Integrability and Chaotic Behavior in Mechanical Billiard Systems

This thesis is devoted to the study of mathematical billiards in the presence of non-constant potentials and their integrability and chaotic behavior. Classical examples of integrable billiards are free billiards in circles and ellipses. In the presence of specific potentials (such as Kepler potential and harmonic (Hooke) potential), there are various known integrable billiard systems. These integrable examples have been found independently in different contexts. In Chapter 2, we illustrate how some of these integrable billiard systems are related to each other by conformal transformations. As an application, we obtain infinitely many billiard systems defined in central force problems which are integrable on a particular energy level. We then explain that the classical Hooke-Kepler correspondence extends to the correspondence between integrable Hooke and Kepler billiards. As a result, we show that any focused conic sections give rise to integrable Kepler billiards which give new examples of integrable Kepler billiards. The conformal transformation technique is applied to Stark-type problems and Euler\u27s two-center problem and provides new examples of integrable mechanical billiards. In Chapter 3 we show that integrable Kepler and Hooke billiard systems on the plane have the corresponding integrable billiard systems on surfaces of constant curvatures. We also establish the integrability of a class of billiard systems defined in the Lagrangian problem, which is the superposition of two Kepler problems and a Hooke problem, on the sphere, in the plane, and in the hyperbolic plane. These results are obtained by the method of projective dynamics and projective billiards. A toy model of billiard systems with a central force problem in the plane and with a line as the reflection wall was proposed by L. Boltzmann to illustrate his ergodic hypothesis. Later, it has been found that not all such systems are ergodic, and it becomes a question whether some of such systems are ergodic. In Chapter 4, we compute the billiard mappings of Boltzmann\u27s billiard systems, and we present some numerical studies on their chaotic behavior and ergodicity. We found some numerical evidence suggesting that some of these systems might be ergodic

Kustaanheimo-Stiefel Transformation, Birkhoff-Waldvogel Transformation and Integrable Mechanical Billiards

The three-dimensional Kepler problem is related to the four-dimensional isotropic harmonic oscillators by the Kustaanheimo-Stiefel Transformations. In the first part of this paper, we study how certain integrable mechanical billiards are related by this transformation. This in part illustrates the rotation-invariance of integrable reflection walls in the three-dimensional Kepler billiards found till so far. The second part of this paper deals with Birkhoff-Waldvogel's Transformation of the three-dimensional problem with two Kepler centers. In particular, we establish an analogous theory of Levi-Civita planes for Birkhoff-Waldvogel's Transformation and showed the integrability of certain three-dimensional 2-center billiards via a different approach.Comment: 25 pages, no figure

Concave Toric Domains in Stark-type Mechanical Systems

It has been shown in [Frauenfelder, 2023] that the bounded component of the energy surface of the planer Stark problem after the Levi-Civita transformation are concave toric domains. In this paper, we present a different approach on the problem of determining concave toric domains in a family of integrable natural mechanical problems in the plane based on the computation of action-variables. We give criteria on the potentials for the bounded components of the energy hypersurfaces to be concave toric domains and apply these criteria to a class of problems.Comment: 12 pages, no figur

Entrepreneurial Mentoring: Informal Mentors and Mentoring Relationship with Entrepreneurs

entoring plays a crucial role in entrepreneurial business development. Studies have shown that mentors support entrepreneurs in identifying goals and achieving objectives. This study clarifies the attributes of ‘informal’ mentoring, wherein the entrepreneur (protégé) and the mentor become partners without a contract. Through an analysis based on a detailed questionnaire survey and indepth semi-structured interviews targeting entrepreneurs, the attributes of informal mentors, frequent mentoring activities, and mentoring outcomes at each entrepreneurial stage are discussed. The results reveal the most common feature of informal mentors who support entrepreneurs and small business managers: male business managers (CEOs/executives and entrepreneurs) who are friends with entrepreneurs and demonstrate the effectiveness of friends as informal mentors, which positively influences various mentoring outcomes. Additionally, the mentoring effects perceived by protégés are distinct in each entrepreneurial stage, and informal mentors are capable of supporting their protégés’ individual stages. This study contributes to identifying and categorizing methods for developing positive mentoring relationships: respectful behavior, bonding activity, and mutually beneficial partnerships