[Paper Review] Normalizers of planar systems with known first integrals

Marco Sabatini|ArXiv.org|Mar 17, 2006

Advanced Differential Equations and Dynamical Systems7 references3 citations

TL;DR

This paper presents a constructive method to compute a normalizer for planar differential systems with a known first integral, enabling explicit computation of the period function's derivative via an integral formula. The key contribution is a canonical normalizer vector field proportional to ∇H / |∇H|², whose associated function μ equals the divergence of the normalizer, allowing direct analysis of period function monotonicity and isochronicity in center-type systems.

ABSTRACT

Given a planar differential system with a first integral, we show how to find a normalizer. For systems with a center, we give an integral formula for the derivative of its period function.

Motivation & Objective

• To develop a systematic method for constructing normalizers of planar vector fields when a first integral is known.
• To enable explicit computation of the derivative of the period function T(z) along trajectories of the normalizer.
• To provide a formula for μ, the normalizing function, in terms of the gradient and Hessian of the first integral H.
• To extend the applicability of normalizer-based methods to systems where standard normalizers (e.g., separable Hamiltonian forms) may fail due to vanishing derivatives.

Proposed method

• Construct a canonical normalizer vector field W = ∇H / |∇H|² for a planar system with a first integral H.
• Derive the associated normalizing function μ using the identity μ = [V, W] · V / |V|².
• Show that μ coincides with the divergence of the normalizer vector field W.
• Prove that the derivative of the period function T along W is T’(H), enabling monotonicity analysis via integration of μ over cycles.
• Establish that the normalizer preserves the level sets of H and is transverse to the flow of V.
• Generalize the construction to non-Hamiltonian systems by expressing the system as a reparametrized Hamiltonian system using a reciprocal integrating factor (RIF).

Experimental results

Research questions

• RQ1 How can one explicitly construct a normalizer for a planar system with a known first integral H?
• RQ2 What is the explicit expression for the normalizing function μ in terms of H and its derivatives?
• RQ3 Can the derivative of the period function T be computed via integration of μ along periodic orbits?
• RQ4 How does the proposed normalizer compare to existing ones (e.g., in separable Hamiltonian systems) in terms of domain of applicability?
• RQ5 Under what conditions does the normalizer remain well-defined even when partial derivatives of H vanish?

Key findings

• A canonical normalizer is constructed as W = ∇H / |∇H|², which is transverse to the vector field V and satisfies [V, W] = μV with μ = div(W).
• The derivative of the period function along W is T’(H), providing a direct method to analyze monotonicity of T.
• The normalizer remains well-defined even when G’(x) or F’(y) vanish in separable Hamiltonian systems, unlike standard normalizers.
• The function μ is explicitly given by μ = Λ_H / |∇H|⁴, where Λ_H is a quadratic form involving second derivatives of H.
• For systems with a reciprocal integrating factor κ, the normalizer is given by W = (−Q, P) / (|V||∇H|), with μ expressed as a rational function of P, Q, and their derivatives.
• The method generalizes to reparametrized Hamiltonian systems, and μ transforms under reparametrization as μ̄ = μ − ∂_W ln κ.