arXiv subjects

Sergio Charles

Publications and source records attributed to Sergio Charles.

3 recordsLinked to original sources

arXiv2019

The Lagrangian Mechanics and Pseudo-Periodicity of The Many-Body Planar Pendulum System

We present the Euler--Langrage equations for a many-body system of coupled planar pendulums. Hence, imposing initial condition data, the equations of motion are linearized and later developed in an idealized model for the pseudo-periodicity of the system as a function of the number of pendulums $N$. The result is empirically corroborated by comparing the model with data obtained via a numerical simulation, and by employing Kane's Method integrator in Python.

math.DS↗

arXiv2019

The Existence of Infinitely Many Geometrically Distinct Non-Constant Prime Closed Geodesics on Riemannian Manifolds

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold $M$. That is, we show that any Riemannian metric on $M$ admits infinitely many prime closed geodesics such that the energy functional $E:ΛM\to\mathbb{R}$ has infinitely many non-degenerate critical points on the free loop space $ΛM$ of Sobolev class $H^1=W^{1,2}$. This result is obtained by invoking a handle decomposition of free loop space and using methods of cellular homology to study its topological invariants.

math.DG↗

arXiv2017

On the Algebro-Geometric Analysis of Meromorphic (1,0)-forms

In this paper, we analyze the theory of meromorphic $(1,0)$-forms $ω\in\mathcal{M}Ω^{(1,0)}(\mathbb{CP}^1).$ Hence, we show that on a compact Riemann surface of genus $g=0,$ isomorphic to $\mathbb{CP}^1,$ every non-constant meromorphic function $f:X\to\mathbb{CP}^1$ has as many zeros as poles, where each is counted according to multiplicities. Such an analysis gives rise to the following result. Invoking the Riemann-Roch theorem for a compact Riemann $X$ with canonical divisor $K,$ it follows that $deg(f)=0$ for any principal divisor $(f):=D$ on $X.$ More precisely, $\ell(D)-\ell(K-D)=deg(D)+1=1$ or $\ell(D)-\ell(K-D)-1=0.$ Furthermore, for a diffeomorphism $η:X\to\mathbb{CP}^1$ of a certain kind, a multistep program is implemented to show $X$ is a compact algebraic variety of dimension one, i.e. a non-singular projective variety. Hence, we adopt a group-theoretic approach and provide a useful heuristic, that is, a set of technical conditions to facilitate the algebro-geometric analysis of simply connected Riemann surfaces $X.$

math.DG↗