Matematica | FUNDAMENTALS OF MATHEMATICAL PHYSICS

Matematica FUNDAMENTALS OF MATHEMATICAL PHYSICS

 0522200046 DIPARTIMENTO DI MATEMATICA EQF7 MATHEMATICS 2022/2023

 OBBLIGATORIO YEAR OF COURSE 1 YEAR OF DIDACTIC SYSTEM 2018 AUTUMN SEMESTER
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY MAT/07 6 48 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 VINCENZO TIBULLO T
ExamDate
ISTITUZIONI DI FISICA MATEMATICA19/06/2023 - 09:00
ISTITUZIONI DI FISICA MATEMATICA10/07/2023 - 09:00
Objectives
KNOWLEDGE AND UNDERSTANDING

TO KNOW AND TO UNDERSTAND THE METHODS OF ANALYTICAL MECHANICS, BOTH IN LAGRANGIAN AND CANONICAL FORMALISM.
TO KNOW AND TO UNDERSTAND VARIATIONAL METHODS AND PRINCIPLES.
TO KNOW AND TO UNDERSTAND LYAPUNOV'S METHODS FOR STABILITY.

APPLYING KNOWLEDGE AND UNDERSTANDING

TO BE ABLE TO APPLY THE METHODS OF LAGRANGIAN AND CANONICAL FORMALISM TO SOLVE SIMPLE MECHANICAL PROBLEMS WITH ONE OR TWO DEGREES OF FREEDOM.
Prerequisites
FOR THE SUCCESSFUL ACHIEVEMENT OF OBJECTIVES, STUDENTS ARE REQUIRED BASIC MATHEMATICAL KNOWLEDGE RELATED TO MATHEMATICAL ANALYSIS AND MOREOVER OF RATIONAL MECHANICS.
Contents
REVIEW OF RATIONAL MECHANICS.
FREE AND CONTRAINED SYSTEMS.
CONSTRAINTS AND THEIR CLASSIFICATION.
POSSIBLE AND VIRTUAL DISPLACEMENTS.
IDEAL CONSTRAINTS.
SYMBOLIC EQUATION OF DYNAMICS AND D'ALEMBERT PRINCIPLE.
SYMBOLIC EQUATION OF STATIC AND PRINCIPLE OF VIRTUAL WORK.
HOLONOMIC SYSTEMS.
INDEPENDENT COORDINATES.
GENERALIZED FORCES.
LAGRANGE EQUATIONS AND APPLICATIONS.
KINETIC ENERGY OF A HOLONOMIC SYSTEM AND STUDY OF LAGRANGE EQUATION.
KINETIC ENERGY THEOREM FOR A HOLONOMIC SYSTEM.
POTENTIAL, GYROSCOPIC AND DISSIPATIVE FORCES.
LAGRANGE EQUATIONS FOR POTENTIAL FORCES.
GENERALIZED POTENTIAL.
LEGENDRE TRANSFORMATIONS.
HAMILTON CANONICAL EQUATIONS.
CYCLIC COORDINATES.
ASSOCIATIVE, COMMUTATIVE, LIE AND POISSON ALGEBRAS.
POISSON BRACKETS.
FIRST INTEGRALS OF MOTION.
VARIATION OF A FUNCTIONAL.
EXTREMA OF A FUNCTIONAL.
NECESSARY CONDITIONS FOR THE MINIMUM OF A FUNCTIONAL.
EULER-LAGRANGE EQUATIONS.
HAMILTON PRINCIPLE.
CONSERVATION LAWS.
NOETHER'S THEOREM.
CANONICAL AND COMPLETELY CANONICAL TRANSFORMATION.
GENERATING FUNCTIONS.
CANONICAL INVARIANTS.
HAMILTON-JACOBI EQUATION.
DEFINITION OF STABILITY FOR A DYNAMIC SYSTEM.
FIRST LYAPUNOV METHOD FOR STABILITY.
SECOND LYAPUNOV METHOD.
DIRICHLET THEOREM.
SMALL OSCILLATIONS AROUND A STABLE EQUILIBRIUM POSITION.
Teaching Methods
THE COURSE COVERS THEORETICAL LESSONS, WHICH WILL BE PRESENTED DURING THE COURSE TOPICS THROUGH LECTURES (32 HOURS/4 CFU) AND CLASSROOM EXERCISES (16 HOURS/2 CFU), DURING WHICH PROVIDE THE MAIN TOOLS NEEDED FOR SOLVING EXERCISES RELATED TO THE CONTENT OF THE THEORETICAL ASPECTS.
Verification of learning
THE EXAM IS AIMED AT EVALUATING THE KNOWLEDGE AND THE ABILITY TO UNDERSTAND THE CONCEPTS EXPOSED DURING LESSONS AND THE ABILITY TO APPLY SUCH KNOWLEDGE AND TO FORMULATE THE DIFFERENTIAL EQUATIONS DESCRIBING THE DYNAMIC OF MATERIAL SYSTEMS.
THE EXAMINATION IS DIVIDED INTO A SELECTIVE WRITTEN TEST AND IN AN ORAL INTERVIEW. THE WRITTEN TEST PROPOSES SIMPLE EXERCISES AND QUESTIONS WITH OPEN ANSWERS. THE ORAL INTERVIEW EVALUATES THE ACQUIRED KNOWLEDGE.
IN THE FINAL EVALUATION, EXPRESSED IN THIRTIETHS, THE ASSESSMENT OF WRITTEN TEST WEIGHS FOR 40%, WHILE THE ORAL INTERVIEW WEIGHS FOR THE REMAINING 60%.
Texts
- MAURO FABRIZIO, ELEMENTI DI MECCANICA CLASSICA, ZANICHELLI
- FELIX GANTMACHER, LEZIONI DI MECCANICA ANALITICA, ED. RIUNITI
- ALBERTO STRUMIA, MECCANICA RAZIONALE - PARTE II, ED. NAUTILUS