A xeometría diferencial dun modelo de movemento da bicicleta

Abstract

[GL] Neste traballo estudaremos o modelo sinxelo de movemento da bicicleta sobre o plano. Consideraremos un segmento de lonxitude prefixada que representará ao cadro da bicicleta. Impoñeremos que o vector velocidade do extremo traseiro sexa paralelo a este. Fixando o percorrido da roda dianteira e a lonxitude do cadro, a traxectoria da roda traseira determínase de forma única a través dunha ecuación diferencial de primeiro orde de tipo Riccati, a ecuación da bicicleta. O traballo consistirá, basicamente, na lectura e comprensión de parte dos contidos (esencialmente en dimensión 2) do artigo [4] con conceptos de xeometría diferencial estudados no grao.[EN] This final degree work focuses on the simple model of movement that the bicycle projects on the plane. We will consider for this aim a predetermined segment with fixed length that will represent the bicycle frame. We will impose that the velocity vector of the rear end is parallel to it. By fixing the travel of the front wheel and the length of the frame, the trajectory of the rear wheel is uniquely determined through a first-order differential equation (Riccati type), the equation of the bike. This work will consist, essentially, of reading and understanding part of the contents (basically in dimension 2) of article [4] with differential geometry concepts studied in the degree.