# Fisica | ANALYTICAL MECHANICS AND STATISTICAL MECHANICS

## Fisica ANALYTICAL MECHANICS AND STATISTICAL MECHANICS

 0512600033 DIPARTIMENTO DI FISICA "E.R. CAIANIELLO" EQF6 PHYSICS 2020/2021

 OBBLIGATORIO YEAR OF COURSE 2 YEAR OF DIDACTIC SYSTEM 2017 ANNUALE
SSD CFU HOURS ACTIVITY TYPE OF ACTIVITY FIS/02 8 64 LESSONS COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS FIS/02 4 48 EXERCISES COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
 ALFONSO ROMANO T
Objectives
THE GENERAL AIMS OF THE COURSE OF “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” ARE:

A)PROVIDE AN INTRODUCTION TO ADVANCED FORMULATIONS OF CLASSICAL MECHANICS.
B)PROVIDE THE BASIC KNOWLEDGE OF THE FUNDAMENTAL STATISTICAL LAWS WHICH GOVERN THE BEHAVIOUR OF MANY-PARTICLE SYSTEMS.

KNOWLEDGE AND UNDERSTANDING:
THE COURSE OF “ANALYTICAL MECHANICS AND STATISTICAL MECHANICS” PROVIDES A DETAILED KNOWLEDGE, AT A THEORETICAL LEVEL AND IN A WAY SUITABLE TO APPLICATIONS, OF THE FOLLOWING TOPICS:
A)LAGRANGIAN AND HAMILTONIAN FORMALISMS AND THEORY OF HAMILTON-JACOBI;
B)PRINCIPLES AND METHODS OF STATISTICAL PHYSICS, WITH A SPECIAL ATTENTION TO THE THEORY OF THE STATISTICAL ENSEMBLES AND ITS APPLICATIONS.
IN BOTH FRAMEWORKS, ATTENTION WILL BE DEVOTED TO THE ANALYTICAL METHODS WHICH ALLOW THE SOLUTION OF SIMPLE MODELS, ALSO IN VIEW OF THE QUANTITATIVE DESCRIPTION OF SOME FUNDAMENTAL PHYSICAL PHENOMENA.

APPLYING KNOWLEDGE AND UNDERSTANDING:
THE COURSE IS EXPECTED TO LEAD TO A FULL COMPREHENSION OF THE THEORETICAL CONCEPTS UNDERLYING ANALYTICAL AND STATISTICAL MECHANICS, MAKING STUDENTS ABLE TO:
A)PRESENT LAGRANGE, HAMILTON AND HAMILTON-JACOBI METHODS AND APPLY THEM TO THE SOLUTION OF COMPLEX PROBLEMS CONCERNING STATICS AND DYNAMICS OF SYSTEMS OF PARTICLES AND RIGID BODIES IN THE PRESENCE OF CONSTRAINTS, WITH A SPECIAL ATTENTION TO THE ROLE PLAYED BY CONSERVATIONS LAWS.
B)PRESENT THE GENERAL THEORY OF THE STATISTICAL ENSEMBLES, WITH APPLICATION TO THE SOLUTION OF EXERCISES REQUIRING A SUITABLE CHOICE OF THE ENSEMBLE, WITH THE RELATED LINK BETWEEN THE MICROSCOPICAL APPROACH AND THE MACROSCOPIC LAWS OF THERMODYNAMICS.
Prerequisites
THE MINIMAL BACKGROUND INCLUDES THE KNOWLEDGE OF THE VECTOR CALCULUS AND THE NEWTONIAN MECHANICS (KINEMATICS, DYNAMICS OF A SINGLE PARTICLE, DYNAMICS OF A SYSTEM OF PARTICLES AND OF RIGID BODIES), AS WELL AS THE KNOWLEDGE OF THE FUNDAMENTALS OF THERMODYNAMICS. CONCERNING MATHEMATICAL ASPECTS, STUDENTS ARE EXPECTED TO KNOW THE LINEAR ALGEBRA AND THE THEORY OF THE FUNCTIONS OF A SINGLE VARIABLE, TOGETHER WITH SOME ELEMENTS OF THE THEORY OF THE FUNCTIONS OF SEVERAL VARIABLES (PARTIAL DERIVATIVES AND DIFFERENTIALS).
Contents
ARGUMENTS OF ANALYTICAL MECHANICS

1) LAGRANGIAN DYNAMICS (LECT. H. 18, EXERC. H. 5):
NEWTON’S LAWS. CONSTRAINTS, DEGREES OF FREEDOM AND GENERALIZED COORDINATES. D’ALEMBERT’S PRINCIPLE AND PRINCIPLE OF VIRTUAL WORKS. LAGRANGE’S EQUATIONS OF MOTION WITH APPLICATIONS. GENERALIZED POTENTIALS: MOTION OF A CHARGED PARTICLE IN AN ELECTROMAGNETIC FIELD. CYCLIC COORDINATES. SYMMETRY PROPERTIES AND CONSERVATIONS LAWS: NOETHER’S THEOREM. HAMILTON’S PRINCIPLE AND ITS EQUIVALENCE WITH LAGRANGE’S EQUATIONS. APPLICATIONS OF THE CALCULUS OF VARIATIONS (BRACHISTOCRONE, GEODETICS, MINIMAL SURFACE OF REVOLUTION). CALCULUS OF VARIATIONS WITH CONSTRAINTS. SUSPENDED CHAIN PROBLEM. SYSTEMS WITH NON-HOLONOMIC CONTRAINTS. LAGRANGE’S MULTIPLIERS.

2) COUPLED OSCILLATION (LECT. H. 6, EXERC. H. 3):
STABILITY OF THE EQUILIBRIUM POSITIONS. EXPANSION OF THE INTERACTION POTENTIAL AND LAGRANGIAN. EIGENVALUE PROBLEM. SIMULTANEOUS DIAGONALIZATION OF THE KINETIC AND THE POTENTIAL ENERGY AS QUADRATIC FORMS. CHARACTERISTIC FREQUENCIES AND NORMAL MODES OF OSCILLATION. LINEAR TRIATOMIC MOLECULE.

3) MOTION UNDER CENTRAL FORCES (LECT. H. 4, EXERC. H. 1):
TWO-BODY PROBLEM AND REDUCTION TO A ONE-BODY PROBLEM IN THE PRESENCE OF A CENTRAL FORCE. EQUATIONS OF MOTION AND SOLUTION VIA CONSERVATION LAWS. EFFECTIVE POTENTIAL AND ORBIT CLASSIFICATION. ORBIT DIFFERENTIAL EQUATION AND INTEGRABLE POTENTIALS SHOWING A POWER-LAW DEPENDENCE ON THE DISTANCE. KEPLER’S PROBLEM: FORCES GOING AS THE INVERSE OF THE SQUARE OF THE DISTANCE.

4) HAMILTONIAN DYNAMICS (LECT. H. 13, EXERC. H. 3):
LEGENDRE’S TRANSFORMATION AND HAMILTON’S EQUATIONS OF MOTIONS. CYCLIC COORDINATES AND ROUTH’S METHOD. THEOREMS ON CONSERVATION LAWS AND PHYSICAL INTERPRETATION OF THE HAMILTONIAN FUNCTION. POISSON BRACKETS, JACOBI IDENTITY AND JACOBI-POISSON’S THEOREM. QUALITATIVE DESCRIPTION OF THE DERIVATION FROM THE HAMILTONIAN FORMALISM OF THE HEISENBERG FORMULATION OF QUANTUM MECHANICS. DERIVATION OF HAMILTON’S EQUATIONS FROM A VARIATIONAL PRINCIPLE. CANONICAL TRANSFORMATIONS. GENERATING FUNCTION AND CONDITIONS OF CANONICITY. SYMPLECTIC APPROACH TO CANONICAL TRANSFORMATIONS.

5) HAMILTON-JACOBI’S METHOD (LECT. H. 6, EXERC. H. 1):
H-J EQUATION AND HAMILTON’S PRINCIPAL FUNCTION. REDUCED H-J EQUATION FOR TIME INDEPENDENT HAMILTONIANS AND HAMILTON’S CHARACTERISTIC FUNCTION USED AS GENERATING FUNCTION. SOLUTION OF THE H-J EQUATION IN THE PRESENCE OF CYCLIC COORDINATES AND METHOD OF SEPARATION OF THE VARIABLES, WITH APPLICATION TO RELEVANT PROBLEMS (HARMONIC OSCILLATOR, MOTION IN THE PRESENCE OF CENTRAL FORCES, ETC.). ACTION-ANGLE VARIABLES

ARGUMENTS OF STATISTICAL MECHANICS

6) BASIC ELEMENTS OF THEORY OF PROBABILITY AND CENTRAL LIMIT THEOREM (2 H)

7) THERMODYNAMICS (8 H):
FIRST AND SECOND PRINCIPLE. ENERGY AND ENTROPY REPRESENTATIONS. STATE EQUATIONS. LEGENDRE TRANSFORMATION AND THERMODYNAMIC POTENTIALS.

8) INTRODUCTION TO STATISTICAL MECHANICS (4 H):
PHASE SPACE. STATISTICAL AND TEMPORAL AVERAGES. ERGODICITY. LIOUVILLE THEOREM

9) FIXED ENERGY SYSTEMS (8 H):
MICROCANONICAL ENSEMBLE. MICROSCOPIC INTERPRETATINO OF THE ENTROPY. TWO-LEVEL SYSTEMS. FREE PARTICLE GAS. GIBBS PARADOX. THEOREM OF THE ENERGY EQUIPARTITION.

10) FIXED TEMPERATURE SYSTEMS (14 H):
CANONICAL ENSEMBLE. THE PARTITION FUNCTION AND ITS CONNECTION WITH HELMHOLTZ FREE ENERGY. APPLICATIONS: TWO-LEVEL SYSTEMS, IDEAL CLASSICAL GAS, INDEPENDENT HARMONIC OSCILLATOR SYSTEMS. CLASSICAL PARAMAGNET.

11) SYSTEMS WITH FIXED TEMPERATURE AND CHEMICAL POTENTIAL (6 H):
GRAND-CANONICAL ENSEMBLE. GRAN-PARTITION FUNCTION AND ITS CONNECTION WITH THE GRAND-CANONICAL POTENTIAL.
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
THE ASSESSMENT OF THE LEVEL OF STUDENT'S LEARNING IS MADE THROUGH A FINAL EXAMINATION CONSISTING, FOR EACH OF THE TWO PARTS OF THE COURSE, OF A WRITTEN TEST FOLLOWED BY AN ORAL DISCUSSION.

THE WRITTEN TEST IN ANALYTICAL MECHANICS CONSISTS IN DETERMINING THE SOLUTION OF TWO EXERCISES ON LAGRANGIAN AND HAMILTONIAN DYNAMICS, AIMING TO EVALUATE STUDENT'S ABILITY OF SUITABLY OPERATING IN BOTH CONTEXTS. THE MAXIMUM SCORES FOR EACH OF THE EXERCISES ARE 18/30 AND 12/30, RESPECTIVELY. IF THE TOTAL SCORE IS NOT LOWER THAN 18/30, STUDENTS ARE ALLOWED TO TAKE THE ORAL EXAMINATION.

THE WRITTEN TEST IN STATISTICAL MECHANICS CONSISTS IN THE SOLUTION OF SEVERAL IN-ITINERE TESTS, EACH REQUIRING THE SOLUTION OF TWO OR THREE EXERCISES, FOCUSING ON DISTINCT PORTIONS OF THE COURSE PROGRAMME. IF THE TOTAL SCORE, AVERAGED OVER ALL THE TESTS, IS NOT LOWER THAN 18/30, STUDENTS ARE ALLOWED TO TAKE THE ORAL EXAMINATION. BELOW THIS THRESHOLD VALUE, STUDENTS ARE REQUIRED TO PASS A SINGLE, GLOBAL WRITTEN TEST FOCUSING ON ALL THE STATISTICAL MECHANICS TOPICS THEORETICALLY ANALYZED IN THE COURSE.

THE FINAL GRADE COMES FROM THE AVERAGE OF THE GRADES OBTAINED IN THE FOUR PARTS OF THE EXAMINATION, WEIGHTED BY THE NUMBER OF CFU ASSOCIATED TO THE TWO PARTS OF THE COURSE (7 FOR THE ANALYTICAL MECHANICS, 5 FOR THE STATISTICAL MECHANICS). A BONUS UP TO A MAXIMUM OF 3 POINTS CAN BE ADDED, DEPENDING ON HOW CLEARLY THE ARGUMENTS ARE SET OUT IN THE ORAL EXAMINATIONS.

THE MINIMUM GRADE OF 18/30 IS OBTAINED WHEN THE STUDENT EXHIBITS A SUFFICIENT KNOWLEDGE OF THE PROCEDURE OF ELIMINATION OF THE CONSTRAINTS LEADING TO THE LAGRANGIAN FORMALISM AND TO THE SUBSEQUENT INTRODUCTION, VIA LEGENDRE TRANSFORMATION, OF THE HAMILTONIAN FORMALISM. A SUFFICIENT KNOWLEDGE OF THE GENERAL SCHEME USED IN STATISTICAL PHYSICS, BASED ON THE INTRODUCTION OF THE STATISTICAL ENSEMBLES, IS ALSO REQUIRED,

THE MAXIMUM GRADE OF 30/30 IS OBTAINED WHEN THE STUDENTS EXHIBITS A COMPLETE AND DEEP KNOWLEDGE OF THE ABOVE-MENTIONED TOPICS AS WELL AS OF ALL THE RELATED APPLICATIONS ANALYZED DURING THE COURSE.

THE MAXIMUM GRADE CUM LAUDE REQUIRES A DEEP AND EXTENDED KNOWLEDGE OF THE THEORY AND THE APPLICATIONS OF ALL THE ARGUMENTS OF THE COURSE, TOGETHER WITH THE ABILITY OF APPLYING THE RELATED PRINCIPLES AND METHODS ALSO WITHIN CONTEXTS GOING BEYOND THE ONES INVESTIGATED 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:
- K. HUANG, “MECCANICA STATISTICA” , ZANICHELLI
- R.K. PATHRIA, "STATISTICAL MECHANICS", BUTTERWORTH-HEINEMANN ED.
- L.D.LANDAU, E.M.LIFSITS, L.P.PITAEVSKIJ, “FISICA STATISTICA I“ ED. RIUNITI
- L.E.REICHL, “A MODERN COURSE IN STATISTICAL PHYSICS” EDWARD ARNOLD PUBLISHERS LTD
- LECTURE NOTES