Introduction to differential equations. Lagrangian mechanics for free and constrained systems. Variational principles. Kinematics and dynamics of rigid bodies. Introduction to Hamiltonian mechanics
Lecture notes and exercises available on the moodle platform
Fasano Marmi, Meccanica Analitica, Bollati Boringhieri
Learning Objectives
knowledge:
Introduction to the mathematical modeling of complex natural phenomena and their treatment with advanced mathematical tools.
Acquired skills (at the end of the course):
Being able of writing the motion equations of mechanical systems, and discussing their solvability.
Being able to set up simple mathematical models for mechanical systems and analyze the relative mathematical problems.
Prerequisites
Courses required: Calculus 1, Geometry 1.
Recommended courses: Physics 1
Teaching Methods
CFU: 12
Total course hours: 300
Personal study and other individual learning: hours 174
Lectures: hours 60
Llaboratory classes: 0
Eexercises: hours 60
Seminars: hours 0
Stages: Hours 0
C course tests: hours 6
Further information
Frequency of lectures and exercises: Recommended
Lecture notes and exercises: UniFi E-Learning: http://el.unifi.it
Office hours Dr. Farina, by prior appointment.
Dipartimento di Matematica e Informatica "Ulisse Dini"
Viale Morgagni, 67/a
50134 - Firenze (FI)
Tel: 055 2751435
E-Mail: farina@math.unifi.it
Type of Assessment
The final exam aims to assess the acquisition of knowledge and skills through a written test lasting 2 hours and an oral examination.
The written test consists of 2 exercises on the on the syllabus taught during the lectures.
The oral test consists of a technical interview with the professor aimed at verifying knowledge, skills and student ability.
Course program
1 Kinematics of rigid systems
1.1. introduction
1.2 Rigid motion
1.3 Instantaneous axis of motion
1.4 Relative kinematic and composition of velocities
1.5 Composition of rigid motions
1.6 Euler angles
1.7 Relative acceleration
2 Differential Equations
2.1 General introduction
2.2 The Cauchy problem
2.2.1 Autonomous Equations
2.2.2 Reversible equations
2.3 Integrable Equations
2.3.1 The “conservative” case
2.3.2 Conservative case: qualitative analysis
2.4 The phase portrait
2.5 Equilibrium and stability
2.5.1 TheLyapounov criterion
2.5.2 Asymptotic stability
2.5.3 The conservative systems
2.6 Period approximation close to the equilibrium points
2.7 Two-dimensional Linear Systems
2.7.1 Damped and forced harmonic and oscillator
2.8 Linear Stability
3 Lagrange equations
3.1 Lagrange's equations for a material point
3.2 The motion in a central field
3.2.1 The equation for r
3.2.2 The Kepler problem
3.2.3 The Kepler orbits
3.2.4 The Kepler’s third Law
4 constrained systems and Lagrangian coordinates
4.1 Holonomic systems
4.1.1 Virtual motion
4.1.2 Virtual displacements as functions of the lagrangean coordinates
4.1.3 Points constrained on a surface
5 The equations of motion
5.1 The constrained point
5.2 The dynamics and static symbolic equation
5.3 The Lagrange equations
5.3.1 Solvability of the Lagrange equations
5.3.2 Invariance of Lagrange equations
5.3.3 Coordinates cyclic
5.3.4 Conservation of energy
5.4 Equilibrium
5.4.1 Stability
5.5 Small Oscillations
5.5.1 Solutions to the small oscillation equations
5.5.2 Proof of spectral Theorem
6 Dynamics of rigid systems
6.1 Cardinal equations
6.2 The fundamental equations for the rigid motion
6.3 Expression of L and T for rigid bodies. The inertia tensor
6.3.1 Angular momentum
6.3.2 Physical meaning of the moments
6.3.3 The kinetic energy
6.4 The precession of inertia
6.4.1 The Euler equations
6.5 Poinsot motion
6.6 The heavy gyroscope
7 Variational principles
7.1 The bracristocrone
7.2 The Euler-Lagrange equation
7.3 The principle of Hamilton
7.4 The principle of least action
7.4.1 Cyclic coordinates
8 The canonical system
8.1 The Poisson brackets
8.2 Variational derivation of the Hamilton equations
8.3 Canonical transformations