Lecture

Harmonic Oscillations: Equilibrium and Stability

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In course

PHYS-101(c): General physics : mechanics (IN)

Le but du cours de physique générale est de donner à l'étudiant les notions de base nécessaires à la compréhension des phénomènes physiques. L'objectif est atteint lorsque l'étudiant est capable de prévoir quantitativement les conséquences de ces phénomène ...

Description

This lecture focuses on harmonic oscillations, specifically discussing equilibrium points, stability analysis, and the concept of natural frequency. The instructor explains the behavior of systems with mass and springs, emphasizing the importance of small angle approximations. The lecture covers the derivation of equations of motion for different systems, such as a mass attached to a spring. Through examples and demonstrations, the instructor illustrates how to determine stable and unstable equilibrium points. The lecture concludes with a discussion on the general applicability of harmonic oscillation models in various physical systems, highlighting the fundamental properties of oscillatory motion.

Instructors (3)

Stefano Rusponi

Education: • 1999 Doctoral degree in Physics obtained at the Physics Department, University of Genova PhD thesis title: “STM study of nanostructures induced by ion sputtering on noble metals”. • 1994 University degree in Physics achieved at the Physics Department, University of Genova. Final mark: 110/110 cum laude Diploma thesis title: “A project for a new method of EELS spectroscopy”. • 1988 High school at the Liceo Scientifico G. P. Vieusseux in Imperia. Final mark: 60/60. Research career plan: • 2016 – present MER: Ecole Polytechnique Fédérale de Lausanne (EPFL) in the group of Prof. Harald Brune • 2003 – 2016: 1er. Assistant: Ecole Polytechnique Fédérale de Lausanne (EPFL) in the group of Prof. Harald Brune • 2000-2003: Assistant: Ecole Polytechnique Fédérale de Lausanne (EPFL) under the direction of Prof. Harald Brune • 1999-2000: Research associate: Max-Planck-Institut of Stuttgart under the direction of Prof. Klaus Kern Miscellaneous of professional activities: a) Review panel • Member of the Elettra proposal review panel • Member of the committee of the EDPY doctoral school in Physics at the EPFL b) Co-worker in the building of the X-Treme beamline: c) Referee for scientific journals: • Nat. Commun., Phys. Rev. Lett., Phys. Rev. B, J. Appl. Phys., Surf. Sci., J. Magn. Magn. Mater. Funding record a) Funding awarded • Quantum Properties of Nanostructures at Surfaces, FNS 200020-157081/1; (01/10/2014 – 31/09/2017); total amount attributed: 832'558 CHF; co-applicant • Controlling magnetic anisotropy by interfacial coupling, FNS 200021_146715/1; (01/01/2014 – 31/12/2016); total amount attributed: 367'800 CHF; co-applicant • Self-assembled bi-metallic magnetic pillar superlattices with enhanced blocking temperature, SER C10.0135; (01/08/2011 – 01/08/2013); total amount attributed: 170'000 CHF; co-applicant • Magnetic and Catalytic Properties of Surface Supported Metallic Nanostructures, FNS 200020-120493/1; (01/04/2008 – 31/03/2010); total amount attributed: 402'669 CHF; co-applicant • Magnetic and Catalytic Properties of Surface Supported Metallic Nanostructures, FNS 200020-112322/1; (01/04/2006 – 31/03/2008); total amount attributed: 347'633 CHF; co-applicant b) Approved proposals for the allocation of beamtime Swiss Light Source (SLS): main proposer: 9 co-proposer: 4 Elettra: main proposer: 5 co-proposer: 1 European Synchrotron Radiation Facility (ESRF): main proposer: 2 co-proposer: 11 Student supervisor • Co-director of PhD thesis: 4 PhD students

Dimitris Mousadakos: Seeking the smallest room temperature magnets; (in progress)
Romana Baltic: Controlling single atom magnetic anisotropy by interfacial coupling; (in progress)
Alberto Cavallin: Growth and magnetism of nanostructures investigated by STM, MOKE, and XMCD; (Oct. 2013), Thèse N°5941
Sergio Vlaic: Magnetism and atomic scale structure of bimetallic nanostructures at surfaces; (Dec. 2012), Thèse N° 5625

• Supervisor of PhD thesis (without co-direction): 4 PhD students

Anne Lehnert: Magnetism of individual adatoms and of epitaxial monolayers; (Jun. 2009), Thèse N° 4411
Geraud Moulas: Growth and magnetism of 2D bimetallic nanostructures; (Dec. 2008), Thèse N° 4231
Philipp Buluschek: Submonolayer growth of cobalt on metallic and insulating surfaces studied by scanning tunneling microscopy and kinetic Monte-Carlo simulations; (Nov. 2007), Thèse N° 3944
Nicolas Weiss: Propriétés magnétiques de nanostructures de Co adsorbées; (Apr. 2004), Thèse N° 2980

• Supervisor of Master thesis: 6 students • Supervisor of semester projects: 9 students • PhD thesis referee: 2 students

Official source

Related concepts (32)

Simple harmonic motion

In mechanics and physics, simple harmonic motion (sometimes abbreviated ()) is a special type of periodic motion an object experiences due to a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is described by a sinusoid which continues indefinitely (if uninhibited by friction or any other dissipation of energy).

Oscillation

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.

Harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: where k is a positive constant. If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency (which does not depend on the amplitude).

Economic equilibrium

In economics, economic equilibrium is a situation in which economic forces such as supply and demand are balanced and in the absence of external influences the (equilibrium) values of economic variables will not change. For example, in the standard text perfect competition, equilibrium occurs at the point at which quantity demanded and quantity supplied are equal. Market equilibrium in this case is a condition where a market price is established through competition such that the amount of goods or services sought by buyers is equal to the amount of goods or services produced by sellers.

Equilibrium point

In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. The point is an equilibrium point for the differential equation if for all . Similarly, the point is an equilibrium point (or fixed point) for the difference equation if for . Equilibria can be classified by looking at the signs of the eigenvalues of the linearization of the equations about the equilibria.

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