Integrability of certain Hamiltonian systems in $2D$ variable curvature spaces

The objective of this work is to examine the integrability of Hamiltonian systems in $2D$ spaces with variable curvature of certain types. Based on the differential Galois theory, we announce the necessary conditions of the integrability. They are given in terms of arithmetic restrictions on values of the parameters describing the system. We apply the obtained results to some examples to illustrate that the applicability of the obtained result is easy and effective. Certain new integrable examples are given. The findings highlight the applicability of the differential Galois approach in studying the integrability of Hamiltonian systems in curved spaces, expanding our understanding of nonlinear dynamics and its potential applications.

💡 Research Summary

The paper investigates the Liouville integrability of Hamiltonian systems defined on two‑dimensional manifolds whose Gaussian curvature varies with position. Starting from a generic Lagrangian with a metric tensor a₁₁(ξ,η), a₁₂(ξ,η), a₂₂(ξ,η) and a potential V(ξ,η), the authors employ Birkhoff’s theorem to introduce isometric coordinates (q₁,q₂) that bring the Lagrangian to the form L = ½Λ(q₁,q₂)(\dot q₁²+ \dot q₂²) – V(q₁,q₂). The function Λ determines the Gaussian curvature κ = ½Λ⁻¹(∂₁₁Λ+∂₂₂Λ).

The paper classifies previous work into three curvature regimes: zero curvature (flat plane), constant curvature (sphere or pseudosphere), and variable curvature. It then focuses on the most general case, introducing the Hamiltonian

H = ½ ( rⁿ Λ₂²(θ) + Λ₁(θ) ) ( p_r² + p_θ² / r² ) + r^m U(θ) , (1)

where m,n∈ℤ and Λ₁, Λ₂, U are meromorphic functions of the angular variable θ. This family contains as special cases the Hamiltonians studied in earlier papers (e.g., H = ½(p_r² + p_θ²/r²) + V, H = ½(p_r² + p_θ²/r²) + r^m U(θ), etc.).

To test integrability, the authors select a particular angle c such that Λ₁′(c)=Λ₂′(c)=U′(c)=0, which yields a two‑dimensional invariant manifold S = {θ=c, p_θ=0}. On S the energy integral reduces to a one‑degree‑of‑freedom Hamiltonian, and the variational equations along a solution on S decouple. The normal variational equations (NVE) for the transverse variables (θ, p_θ) become a second‑order linear ODE

\ddot Θ + a(r,p_r) \dot Θ + b(r,p_r) Θ = 0 .

After the change of independent variable τ = rⁿ and a scaling z = –(A₂c/A₁c) τ, the NVE is transformed into a Riemann‑P equation

d²Θ/dz² + … = 0 ,

characterised by three pairs of exponents (α,α′), (β,β′), (γ,γ′) at the singular points z = 0,1,∞. The differences of these exponents are denoted ρ = α–α′, σ = γ–γ′, τ = β–β′ and are expressed in terms of the original parameters as

ρ = χ₁ , σ = 3/2 , τ = χ₂ ,

with

χ₁ = ½n⁻¹