The class is devoted to a modern study of classical mechanics from a mathematical point of view. The aim of the class is to introduce the tools and techniques of global and numerical analysis, differential geometry and dynamical systems to formalise a model of classical mechanics.
At the end of the class a student should be able to construct a model of physical phenomena of mechanical type, write the equations of motion in Lagrangian and Hamiltonian form and analyse the dynamical aspects of the problem.
• Introduction. At the beginning of the course we will quickly review some basic aspects of dynamical systems using the modern tools of differential geometry and global analysis. Vector fields on a manifolds, flow and conjugation of vector fields. First integrals, foliation of the phase space and reduction of order for a ODE. 1-dimensional mechanical systems.
• Newtonian mechanics. The structure of the Galilean space-time and the axioms of mechanics. Systems of particles: cardinal equations. Conservative force fields. Mass particle in a central field force and the problem of two bodies.
• Variational principles. Introduction to the calculus of variations: Hamilton’s principle and the equivalence between Newton and Lagrangian equations for conservative systems. Legendre transformation and Hamilton equations.
• Lagrangian mechanics on manifolds. Constrained systems: d’Alembert principle and Lagrange equations. Models of constraints and their equivalence. Invariance of Lagrange equations for change of coordinates. Jacobi integral. Noether’s Theorem, conserved quantities and Routh’s reduction.
• Hamiltonian mechanics. Hamilton equations, Poisson brackets. Noether’s Theorem from the Hamiltonian point of view.
• Rigid bodies. Orthonormal basis, orthogonal and skew-symmetric matrices. Space and body frame: angular velocities. Cardinal equations in different reference frames. A model for rigid bodies. Euler’s equations.
Some qualitative numerical aspects will also been investigated. The course will also include seminars in geometric mechanics, geometric control theory and applications to robotics and surgical robotics.
|A. Fasano and S. Marmi||Analytical Mechanics: an Introduction.||Oxford University Press||2006||Graduate Texts|
|R. Abraham, J.E. Marsden and T.S. Ratiu||Manifolds, tensor analysis, and applications. (Edizione 3)||Applied Mathematical Sciences, 75 Springer–Verlag||1988||Testo utile nella fase introduttiva e di richiami o approfondimenti di Geometria Differenziale.|
|V.I. Arnol'd||Mathematical Methods of Classical Mechanics||Springer-Verlag||1989||Graduate Texts in Mathematics 60|
The exam will be divided in two part: Part A. consists on a written text with two practical or theoretical questions followed by (Part B.) an oral examination where the written examination is discussed and other aspects are explored.