Alfonso ROMANO | ANALYTICAL MECHANICS

cod. 0512600018

ANALYTICAL MECHANICS

	0512600018
	DIPARTIMENTO DI FISICA "E.R. CAIANIELLO"
	EQF6
	PHYSICS
	2017/2018

PERCORSO COMUNE Cohort 2016/2017

	OBBLIGATORIO
	YEAR OF COURSE 2
	YEAR OF DIDACTIC SYSTEM 2016
	ANNUALE

TEACHING UNITS

SSD	CFU	HOURS	ACTIVITY	TYPE OF ACTIVITY
FIS/02	6	48	LESSONS	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS
FIS/02	3	36	EXERCISES	COMPULSORY SUBJECTS, CHARACTERISTIC OF THE CLASS

PROFESSORS

	ALFONSO ROMANO T Curriculum
	CANIO NOCE Curriculum

	Objectives
	KNOWLEDGE AND UNDERSTANDING: THE AIM OF THE COURSE IS THE ACHIEVEMENT OF AN ADVANCED KNOWLEDGE OF THE LAGRANGIAN AND HAMILTONIAN FORMALISMS, AS WELL AS OF THE SO-CALLED THEORY OF HAMILTON-JACOBI. THE USE OF THESE METHODS AND THEIR APPLICATION TO SPECIFIC PROBLEMS IS EXPECTED TO FAVOR THE ABILITY OF STUDENTS TO DEVELOP RIGOROUS APPROACHES IN THEIR STUDIES. APPLYING KNOWLEDGE AND UNDERSTANDING: THE COURSE IS EXPECTED TO LEAD TO A FULL COMPREHENSION OF THE THEORETICAL CONCEPTS UNDERLYING ANALYTICAL MECHANICS, MAKING STUDENTS ABLE TO APPLY LAGRANGE, HAMILTON AND HAMILTON-JACOBI METHODS 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.

KNOWLEDGE AND UNDERSTANDING:
THE AIM OF THE COURSE IS THE ACHIEVEMENT OF AN ADVANCED KNOWLEDGE OF THE LAGRANGIAN AND HAMILTONIAN FORMALISMS, AS WELL AS OF THE SO-CALLED THEORY OF HAMILTON-JACOBI. THE USE OF THESE METHODS AND THEIR APPLICATION TO SPECIFIC PROBLEMS IS EXPECTED TO FAVOR THE ABILITY OF STUDENTS TO DEVELOP RIGOROUS APPROACHES IN THEIR STUDIES.

APPLYING KNOWLEDGE AND UNDERSTANDING:
THE COURSE IS EXPECTED TO LEAD TO A FULL COMPREHENSION OF THE THEORETICAL CONCEPTS UNDERLYING ANALYTICAL MECHANICS, MAKING STUDENTS ABLE TO APPLY LAGRANGE, HAMILTON AND HAMILTON-JACOBI METHODS 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.

	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 FAR AS MATHEMATICAL ASPECTS ARE CONCERNED, 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).

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 FAR AS MATHEMATICAL ASPECTS ARE CONCERNED, 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
	LAGRANGIAN DYNAMICS: 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. COUPLED OSCILLATION: 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. MOTION UNDER CENTRAL FORCES: 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. HAMILTONIAN DYNAMICS: 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. HAMILTON-JACOBI’S METHOD: 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 WEIERSTRASS EQUATION WITH RELEVANT APPLICATIONS (MOTION UNDER GRAVITY, SIMPLE PENDULUM, MOTION UNDER A CENTRAL FORCE). LAGRANGIAN AND HAMILTONIAN FORMULATION FOR CONTINUOUS SYSTEMS AND FIELDS: HOMOGENEOUS BAR, HARMONIC APPROXIMATION AND NORMAL MODES IN THREE DIMENSIONS. LAGRANGIAN FOR CONTINUOUS SYSTEMS AND HAMILTON PRINCIPLE. STRESS ENERGY TENSOR AND CONSERVATION LAWS. HAMILTONIAN FORMULATION FOR CONTINUOUS SYSTEMS. NOETHER'S THEOREM. COORDINATE AND FIELD TRANSFORMATIONS. NOETHER CURRENTS AND CONSERVATION LAWS. GAUGE TRANSFORMATIONS. INTRODUCTION TO THE TENSOR CALCULUS: FORMAL DEFINITION OF TENSOR. OPERATIONS WITH TENSORS AND INDEX CONTRACTION, METRIC TENSOR.

Contents

LAGRANGIAN DYNAMICS:
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.
COUPLED OSCILLATION:
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.
MOTION UNDER CENTRAL FORCES:
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.
HAMILTONIAN DYNAMICS:
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.
HAMILTON-JACOBI’S METHOD:
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
WEIERSTRASS EQUATION WITH RELEVANT APPLICATIONS (MOTION UNDER GRAVITY, SIMPLE PENDULUM, MOTION UNDER A CENTRAL FORCE).
LAGRANGIAN AND HAMILTONIAN FORMULATION FOR CONTINUOUS SYSTEMS AND FIELDS:
HOMOGENEOUS BAR, HARMONIC APPROXIMATION AND NORMAL MODES IN THREE DIMENSIONS. LAGRANGIAN FOR CONTINUOUS SYSTEMS AND HAMILTON PRINCIPLE. STRESS ENERGY TENSOR AND CONSERVATION LAWS. HAMILTONIAN FORMULATION FOR CONTINUOUS SYSTEMS. NOETHER'S THEOREM. COORDINATE AND FIELD TRANSFORMATIONS. NOETHER CURRENTS AND CONSERVATION LAWS. GAUGE TRANSFORMATIONS.
INTRODUCTION TO THE TENSOR CALCULUS:
FORMAL DEFINITION OF TENSOR. OPERATIONS WITH TENSORS AND INDEX CONTRACTION, METRIC TENSOR.

	Teaching Methods
	ABOUT TWO THIRDS OF THE COURSE ARE DEVOTED TO THEORETICAL LECTURES FOCUSING ON THE PRESENTATON OF THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF CLASSICAL MECHANICS, AS WELL AS TO THE HAMILTON-JACOBI METHOD. THE REMAINING PART OF THE COURSE CONSISTS OF LECTURES FOCUSED ON THE APPLICATION OF THE THEORETICAL METHODS TO THE SOLUTION OF EXERCISES AND PROBLEMS INVOLVING SYSTEMS OF PARTICLES WITH CONSTRAINTS.

Teaching Methods

ABOUT TWO THIRDS OF THE COURSE ARE DEVOTED TO THEORETICAL LECTURES FOCUSING ON THE PRESENTATON OF THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS OF CLASSICAL MECHANICS, AS WELL AS TO THE HAMILTON-JACOBI METHOD. THE REMAINING PART OF THE COURSE CONSISTS OF LECTURES FOCUSED ON THE APPLICATION OF THE THEORETICAL METHODS TO THE SOLUTION OF EXERCISES AND PROBLEMS INVOLVING SYSTEMS OF PARTICLES WITH CONSTRAINTS.

	Verification of learning
	THE ASSESSMENT OF THE LEVEL OF STUDENT'S LEARNING IS MADE THROUGH A FINAL EXAM CONSISTING OF A WRITTEN TEST FOLLOWED BY AN ORAL DISCUSSION. A THRESHOLD LEVEL MUST BE REACHED IN THE WRITTEN TEST IN ORDER TO OBTAIN THE ADMISSION TO THE ORAL DISCUSSION.

	Texts
	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 P. G. BERGMANN, “INTRODUCTION TO THE THEORY OF RELATIVITY”, DOVER L. LANDAU, E. LIFSHITZ, “FISICA TEORICA 2: TEORIA DEI CAMPI”, EDITORI RIUNITI J. D. JACKSON, “CLASSICAL ELECTRODYNAMICS”, WILEY

Texts

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
P. G. BERGMANN, “INTRODUCTION TO THE THEORY OF RELATIVITY”, DOVER
L. LANDAU, E. LIFSHITZ, “FISICA TEORICA 2: TEORIA DEI CAMPI”, EDITORI RIUNITI
J. D. JACKSON, “CLASSICAL ELECTRODYNAMICS”, WILEY

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	TEACHER'S E-MAIL ADDRESS: ALROMANO@UNISA.IT

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Alfonso ROMANO | ANALYTICAL MECHANICS