Mathematica in Theoretical Physics: Selected Examples from Classical Mechanics to Fractals

Variational methods have a long and remarkable role in theoretical physics. Few of our students when first exposed to them fail to admire their elegance and efficacy in the formulation and solution of physical problems. In this paper we apply the variational approach that leads to the Euler-Lagrange equations to the determination of the shape of the surface of a mass of water that partially fills a cylindrical bucket that rotates with constant angular velocity (Newton's bucket). Here this approach will lead us to the principle of minimization of the effective potential energy associated with the system. The effect of an external pressure on the equilibrium shape is also taken into account and two models, the constant pressure model and the linear model are discussed. The level of the discussion is kept accessible to undergraduates taking an intermediate level course in classical mechanics.

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Introduction

10.1093/oso/9780198794899.003.0001 ◽

2017 ◽

Author(s):

Dusa McDuff ◽

Dietmar Salamon

Keyword(s):

Mirror Symmetry ◽

Complex Geometry ◽

Equations Of Motion ◽

Classical Mechanics ◽

Hamiltonian Formalism ◽

Theoretical Physics ◽

Lagrange Equations ◽

Low Dimensional Topology ◽

The Subject ◽

Low Dimensional

Symplectic topology has a long history. It has its roots in classical mechanics and geometric optics and in its modern guise has many connections to other fields of mathematics and theoretical physics ranging from dynamical systems, low-dimensional topology, algebraic and complex geometry, representation theory, and homological algebra, to classical and quantum mechanics, string theory, and mirror symmetry. One of the origins of the subject is the study of the equations of motion arising from the Euler–Lagrange equations of a one-dimensional variational problem. The Hamiltonian formalism arising from a Legendre transformation leads to the notion of a ...

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Exploring Classical Mechanics

10.1093/oso/9780198853787.001.0001 ◽

2020 ◽

Author(s):

G. L. Kotkin ◽

V. G. Serbo

Keyword(s):

Quantum Mechanics ◽

Statistical Mechanics ◽

Classical Mechanics ◽

Theoretical Physics ◽

English Edition ◽

Mechanical Part ◽

Russian Edition ◽

The Given

This book was written by the working physicists for students and teachers of physics faculties of universities. Its contents correspond roughly to the corresponding course in the textbooks Mechanics by L. D. Landau and E. M. Lifshitz (1976) and Classical Mechanics by H. Goldstein, Ch. Poole, and J. Safko (2000). As a rule, the given solution of a problem is not finished with obtaining the required formulae. It is necessary to analyse the results, and this is of great interest and by no means a mechanical part of the solution. The authors consider classical mechanics as the first chapter of theoretical physics; the methods and ideas developed in this chapter are literally important for all other sections of theoretical physics. Thus, the authors have indicated wherever this does not require additional amplification, the analogy or points of contact with the problems in quantum mechanics, electrodynamics, or statistical mechanics. The first English edition of this book was published by Pergamon Press in 1971 with the invaluable help by the translation editor D. ter Haar. This second English publication is based on the fourth Russian edition of 2010 as well as the problems added in the publications in Spanish and French. As a result, this book contains 357 problems instead of the 289 problems that appeared in the first English edition.

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A short course on quantum mechanics and methods of quantization

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887815600087 ◽

2015 ◽

Vol 12 (08) ◽

pp. 1560008 ◽

Cited By ~ 3

Author(s):

Elisa Ercolessi

Keyword(s):

Quantum Mechanics ◽

Hilbert Spaces ◽

International Workshop ◽

Classical Mechanics ◽

Theoretical Physics ◽

Feynman Path Integral ◽

Introductory Courses ◽

Feynman Path ◽

Geometric Structures ◽

Short Course

These notes collect the lectures given by the author to the "XXIII International Workshop on Geometry and Physics" held in Granada (Spain) in September 2014. The first part of this paper aims at introducing a mathematical oriented reader to the realm of Quantum Mechanics (QM) and then to present the geometric structures that underline the mathematical formalism of QM which, contrary to what is usually done in Classical Mechanics (CM), are usually not taught in introductory courses. The mathematics related to Hilbert spaces and Differential Geometry are assumed to be known by the reader. In the second part, we concentrate on some quantization procedures, that are founded on the geometric structures of QM — as we have described them in the first part — and represent the ones that are more operatively used in modern theoretical physics. We will discuss first the so-called Coherent State Approach which, mainly complemented by "Feynman Path Integral Technique", is the method which is most widely used in quantum field theory. Finally, we will describe the "Weyl Quantization Approach" which is at the origin of modern tomographic techniques, originally used in optics and now in quantum information theory.

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Teaduslik teooria kui teadusfilosoofia kategooria

Studia Philosophica Estonica ◽

10.12697/spe.2009.2.1.03 ◽

2009 ◽

pp. 32-46

Author(s):

Rein Vihalemm

Keyword(s):

Philosophy Of Science ◽

Scientific Theory ◽

Chemical Structure ◽

Physical Theory ◽

Classical Mechanics ◽

Scientific Discipline ◽

Theoretical Physics ◽

Structure Theory ◽

Characteristic Theory ◽

Ronald Giere

Artiklis arendatakse alternatiivset kontseptsiooni niihästi traditsioonilisele füüsikakesksele teadusliku teooria käsitlusele kui ka seisukohale, et füüsikateooriat ei saa teadusfilosoofias mõista teadusliku teooria mudelina, sest erinevates teadustes on teooriad oma loomult erinevad. Ollakse seisukohal, et teaduslik teooria on ikkagi teadusfilosoofia kategooriana teadusliku distsipliini eripärast sõltumatu. Käsitletakse põhiliselt kahte punkti: (1) miks on teadusfilosoofias põhjust kritiseerida traditsioonilist, füüsika põhjal saadud ettekujutust teaduslikust teooriast? (2) miks ei ole põhjendatud seisukoht, et nt keemias on teaduslik teooria (nt klassikaline keemilise struktuuri teooria) oma loomult füüsikateooriast (nt klassikalisest mehhaanikast) erinev? Traditsioonilise füüsikakeskse lähenemisviisi puhul ei ole piisavalt uuritud, miks õieti füüsikateooria on saanud teadusliku teooria etaloniks. Teoreetilise füüsika keskne teadusekontseptsioon on olnud ühekülgselt orienteeritudmatemaatikale ja loogikale. Kui aga lähtuda teooriast kui mudelite populatsioonist, nagu Ronald Giere on seda teinud ka klassikalise mehhaanika - füüsikakeskse teooriakäsituse traditsioonilise näite - korral, siis osutub see teooriakäsitus tõepoolest invariantseks teatavale tunnetustüübile, mida on alust nimetada teaduslikuks ja mis on selgesti omane ka nt keemiale.In this paper an alternative conception is proposed both to (1) the traditional physics-based conception of scientific theory, and (2) the view that a physical theory cannot be regarded as the model for scientific theory in philosophy of science in general because scientific theories are discipline-dependent. It is argued that scientific theory as a category of philosophy of science is independent of a particular scientific discipline. The main focus of the paper is on two questions: (1) Why is the traditional physics-based conception of scientific theory subject to criticism in philosophy of science? (2) Why is it unjustified to consider a scientific theory in chemistry (e.g., the classical chemical structure theory, which is probably the most characteristic theory of chemistry) to be different in character from a physical theory (e.g., classical mechanics, which is a prototypical example of a theory in the philosophy of science)? In case of the traditional physics-based approach not enough research has been done as to why theories of physics have become the etalon of scientific theory. The accepted view of science, centered on theoretical physics, has been one-sidedly oriented towards mathematics and logic. But when proceeding from a conception of a scientific theory as a population of models, as Ronald Giere has done for the case of classical mechanics - the traditional example of a physical theory -, this conception of a theory really does prove the invariance of a certain type of cognition that can justifiably be called scientific and that is clearly characteristic of, e.g., chemistry as well.

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О ФОРМИРОВАНИИ УНИВЕРСАЛЬНЫХ КОМПЕТЕНЦИЙ ПРИ РЕШЕНИИ ЗАДАЧ ТЕОРЕТИЧЕСКОЙ ФИЗИКИ В МОДУЛЕ «КЛАССИЧЕСКАЯ МЕХАНИКА»

Pedagogical Review ◽

10.23951/2307-6127-2020-2-121-129 ◽

2020 ◽

Author(s):

Olesya Azorkina ◽

Elena Kirillova

Keyword(s):

Mathematical Analysis ◽

Teaching Methods ◽

Classical Mechanics ◽

The Body ◽

Theoretical Physics ◽

Creative Potential ◽

Intellectual Activity ◽

Decision Algorithm ◽

Level Of Understanding ◽

Physics Knowledge

В работе изучаются некоторые аспекты формирования универсальных компетенций в курсе «Теоретическая физика» (модуль «Классическая механика») для бакалавриата педагогических вузов. В качестве примера выбран фрагмент курса, связанный с нахождением закона движения тела. При решении задач активно используются знания из математических курсов. Актуальность темы заключается в том, что курс теоретической физики объединяет дисциплины естественно-научного и математического циклов, поэтому важно суметь в полной мере использовать накопленные знания для подъема на новый, интегральный уровень освоения учебной информации. Новизна подхода состоит в том, что он дает возможность одновременно формировать научное, исследовательское мышление обучающихся; навыки методики преподавания будущими педагогами физико-математических дисциплин; развивать творческий потенциал обучающихся. Показано, как при решении выбранных задач у студентов формируется набор группы УК-1.The paper studies some aspects of the formation of universal competencies (UC) in the course of theoretical physics for students of pedagogical universities. As an example, a fragment of the course of classical mechanics is selected, associated with finding the law of motion of the body in Newton’s formulation. Four types of tasks are considered, which differ mainly in the nature of the time dependence of the resulting force. When solving the problems of the module “Classical Mechanics” of a course in theoretical physics, knowledge from mathematical courses (Mathematical Analysis, Differential Equations) is actively used, which stimulates the students’ analytical and creative intellectual activity. The relevance of the topic is that the course of theoretical physics combines the disciplines of the natural science and mathematical cycles, and it is important to be able to fully use the knowledge accumulated by students to reach a new, integral level of understanding of educational information. The novelty of this approach is that it makes it possible to simultaneously form the scientific, research thinking of students; mastering the teaching methods of physical and mathematical disciplines (which is important for future teachers); to develop the creative potential of students when choosing a particular decision algorithm. The article shows how, when solving selected problems, students form a set of competencies of the UС-1 group.

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Lagrangian and Hamiltonian Dynamics

10.1093/oso/9780198822370.001.0001 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Scientific Method ◽

Hamiltonian Dynamics ◽

Laws Of Nature ◽

Classical Mechanics ◽

Ancient Greece ◽

Theoretical Physics ◽

Analytical Mechanics ◽

Classical Physics ◽

Active Research ◽

Natural Philosophers

This book explores the fascinating subject of classical mechanics, which is the pinnacle of nineteenth-century physics, from a fresh and exciting viewpoint. With its foundations laid down in ancient Greece, classical physics was truly born in the 1700s with Sir Isaac Newton’s discoveries and quickly developed into the modern scientific method that is commonplace today. After the Newtonian revolution, others reformulated classical mechanics into different descriptions and new formalisms, each uncovering novel aspects of the mathematical and geometrical laws of nature. Over the last 400 years, classical physics has been used to engineer bridges, railways, engines, antennas, planes and much, much more. Classical mechanics is still a vibrant field of active research in theoretical physics and, to this day, captures the excitement of many physicists. Classical mechanics persists today due to its incredible practicality and as the physical embodiment of many fields of abstract mathematics. In this book, the reader journeys from Newton’s three laws of motion to analytical mechanics and Lagrangian and Hamiltonian dynamics, as well as the formulations of Jacobi and many other hard-working natural philosophers who lend their names to classical mechanics.

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