Mecânica Geométrica

Objectives

Computing the Levi-Civita connection of a Riemannian manifold. Defining mechanical system on a Riemannian manifold and computing the corresponding trajectories. Being familiar with some classical examples of conservative systems, e.g. the rigid body with one fixed point. Identifying holonomic and nonholonomic constraints. Writing the Euler-Lagrange equations and applying Noether's Theorem. Being familiar with the basic notions of symplectic geometry and Poisson geometry. Identifying completely integrable systems. Using symmetry to perform reduction on Poisson manifolds. Being familiar with the basic notions of Geometric Control Theory (controlabiliy, accessibility and stabilization), including nonholonomic systems. Applying the Maximum Principle to compute optimal controls. Being familiar with the basic notions of Lorentzian geometry and the Einstein equation, the Schwarzschild solution and its relation with black holes, and the Friemann-Lemaitre-Robertson-Walker cosmological models.

Syllabus

Elements of Differential Geometry: Differentiable manifolds. Connections and parallelism. Riemannian Manifolds and Levi-Civita connection. Mechanical Systems on Riemannian Manifolds: Definition and classical examples. Conservative systems. Holonomic and nonholonomic constraints. Hamiltonian Mechanics: Lagrangean mechanics on manifolds. Noether's Theorem. Legendre's Transformation. Symplectic geometry and Poisson geometry. Completely integrable systems. Symmetry and reduction. Geometric Control: Controlability, accessibility and stabilization. Control of nonholonomic systems. Optimal control and the Maximum Principle. Relativity: Minkowski's spacetime. Lorentzian manifolds and Einstein's equation. The Schwarzschild solution. Black holes. Cosmology.

Prerequisites

Basic knowledge of Riemannian Geometry.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.