arXiv subjects

Teresinha J. Stuchi

Publications and source records attributed to Teresinha J. Stuchi.

3 recordsLinked to original sources

arXiv2024

Non-integrability of the axisymmetric Bianchi IX cosmological model via Differential Galois Theory

We investigate the integrability of an anisotropic universe with matter and cosmological constant formulated as Bianchi IX models. The presence of the cosmological constant causes the existence of a critical point in the finite part of the phase space. The separatrix associated to this Einstein's static universe is entirely contained in an invariant isotropic plane forgetting the singularity at the origin. This invariant plane of isotropy is an integrable sub-space of the Taub type. In this paper we analyse the differential Galois group of the second order variational equations to this plane in order to apply the integrability theorem of the second author with Ramis and Sim\' o. The main result is that the model is non-integrable by meromorphic functions.

math.DS↗

arXiv2021

A note on the integrability of exceptional potentials via polynomial bi-homogeneous potentials

This paper is concerned with the polynomial integrability of the two-dimensional Hamiltonian systems associated to complex homogeneous polynomial potentials of degree $k$ of type $V_{k,l}=α(q_2-i q_1)^l (q_2+iq_1)^{k-l}$ with $α\in\mathbb{C}$ and $l=0,1,\dots, k$, called exceptional potentials. Hietarinta \cite{Hietarinta1983} proved that the potentials with $l=0,1,k-1,k$ and $l=k/2$ for $k$ even are polynomial integrable. We present an elementary proof of this fact in the context of the polynomial bi-homogeneous potentials, as was introduced by Combot et al. \cite{Combot2020}. In addition, we take advantage of the fact that we can exchange the exponents to derive an additional first integral for $V_{7,5}$, unknown so far. The paper concludes with a Galoisian analysis for $l=k/2$.

math.DS↗

arXiv2017

About simple variational splines from the Hamiltonian viewpoint

In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curve's acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case $Q$ is the $2$-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.

math.SG↗