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ANALYTICAL MECHANICS AND STATISTICAL MECHANICS

Alfonso ROMANO ANALYTICAL MECHANICS AND STATISTICAL MECHANICS

0512600033
DEPARTMENT OF PHYSICS "E. R. CAIANIELLO"
EQF6
PHYSICS
2023/2024

OBBLIGATORIO
YEAR OF COURSE 2
YEAR OF DIDACTIC SYSTEM 2017
FULL ACADEMIC YEAR
CFUHOURSACTIVITY
864LESSONS
448EXERCISES
ALFONSO ROMANO T Curriculum
Objectives
THE GENERAL AIMS OF THE COURSE OF “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” ARE:
A) TO PROVIDE AN INTRODUCTION TO ADVANCED FORMULATIONS OF CLASSICAL MECHANICS.
B) TO PROVIDE THE BASIC KNOWLEDGE OF THE STATISTICAL LAWS WHICH GOVERN THE BEHAVIOUR OF THE MANY-PARTICLE SYSTEMS.

KNOWLEDGE AND UNDERSTANDING:
THE COURSE OF “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” AIMS AT FOSTERING A THOROUGH AND APPLICATION-ORIENTED KNOWLEDGE OF THE FOLLOWING TOPICS:
A)LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF CLASSICAL MECHANICS, ALSO INCLUDING THE HAMILTON-JACOBI METHOD;
B)PRINCIPLES AND METHODS OF STATISTICAL PHYSICS, DEVELOPED WITHIN THE GENERAL THEORY OF THE STATISTICAL ENSEMBLES.
IN BOTH CONTEXTS, THE FOCUS WILL BE ON ANALYTICAL METHODS THAT ALLOW A QUANTITATIVE DESCRIPTION OF THE PHYSICAL PHENOMENA OF INTEREST AND ON SOME SIMPLE MODELS TO WHICH THESE METHODS CAN BE APPLIED. THE COURSE ALSO AIMS TO ACCUSTOM STUDENTS TO RIGOROUS REASONING THROUGH THE USE OF VARIOUS APPLICATIONS AND DEMONSTRATION TECHNIQUES.

APPLYING KNOWLEDGE AND UNDERSTANDING:
AT THE END OF THE COURSE, STUDENTS ARE EXPECTED TO HAVE ASSIMILATED THE THEORETICAL KNOWLEDGE PROVIDED, TO BE ABLE TO EXPRESS IT CLEARLY AND CONSCIOUSLY, AND TO BE ABLE TO APPLY IT IN SOLVING EXERCISES. IN PARTICULAR, THE STUDENT WILL BE ABLE TO SOLVE
A) EXERCISES CONCERNING THE STATICS AND DYNAMICS OF SYSTEMS OF PARTICLES AND OF RIGID BODIES SUBJECT TO CONSTRAINTS, ANALYSED THROUGH THE USE OF THE LAGRANGE EQUATIONS, THE HAMILTON EQUATIONS OR THE HAMILTON-JACOBI EQUATION, SUPPLEMENTED WHERE POSSIBLE BY THE IDENTIFICATION OF THE UNDERLYING CONSERVATION LAWS;
B) STATISTICAL PHYSICS EXERCISES, TO BE ANALYSED THROUGH AN APPROPRIATE CHOICE OF THE STATISTICAL ENSEMBLE IN WHICH TO OPERATE AND BY LINKING THE MICROSCOPIC APPROACH USED TO THE MACROSCOPIC LAWS OF THERMODYNAMICS.
Prerequisites
THE MINIMAL BACKGROUND REQUIRED TO ATTEND FRUITFULLY THE COURSES INCLUDES KNOWLEDGE OF VECTOR CALCULUS, MECHANICS OF SYSTEMS OF PARTICLES AND OF RIGID BODIES, AS WELL AS KNOWLEDGE OF THERMODYNAMICS, AS COVERED IN THE COURSES OF THE FIRST YEAR OF THE DEGREE COURSE IN PHYSICS. AS FAR AS MATHEMATICS IS CONCERNED, STUDENTS ARE EXPECTED TO HAVE A GOOD KNOWLEDGE OF THE BASIC NOTIONS OF LINEAR ALGEBRA AND OF THE ANALYSIS OF THE FUNCTIONS OF A SINGLE VARIABLE. ELEMENTARY NOTIONS CONCERNING THEFUNCTIONS OF SEVERAL VARIABLES, WITH PARTICULAR REGARD TO THE CONCEPTS OF PARTIAL DERIVATIVE, TOTAL DIFFERENTIAL AND EXACT DIFFERENTIAL, ARE PROVIDED AS PART OF THE COURSE.
Contents
ANALYTICAL MECHANICS

LAGRANGIAN DYNAMICS (LECT. H. 18, EXERC. H. 6):
CONSTRAINTS. DEGREES OF FREEDOM AND GENERALIZED COORDINATES. D’ALEMBERT’S PRINCIPLE AND PRINCIPLE OF VIRTUAL WORKS. LAGRANGE’S EQUATIONS OF MOTION. CYCLIC COORDINATES. SYMMETRY PROPERTIES AND CONSERVATIONS LAWS: NOETHER’S THEOREM. HAMILTON’S PRINCIPLE. APPLICATIONS OF THE CALCULUS OF VARIATIONS. SYSTEMS WITH NON-HOLONOMIC CONTRAINTS. LAGRANGE’S MULTIPLIERS.

COUPLED OSCILLATION (LECT. H. 6, EXERC. H. 4):
LAGRANGIAN OF COUPLED OSCILLATION AROUND A STABLE EQUILIBRIUM CONFIGURATIONSIMULTANEOUS DIAGONALIZATION OF THE KINETIC AND THE POTENTIAL ENERGY QUADRATIC FORMS. NORMAL MODES OF OSCILLATION. LINEAR TRIATOMIC MOLECULE.

MOTION UNDER CENTRAL FORCES (LECT. H. 4, EXERC. H. 1):
TWO-BODY PROBLEM. LAGRANGIAN, EQUATIONS OF MOTION AND SOLUTION VIA CONSERVATION LAWS. EFFECTIVE POTENTIAL AND ORBIT CLASSIFICATION. ANALYTICAL SOLUTION OF ORBIT DIFFERENTIAL EQUATION. COMPLETE SOLUTION OF THE KEPLER’S PROBLEM.

HAMILTONIAN DYNAMICS (LECT. H. 13, EXERC. H. 4):
LEGENDRE’S TRANSFORMATION AND HAMILTON’S EQUATIONS OF MOTIONS. CYCLIC COORDINATES. POISSON BRACKETS, JACOBI IDENTITY AND JACOBI-POISSON’S THEOREM. DERIVATION OF HAMILTON’S EQUATIONS FROM A VARIATIONAL PRINCIPLE. CANONICAL TRANSFORMATIONS. GENERATING FUNCTION AND CONDITIONS OF CANONICITY.

HAMILTON-JACOBI’S METHOD (LECT. H. 6, EXERC. H. 2):
H-J EQUATION AND HAMILTON’S PRINCIPAL FUNCTION. REDUCED H-J EQUATION FOR TIME INDEPENDENT HAMILTONIANS AND HAMILTON’S CHARACTERISTIC FUNCTION. SOLUTION IN THE PRESENCE OF CYCLIC COORDINATES AND METHOD OF SEPARATION OF THE VARIABLES. APPLICATION TO RELEVANT PROBLEMS (HARMONIC OSCILLATOR, MOTION IN THE PRESENCE OF CENTRAL FORCES, ETC.). ACTION-ANGLE VARIABLES


STATISTICAL MECHANICS

ADVANCED THERMODYNAMICS (LECT. H. 8, EXERC. H. 2):
ENTROPY AND ENERGY REPRESENTATIONS AND CHARACTERISATION OF THE EQUILIBRIUM STATES OF INTERACTING SYSTEMS. FUNDAMENTAL EQUATIONS OF THERMODYNAMICS IN THE TWO REPRESENTATIONS. GIBBS-DUHEM EQUATION. LEGENDRE TRANSFORMATION. THERMODYNAMIC POTENTIALS AND MINIMUM PRINCIPLES. MAXWELL'S RELATIONS. RECALLS OF KINETIC THEORY OF GASES.

STATISTICAL MECHANICS, INTRODUCTORY CONCEPTS (LECT. H. 3):
EQUILIBRIUM STATES. MENTAL COPIES AND STATISTICAL ENSEMBLES. PHASE SPACE AND DENSITY FUNCTION. EQUIPROBABILITY OF MICROSTATES AT FIXED ENERGY. ERGODIC HYPOTHESIS. LIOUVILLE'S THEOREM.

MICROCANONICAL ENSEMBLE (LECT. H. 8, EXERC. H. 3):
SYSTEMS AT FIXED ENERGY. DENSITY FUNCTION. LINK WITH THERMODYNAMICS. TWO-LEVEL SYSTEMS AND THE PROBLEM OF NEGATIVE TEMPERATURES. SPIN SYSTEM IN A MAGNETIC FIELD. VOLUME OF A HYPERSPHERE IN N DIMENSIONS. GAS OF FREE PARTICLES. GIBBS PARADOX. SYSTEM OF HARMONIC OSCILLATORS. THEOREM OF EQUIPARTITION OF ENERGY.

CANONICAL ENSEMBLE (LECT. H. 9, EXERC. H. 4):
SYSTEMS AT FIXED TEMPERATURE. PROBABILITY AND BOLTZMANN FACTOR. PARTITION FUNCTION AND STATISTICAL AVERAGES. FLUCTUATIONS AND EQUIVALENCE BETWEEN THE MICROCANONICAL AND THE CANONICAL ENSEMBLE. APPLICATIONS: TWO-LEVEL SYSTEMS; FREE PARTICLE GAS; FREE PARTICLE GAS IN THE ULTRARELATIVISTIC LIMIT; CLASSICAL AND QUANTUM HARMONIC OSCILLATORS; THE EINSTEIN'S MODEL FOR THE SPECIFIC HEAT; CLASSICAL PARAMAGNET AND CURIE’S LAW.

GRAN-CANONICAL ENSEMBLE (LECT. H. 3, EXERC. H. 1):
SYSTEMS AT FIXED TEMPERATURE AND CHEMICAL POTENTIAL. GRAN-PARTITION FUNCTION AND LINK WITH THE GRAN-CANONICAL POTENTIAL. APPLICATIONS.

QUANTUM STATISTICAL PHYSICS: DERIVATION OF THE BOSE AND FERMI STATISTICS AND MAXWELL-BOLTZMANN CLASSIC LIMIT (LECT. H. 3).

ISING MODEL: EXACT SOLUTION IN THE ONE-DIMENSIONAL CASE AND STUDY OF THE MAGNETIZATION, MEAN-FIELD SOLUTION AND BETHE APPROXIMATION. (LECT. H. 4)
Teaching Methods
ABOUT TWO THIRDS OF THE COURSE ARE DEVOTED TO THEORETICAL LECTURES FOCUSING ON THE PRESENTATION OF THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF CLASSICAL MECHANICS, AS WELL AS OF THE PRINCIPLES AND METHODS OF STATISTICAL MECHANICS.

THE REMAINING PART OF THE COURSE CONSISTS OF LECTURES FOCUSING ON THE APPLICATION OF SUITABLE METHODS OF SOLUTION TO EXERCISES AND PROBLEMS RELATED TO THE TOPICS THEORETICALLY INVESTIGATED.
Verification of learning
VERIFICATION OF THE LEVEL OF LEARNING ACHIEVED BY THE STUDENT IS DONE THROUGH A FINAL EXAMINATION STRUCTURED IN TWO STAGES, RELATING ONE TO THE VERIFICATION OF KNOWLEDGE ACQUIRED IN THE FIELD OF ANALYTICAL MECHANICS AND THE OTHER TO THE VERIFICATION OF KNOWLEDGE ACQUIRED IN THE FIELD OF STATISTICAL MECHANICS.

FOR THE FIRST PART THERE IS A WRITTEN TEST FOLLOWED BY AN ORAL EXAMINATION. THE WRITTEN TEST INVOLVES THE SOLUTION OF TWO EXERCISES, ONE ON SYSTEMS SUBJECT TO CONSTRAINTS TO BE SOLVED WITHIN THE FRAMEWORK OF LAGRANGIAN FORMALISM, AND ONE RELATED TO TOPICS IN HAMILTONIAN DYNAMICS, AND IS THEREFORE AIMED AT ASSESSING THE STUDENT'S ABILITY TO WORK EFFECTIVELY IN BOTH AREAS. NO BOOKS, NOTES OR COMPUTER AIDS MAY BE USED DURING THE TEST. THE MAXIMUM SCORE THAT CAN BE ACHIEVED BY SOLVING EACH OF THE TWO EXERCISES EXACTLY IS 18/30 AND 12/30, RESPECTIVELY. TO BE ADMITTED TO THE ORAL TEST, THE STUDENT MUST HAVE OBTAINED AN OVERALL MARK IN THE WRITTEN TEST OF NOT LESS THAN 18/30.

FOR THE SECOND PART THERE IS A WRITTEN TEST CONSISTING OF TWO OPEN-ENDED QUESTIONS ON THEORETICAL TOPICS DEVELOPED DURING THE COURSE AND TWO APPLICATIONS OF COMPUTATIONAL TECHNIQUES RELATED TO THE METHODS DEVELOPED IN THE STUDY OF STATISTICAL ENSEMBLES. AGAIN NO BOOKS, NOTES OR COMPUTER AIDS MAY BE USED DURING THE TEST. IN CASE THE STUDENT DELIVERS SOLUTIONS FOR WHICH THE TEACHER NEEDS CLARIFICATION, A SHORT ORAL INTERVIEW ON THE TOPICS OF THE WRITTEN TEST CAN BE SCHEDULED.

THE WRITTEN AND ORAL TESTS DESCRIBED ABOVE AIM AT A THOROUGH VERIFICATION OF THE THEORETICAL KNOWLEDGE ACQUIRED, THE DEGREE OF AUTONOMY ACHIEVED AND THE STUDENT'S EXPOSITORY SKILLS. THE FINAL GRADE RESULTS FROM THE AVERAGE OF THE SCORES ACQUIRED ON THE ANALYTICAL MECHANICS AND STATISTICAL MECHANICS PARTS, RESPECTIVELY, WEIGHTED ACCORDING TO THE NUMBER OF CFU ASSOCIATED WITH THEM (7 FOR THE ANALYTICAL MECHANICS PART, 5 FOR THE STATISTICAL MECHANICS PART). THE ASSIGNMENT OF THE GRADE ON THE SCALE RANGING FROM 18/30 TO 30/30 IS MADE BY ASSESSING ON THE ONE HAND THE STUDENT'S MASTERY OF THE THEORETICAL ASPECTS CONCERNING THE MAIN TOPICS COVERED IN THE COURSE, AND ON THE OTHER HAND THE ABILITY TO APPLY LAGRANGIAN AND HAMILTONIAN TECHNIQUES, AS WELL AS THE METHODS OF STATISTICAL MECHANICS, TO THE RESOLUTION OF THE PROPOSED EXERCISES. A BONUS OF UP TO A MAXIMUM OF 3 POINTS MAY BE AWARDED BASED ON THE DEGREE OF AUTONOMY AND CLARITY OF EXPOSITION SHOWN BY THE STUDENT IN THE COURSE OF THE TESTS.

THE FINAL GRADE IS CALCULATED AS THE AVERAGE OF THE SCORES OBTAINED ON THE ANALYTICAL MECHANICS AND STATISTICAL MECHANICS PARTS, RESPECTIVELY, WEIGHTED ACCORDING TO THE NUMBER OF CFU ASSOCIATED WITH THEM (7 FOR THE ANALYTICAL MECHANICS PART, 5 FOR THE STATISTICAL MECHANICS PART). A BONUS OF UP TO A MAXIMUM OF 3 POINTS MAY BE AWARDED ON THE BASIS OF THE STUDENT'S MASTERY OF THE SUBJECT AND THE DEGREE OF CLARITY IN THE EXPOSITION SHOWN DURING THE TWO ORAL TESTS.

THE MARK ON THE SCALE FROM 18/30 TO 30/30 IS AWARDED BY ASSESSING, ON THE ONE HAND, THE STUDENT'S MASTERY OF THE THEORETICAL ASPECTS OF THE MAIN TOPICS DEALT WITH IN THE COURSE AND, ON THE OTHER HAND, THE ABILITY TO APPLY LAGRANGIAN AND HAMILTONIAN TECHNIQUES, AS WELL AS THE METHODS OF STATISTICAL MECHANICS, TO THE RESOLUTION OF THE PROPOSED EXERCISES.
THE MINIMUM FINAL MARK OF 18/30 IS REACHED WHEN THE STUDENT DEMONSTRATES AN ACCEPTABLE KNOWLEDGE OF THE ANALYTICAL MECHANICS, BOTH AT A THEORETICAL AND APPLICATIVE LEVEL, CONCERNING IN PARTICULAR THE PROCEDURE OF ELIMINATION OF CONSTRAINTS THAT ALLOWS THE PASSAGE FROM THE NEWTONIAN APPROACH TO THE LAGRANGIAN ONE AND THEN, THROUGH LEGENDRE TRANSFORMATION, TO THE HAMILTONIAN ONE. AS CONCERNS STATISTICAL MECHANICS, THE STUDENT IS EXPECTED TO KNOW THE MAIN FEATURES OF THE VARIOUS STATISTICAL ENSEMBLES, TOGETHER WITH THEIR CONNECTIONS TO THE THERMODYNAMIC PROPERTIES OF THE ANALYSED SYSTEM.

THE MAXIMUM FINAL MARK OF 30/30 IS AWARDED WHEN THE STUDENT DEMONSTRATES A COMPLETE AND THOROUGH KNOWLEDGE OF THE TOPICS LISTED ABOVE, TOGETHER WITH ALL THE APPLICATIONS ANALYSED DURING THE COURSE.

THE MAXIMUM GRADE CUM LAUDE IS AWARDED WHEN THE STUDENT, BESIDES DEMONSTRATING A COMPLETE CONTROL OF BOTH THE THEORETICAL AND APPLICATIVE ASPECTS OF THE TOPICS STUDIED, SHOWS A HIGH CAPACITY FOR AUTONOMOUS ELABORATION, EVEN IN CONTEXTS OTHER THAN THOSE PROPOSED IN THE COURSE.
Texts
ANALYTICAL MECHANICS:
- H. GOLDSTEIN, C. P. POOLE, J. L. SAFKO, “MECCANICA CLASSICA”, ZANICHELLI
- F.R. GANTMACHER, “LEZIONI DI MECCANICA ANALITICA”, EDITORI RIUNITI
- L. LANDAU, E. LIFSHITZ, “MECCANICA“, EDITORI RIUNITI

STATISTICAL MECHANICS:
- H. CALLEN, "THERMODYNAMICS AND AN INTRODUCTION TO THERMOSTATISTICS", JOHN WILEY & SONS
- K. HUANG, “MECCANICA STATISTICA” , ZANICHELLI
- R.K. PATHRIA, "STATISTICAL MECHANICS", BUTTERWORTH-HEINEMANN ED.
- W. GREINER, L. NEISE, H. STOECKER, “THERMODYNAMICS AND STATISTICAL MECHANICS“, SPRINGER
- L.E.REICHL, “A MODERN COURSE IN STATISTICAL PHYSICS” EDWARD ARNOLD PUBLISHERS LTD
- LECTURE NOTES
More Information
TEACHER'S E-MAIL ADDRESS:
ALROMANO@UNISA.IT
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