invariants in such systems, especially when genuine nonlinearity fails?

Key findings

• The generalized Richness condition (Φ) is invariant under rotations of the plane and reduces to the classical condition (R) when the system is in evolution form.
• Under the generalized Richness condition, the orthogonal derivative $ w_i = L_{v_i^ot} r_i $ satisfies the Riccati equation $ L_{v_i}( ext{exp}(-G_i)w_i) + ext{exp}(G_i)rac{ heta_{i,i}}{ an( heta_i - heta_j)}( ext{exp}(-G_i)w_i)^2 = 0 $, where $ G_i $ is defined via $ rac{ heta_{j,i}}{ an( heta_i - heta_j)} $.
• The system is Rich in the new sense if and only if it can be written in conservation law form $ (g_i)_x + (h_i)_y = 0 $, via a regular change of variables and multiplication by an invertible matrix.
• For the geodesic flow on the 2-torus in conformal coordinates, the resulting quasi-linear system satisfies the generalized Richness condition, even though it is not in evolution form.
• The polynomial $ P = ext{det}(eta A - eta B) $ is non-vanishing identically, ensuring the system remains strictly hyperbolic even when $ A $ and $ B $ degenerate at some points.
• The example with $ n=3 $ shows that the system can be written in conservation law form, and thus satisfies the generalized Richness condition, even when both $ A $ and $ B $ are singular at some points.