Vincenzo TIBULLO | FUNDAMENTALS OF MATHEMATICAL PHYSICS
Vincenzo TIBULLO FUNDAMENTALS OF MATHEMATICAL PHYSICS
cod. 0522200046
FUNDAMENTALS OF MATHEMATICAL PHYSICS
| 0522200046 | |
| DEPARTMENT OF MATHEMATICS | |
| EQF7 | |
| MATHEMATICS | |
| 2024/2025 |
| OBBLIGATORIO | |
| YEAR OF COURSE 1 | |
| YEAR OF DIDACTIC SYSTEM 2018 | |
| AUTUMN SEMESTER |
| SSD | CFU | HOURS | ACTIVITY | |
|---|---|---|---|---|
| MAT/07 | 6 | 48 | LESSONS |
| Exam | Date | Session | |
|---|---|---|---|
| ISTITUZIONI DI FISICA MATEMATICA | 09/01/2025 - 09:00 | SESSIONE ORDINARIA | |
| ISTITUZIONI DI FISICA MATEMATICA | 09/01/2025 - 09:00 | SESSIONE DI RECUPERO | |
| ISTITUZIONI DI FISICA MATEMATICA | 30/01/2025 - 09:00 | SESSIONE ORDINARIA | |
| ISTITUZIONI DI FISICA MATEMATICA | 30/01/2025 - 09:00 | SESSIONE DI RECUPERO | |
| ISTITUZIONI DI FISICA MATEMATICA | 20/02/2025 - 09:00 | SESSIONE ORDINARIA | |
| ISTITUZIONI DI FISICA MATEMATICA | 20/02/2025 - 09:00 | SESSIONE DI RECUPERO |
| Objectives | |
|---|---|
| GENERAL OBJECTIVE THE COURSE AIMS TO INTRODUCE STUDENTS TO THE STUDY OF ANALYTICAL MECHANICS, BOTH IN LAGRANGIAN AND HAMILTONIAN FORMALISM, AND TO THE INTRODUCTION TO THE STUDY OF DYNAMIC SYSTEMS AND STABILITY. KNOWLEDGE AND UNDERSTANDING THE MAIN AIM OF THE COURSE IS TO DERIVE THE LAGRANGE EQUATIONS, STUDY THEIR PROPERTIES AND STUDY THE CONSEQUENCES DERIVING FROM THE PRESENCE OF SYMMETRIES, DERIVE THE HAMILTON EQUATIONS USING THE LEGENDRE TRANSFORM AS A TOOL, STUDY THEIR PROPERTIES, INTRODUCE VARIATIONAL CALCULUS AND VARIATIONAL FORMULATION OF MECHANICS, STUDY THE GROUP OF CANONICAL TRANSFORMATIONS AND INTRODUCE THE SYMPLECTIC CALCULUS, INTRODUCE THE STABILITY OF DYNAMIC SYSTEMS ACCORDING TO LYAPUNOV AND THE CONDITIONS FOR STABILITY AND INSTABILITY. ABILITY TO APPLY KNOWLEDGE AND UNDERSTANDING DURING THE COURSE, MECHANICAL PROBLEMS WILL BE PROPOSED FOR WHICH IT WILL BE NECESSARY TO IDENTIFY THE NUMBER OF DEGREES OF FREEDOM, CHOOSE THE LAGRANGIAN COORDINATES AND WRITE THE LAGRANGE EQUATIONS, DETERMINE THE EQUILIBRIUM POSITIONS AND DETERMINE THEIR STABILITY. ADDITIONALLY STUDENTS WILL BE ABLE TO DETERMINE WHETHER A HAMILTONIAN COORDINATE TRANSFORMATION IS CANONICAL OR GENERATE CANONICAL TRANSFORMATIONS VIA A GENERATING FUNCTION. AUTONOMY OF JUDGEMENT TEACHING ALSO AIMS TO PROMOTE STUDENTS' INDEPENDENT JUDGMENT. THE DEVELOPMENT OF CRITICAL THINKING AND AUTONOMOUS REASONING ARE AMONG THE TRANSVERSAL PROCESSES UNDERLYING THE PLANNED ACTIVITIES. STUDENTS ARE ENCOURAGED TO APPROACH LESSONS WITH A CRITICAL AND INTERACTIVE SENSE, STIMULATED TO MAKE DECISIONS, SELECT, EVALUATE AND DISCERN IN LINE WITH THEIR OBJECTIVES. ALL THIS CONTRIBUTES TO MAKING THEM MORE RESPONSIBLE AND PROMOTING THEIR INDEPENDENT JUDGEMENT. COMMUNICATION SKILLS THE STUDENT WILL BE ABLE TO: - COMMUNICATE USING TECHNICAL-SCIENTIFIC LANGUAGE, - COMMUNICATE USING SYMBOLIC MATHEMATICAL LANGUAGE, - REPRESENT AND COMMUNICATE, THROUGH TEXTS OR VIRTUAL TOOLS, THE RESULTS OF RESEARCH OR OWN ELABORATIONS. LEARNING ABILITY TEACHING ALSO AIMS TO DEVELOP SKILLS THAT ARE TRANSFERABLE TO OTHER CONTEXTS SUCH AS ABSTRACTION, CRITICAL THINKING, CREATIVE THINKING, COMMUNICATION SKILLS, COLLABORATION, COOPERATION. AT THE END OF THE COURSE THE STUDENT WILL BE ABLE TO: - HAVE A PROACTIVE ATTITUDE TOWARDS A PROBLEM OF ANY KIND - ORGANIZE YOUR STUDY AND KNOW HOW TO FRAME A TOPIC - APPROACH PROBLEMS CRITICALLY - UNDERSTAND MORE ADVANCED TEXTS - COMMUNICATE YOUR IDEAS - FEEL MORE AUTONOMOUS - ARGUE - REVIEW AND SELECT |
| 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 (4 HOURS) FREE AND CONTRAINED SYSTEMS. CONSTRAINTS AND THEIR CLASSIFICATION. POSSIBLE AND VIRTUAL DISPLACEMENTS. IDEAL CONSTRAINTS. LAGRANGE EQUATIONS (17 HOURS) 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. CONSERVATION LAWS. NOETHER'S THEOREM. HAMILTON EQUATIONS (5 HOURS) LEGENDRE TRANSFORMATIONS. HAMILTON CANONICAL EQUATIONS. CYCLIC COORDINATES. POISSON BRACKETS. FIRST INTEGRALS OF MOTION. VARIATIONAL FORMULATION OF MECHANICS (4 HOURS) VARIATION OF A FUNCTIONAL. EXTREMA OF A FUNCTIONAL. NECESSARY CONDITIONS FOR THE MINIMUM OF A FUNCTIONAL. EULER-LAGRANGE EQUATIONS. HAMILTON PRINCIPLE. CANONICAL TRANSFORMATIONS (12 ORE) CANONICAL AND COMPLETELY CANONICAL TRANSFORMATION. GENERATING FUNCTIONS. TRANSFORMATIONS PRESERVING POISSON PARENTHESES. SIMPLECTIC MATRICES, SIMPLECTIC INNER PRODUCT, SIMPLECTIC GRADIENT. TRANSFORMATIONS WITH SIMPLECTIC JACOBIAN. HAMILTON-JACOBI EQUATION. DYNAMICAL SYSTEMS AND STABILITY (6 HOURS) 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 ASSESSING THE KNOWLEDGE AND ABILITY TO UNDERSTAND THE CONCEPTS EXPOSED DURING THE LESSONS AND THE ABILITY TO APPLY SUCH KNOWLEDGE AND FORMULATE THE DIFFERENTIAL EQUATIONS THAT DESCRIBE THE DYNAMICS OF MATERIAL SYSTEMS. THE EXAM CONSISTS OF A SELECTIVE WRITTEN TEST AND AN ORAL INTERVIEW. THE WRITTEN TEST PROPOSES A SINGLE EXERCISE IN WHICH, HAVING DESCRIBED A MECHANICAL SYSTEM WITH ONE OR TWO DEGREES OF FREEDOM, IT IS NECESSARY TO CHOOSE THE APPROPRIATE LAGRANGE COORDINATES, TO WRITE THE LAGRANGE EQUATIONS FOR THE SYSTEM, TO DETERMINE ANY POSITIONS OF EQUILIBRIUM, IN THE PARTICULAR CASE IN WHICH SOME CONDITIONS ON THE SYSTEM PARAMETERS ARE VALID, AND DETERMINE THE STABILITY OF THE IDENTIFIED EQUILIBRIUM POSITIONS. THE TIME AVAILABLE IS ONE HOUR AND THIRTY MINUTES AND THE GRADE IS ASSIGNED WITH A BAND RANGING FROM "A", AS THE MAXIMUM RATING, TO "D", AS THE MINIMUM SUFFICIENT RATING, OR ALTERNATIVELY "FAIL". PASSING THE TEST WITH A MINIMUM "D" GRADE IS A NECESSARY CONDITION TO ACCESS THE ORAL INTERVIEW. THE ORAL INTERVIEW EVALUATES THE KNOWLEDGE ACQUIRED. THE DURATION IS ON AVERAGE THIRTY MINUTES AND THE QUESTIONS ARE RELATED TO ALL THE TOPICS OF THE COURSE, THE DEFINITIONS, THEOREMS AND THEIR PROOFS. PARTICULAR ATTENTION IS ATTRIBUTED TO THE PROPERTY OF LANGUAGE, THE RIGOR AND PRECISION OF THE DEMONSTRATIONS AND THE DEGREE OF DEPTH OF THE UNDERSTANDING OF THE TOPICS SHOWN. THE ASSESSMENT OF THE ORAL INTERVIEW CORRESPONDS TO THE FINAL EVALUATION AND TAKES INTO ACCOUNT THE EVALUATION OF THE WRITTEN TEST. THE FINAL RATING IS EXPRESSED IN THIRTIETH AND RANGES FROM 30, AS THE MAXIMUM RATING, TO 18, AS THE MINIMUM SUFFICIENT RATING, OR ALTERNATIVELY "FAILURE TO PASS". THE EVALUATION OF THE WRITTEN TEST IS TAKEN INTO ACCOUNT WITH A WEIGHT EQUAL TO ONE THIRD OF THE OVERALL EVALUATION. LAUDE IS AWARDED WHEN THE MAXIMUM SCORE IS REACHED BY SHOWING NOTABLE DEPTH OF UNDERSTANDING OF THE TOPICS AND INDEPENDENT JUDGMENT. |
| Texts | |
|---|---|
| REFERENCE TEXTS: - TEACHER'S NOTES ADDITIONAL TEXTS: - MAURO FABRIZIO, ELEMENTI DI MECCANICA CLASSICA, ZANICHELLI - FELIX GANTMACHER, LEZIONI DI MECCANICA ANALITICA, ED. RIUNITI - ALBERTO STRUMIA, MECCANICA RAZIONALE - PARTE II, ED. NAUTILUS - HANDOUTS BENETTIN, GALGANI, GIORGILLI - ANTONIO FASANO, STEFANO MARMI, ANALYTICAL MECHANICS, OXFORD UNIVERSITY PRESS |
| More Information | |
|---|---|
| THE COURSE IS HELD IN ITALIAN. EMAIL ADDRESS OF THE TEACHER: VTIBULLO@UNISA.IT |
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