A smooth pseudo-gradient for the Lagrangian action functional

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.

On invariant sets in Lagrangian graphs

In this exposition, we show that a Hamiltonian is always constant on a compact invariant connected subset which lies in a Lagrangian graph provided that the Hamiltonian and the graph are smooth enough. We also provide some counterexamples for the case that the Hamiltonians are not smooth enough.

On the Lyapunov instability in Lagrangian dynamics

In the context of mechanical Lagrangian dynamics, we prove a new Lyapunov instability criterion for a non strict local minimum equilibrium point of a smooth potential where the sufficient condition for instability is the existence of a smooth solution of a certain linear PDE derived from the mechanical Lagrangian governing the dynamics. In the presence of a magnetostatic field, we also give an additional sufficient condition for the motion of a charged particle to be Lyapunov unstable.

Thermodynamics of smooth models of pseudo-Anosov homeomorphisms

We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric $t$-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the Central Limit Theorem.

光滑伪旋转在测度理论上与有理无关角度的圆旋转等价

Smooth Pseudo-Rotations Measure-Theoretically Isomorphic to Circle Rotations of Rationally Independent Angle

本文研究了光滑流形上伪旋转与圆旋转的测度等价性，证明了在特定条件下，伪旋转可与无理旋转在测度上等价，并探讨了其在闭环面中的存在性。

Application of Lagrangian mechanics equations for finding of the minimum distance between smooth submanifolds in N-dimensional Euclidean space -- Part II

The method of finding the minimal distance between smooth non crossing submanifolds in N-dimensional Euclidean space are presented. It based on solution of the equations that describe the dynamics of the pair of material points. The dynamical system can be presented as a natural mechanical system determined by Riemannian geometry on the manifold and chosen potential energy. Such an approach makes it possible to find Lyapunov function of the considered system and to formulates the requirements on the form of potential energy that brings to the convergence of the method.