ANALYTICAL MECHANICS
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- VINCENZO COSCIA
- Credits
- 6
- Didactic period
- Primo Semestre
- SSD
- MAT/07
Training objectives
- Aim of the course is to acquire basic notions and methods of Analytical Mechanics, especially on those topics that are used in the classes of Relativity and of basic Quantum Mechanics.
The main tools the students will learn concern:
- introductory multilinear and tensor algebra, mainly referred to euclidean tensors and to their geometric applications
- kinematic of material systems with emphasis to the properties of space-time transformation
- quick recall of the principles of Newtonian mechanics of particles and rigid bodies systems
- Lagrangian mechanics of holonomic systems, with a brief introduction to differential geometry and to the calculus of variations. Covariance properties of Lagrange equations
- Legendre transformations and Hamiltonian mechanics. Canonical structure of the equations of motion. Poisson brackets and conservation laws.
- Equilibrium and stability
At the end of the class the student will be able to:
- perform simple operations in vector and tensor algebra, determine the transformation properties among general coordinate systems
- write down the Lagrange equations of holonomic systems
find out qualitative properties of the motion in Lagrangian formalism, use the first integrals of motion to determine the conservations laws
- derive the Hamilton equations of motion, investigate the properties of the Poisson's parenthesis, find out the the relations between symmetry and invariance properties
- determine the equilibrium configurations and study their stability properties Prerequisites
- Working knowledge of linear and vector algebra, differential and integral calculus with many coordinates.
In order to take the exam of Analytical Mechanics it is required to have passed the exam of General Physics I. Course programme
- The course consists in 60 class hours (6 CFU), part of theory and part of problem solving practice. In details:
- Course motivation, difficulties and overcoming in Newtonian mechanics (2 hours)
- Briefs on linear algebra. Vector spaces, linear functionals and duals. Contravariant components. Euclidean end properly euclidean spaces. Covariant components (4 hours)
- Bilinear transformations algebra. Tensor spaces. Contravariant, covariant and mixed rank 2 tensors. Transformation properties of rank 2 tensors. Euclidean tensors. Metric tensor and its properties (4 hours)
- Eigenvalues, eigenvectors and spectral properties of rank 2 tensors. Orthogonal transformations. Properties of orthogonal transformations and special orthogonal group in R^3 (4 hours)
- Punctual spaces. Frames. Orthogonal coordinate systems. Properties and calculus in punctual spaces (2 hours)
- Kinematic of material systems. Constrained systems and classification of constraints. Degree of freedom. Holonomic systems and lagrangian coordinates. Applications to rigid bodies kinematics. Relative kinematics (8 hours)
- Principles of Newtonian mechanics. Law of force. Fundamental equation and its covariance properties. Balance equations. Applications (6 hours)
- D’Alembert-Lagrange equation. Ideal constraints. Lagrangian mechanics. Lagrange equations for holonomic bilateral systems. Covariance properties of the lagrangian function. Conservation laws and first integrals (8 hours)
- Introductory differential geometry. Configuration space as a differentiable manifold, tangent and cotangent spaces. Riemannian metrics (8 hours)
- Introductory variations calculus. Lagrange equations as extremals of the action (4 hours)
- Equilibrium configurations. Stability of motion and of equilibria. Sufficient conditions for the equilibrium stability. Normal modes (4 hours)
- Hamilton equations and their properties. Poisson brackets and canonical structure. Symmetry and conservation laws. Classical analytical mechanics in the foundation of quantum mechanics (8 hours) Didactic methods
- Class lectures that will include both theoretical presentations and exercises.
Learning assessment procedures
- Written/oral examination. Passing the written test is mandatory to access the final (oral) exam.
The written examination consists in the solution of a simple problem on different point. Specifically:
Lagrange & Hamilton equations, Equilibrium, Stability, Small oscillations, Conservation laws
The final examination has the aim to ascertain an operative knowledge of the whole contents of the course. The final grade will take into account in a critical and non-automatic way the performances in both (written and oral) examinations. Reference texts
- Lecture notes
Fasano A., Marmi S., Analytical mechanics, Oxford University Press, 2006
Jeevanjee N., An Introduction to Tensors and Group Theory for Physicists, Birkhäuser, 2011
Johns O.D., Analytical Mechanics for Relativity and Quantum Mechanics, Oxford Graduate Texts, 2005
F.R. Gantmacher, Lezioni di Meccanica Analitica, Editori Riuniti, 1980
A. Romano, M. Furnari, The Physical and Mathematical Foundations of the Theory of Relativity, Birkhäuser, 2019
B. Ferretti, Le radici classiche della meccanica quantica, Bollati Boringhieri, 1980
