Classical Mechanics

Classical Mechanics

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The principles of mechanics successfully described many other phenomena encountered in the world. Conservation laws involving energy, momentum and angular momentum provided a second parallel approach to solving many of the same problems. In this course, we will investigate both approaches: Force and conservation laws.

Our exploration of the theoretical underpinnings of modern physics begins with classical mechanics, the mathematical physics worked out by Isaac Newton (1642--1727) and later by Joseph Lagrange (1736--1813) and William Rowan Hamilton (1805--1865). We will start with a discussion of the allowable laws of physics and then delve into Newtonian mechanics. We then study three formulations of classical mechanics respectively by Lagrange, Hamiltonian and Poisson. Throughout the lectures we will focus on the relation between symmetries and conservation laws. The last two lectures are devoted to electromagnetism and the application of the equations of classical mechanics to a particle in electromagnetic fields. (Image credit: The International Astronomical Union/Martin Kornmesser)

This is a second course in classical mechanics, given to final year undergraduates. They were last updated in January 2015. Individual chapters and problem sheets are available below. The full set of lecture notes, weighing in at around 130 pages, can be downloaded here:

In this documentation many components of the physics/mechanics module willbe discussed. sympy.physics.mechanics has been written to allow for creation ofsymbolic equations of motion for complicated multibody systems.

This module derives the vector-related abilities and related functionalitiesfrom sympy.physics.vector. Please have a look at the documentation ofsympy.physics.vector and its necessary API to understand the vector capabilitiesof sympy.physics.mechanics.

Together with the rest of SymPy, this module performs steps 4 and 5,provided that the user can perform 1 through 3 for the module. That is to say,the user must provide a complete representation of the freebody diagrams that themselves represent the system, with which this code canprovide equations of motion in a form amenable to numerical integration. Step5 above amounts to arduous algebra for even fairly simple multi-body systems.Thus, it is desirable to use a symbolic math package, such as SymPy, toperform this step. It is for this reason that this module is a part of SymPy.Step 4 amounts to this specific module, sympy.physics.mechanics.

We now know that there is much more to classical mechanics than previously suspected. Derivations of the equations of motion, the focus of traditional presentations of mechanics, are just the beginning. This innovative textbook, now in its second edition, concentrates on developing general methods for studying the behavior of classical systems, whether or not they have a symbolic solution. It focuses on the phenomenon of motion and makes extensive use of computer simulation in its explorations of the topic. It weaves recent discoveries in nonlinear dynamics throughout the text, rather than presenting them as an afterthought. Explorations of phenomena such as the transition to chaos, nonlinear resonances, and resonance overlap to help the student develop appropriate analytic tools for understanding. The book uses computation to constrain notation, to capture and formalize methods, and for simulation and symbolic analysis. The requirement that the computer be able to interpret any expression provides the student with strict and immediate feedback about whether an expression is correctly formulated.

Classical mechanics is the study of the motion of bodies based upon Isaac Newton's famous laws of mechanics. There are no new physical concepts in classical mechanics that are not already extant in other areas of physics. What classical mechanics does is mathematically reformulate Newtonian physics to address a huge range of problems ranging from molecular dynamics to the motion of celestial bodies.

There is an extraordinarily large number of textbooks in theoretical mechanics, because it is a fairly old and well-studied subject. You need any textbook on classical mechanics that you can understand and that talks about "Lagrangians" early on. (Books that only talk about accelerations, forces, and torques may be quite advanced but they do not cover the subject of theoretical mechanics.)

Continued advances in the precision manufacturing of new structures at the nanometer scale have provided unique opportunities for device physics. This book sets out to summarize those elements of classical mechanics most applicable for scientists and engineers studying device physics. It provides sophisticated quantum models of device behavior along with problems that the reader may wish to solve, though some are more challenging than others. Supplementary MATLAB® materials are available for all figures generated numerically.

As a limit theory of quantum mechanics, classical dynamics comprises a large variety of phenomena, from computable (integrable) to chaotic (mixing) behavior. This book presents the KAM (Kolmogorov-Arnold-Moser) theory and asymptotic completeness in classical scattering. Including a wealth of fascinating examples in physics, it offers not only an excellent selection of basic topics, but also an introduction to a number of current areas of research in the field of classical mechanics. Thanks to the didactic structure and concise appendices, the presentation is self-contained and requires only knowledge of the basic courses in mathematics.

8.01L Physics I (, ) Prereq: NoneUnits: 3-2-7Credit cannot also be received for 8.01, 8.011, 8.012, ES.801, ES.8012Introduction to classical mechanics (see description under 8.01). Includes components of the TEAL (Technology-Enabled Active Learning) format. Material covered over a longer interval so that the subject is completed by the end of the IAP. Substantial emphasis given to reviewing and strengthening necessary mathematics tools, as well as basic physics concepts and problem-solving skills. Content, depth, and difficulty is otherwise identical to that of 8.01. The subject is designated as 8.01 on the transcript.Fall: P. Jarillo-HerreroIAP: P. Jarillo-HerreroNo required or recommended textbooks

8.04 Quantum Physics I () Prereq: 8.03 and (18.03 or 18.032)Units: 5-0-7Credit cannot also be received for 8.041 Lecture: MW9.30-11 (6-120) Recitation: TR10 (4-257) or TR11 (4-257) or TR1 (26-322) or TR2 (26-322) +finalExperimental basis of quantum physics: photoelectric effect, Compton scattering, photons, Franck-Hertz experiment, the Bohr atom, electron diffraction, deBroglie waves, and wave-particle duality of matter and light. Introduction to wave mechanics: Schroedinger's equation, wave functions, wave packets, probability amplitudes, stationary states, the Heisenberg uncertainty principle, and zero-point energies. Solutions to Schroedinger's equation in one dimension: transmission and reflection at a barrier, barrier penetration, potential wells, the simple harmonic oscillator. Schroedinger's equation in three dimensions: central potentials and introduction to hydrogenic systems.A. HarrowTextbooks (Spring 2023)

8.041 Quantum Physics I () Prereq: 8.03 and (18.03 or 18.032)Units: 2-0-10Credit cannot also be received for 8.04Blended version of 8.04 using a combination of online and in-person instruction. Covers experimental basis of quantum physics: photoelectric effect, Compton scattering, photons, Franck-Hertz experiment, the Bohr atom, electron diffraction, deBroglie waves, and wave-particle duality of matter and light. Introduction to wave mechanics: Schroedinger's equation, wave functions, wave packets, probability amplitudes, stationary states, the Heisenberg uncertainty principle, and zero-point energies. Solutions to Schroedinger's equation in one dimension: transmission and reflection at a barrier, barrier penetration, potential wells, the simple harmonic oscillator. Schroedinger's equation in three dimensions: central potentials and introduction to hydrogenic systems.V.. Vuletic

8.044 Statistical Physics I ()Prereq: 8.03 and 18.03Units: 5-0-7Lecture: TR11-12.30 (6-120) Recitation: MW10 (26-204) or MW11 (26-204) or MW2 (26-322) or MW3 (26-322) +finalIntroduction to probability, statistical mechanics, and thermodynamics. Random variables, joint and conditional probability densities, and functions of a random variable. Concepts of macroscopic variables and thermodynamic equilibrium, fundamental assumption of statistical mechanics, microcanonical and canonical ensembles. First, second, and third laws of thermodynamics. Numerous examples illustrating a wide variety of physical phenomena such as magnetism, polyatomic gases, thermal radiation, electrons in solids, and noise in electronic devices. Concurrent enrollment in 8.04 is recommended.R. FletcherTextbooks (Spring 2023)

8.051 Quantum Physics II ()Prereq: 8.04 and permission of instructorUnits: 2-0-10Credit cannot also be received for 8.05Lecture: MW10 (2-105) +finalBlended version of 8.05 using a combination of online and in-person instruction. Together with 8.06 covers quantum physics with applications drawn from modern physics. General formalism of quantum mechanics: states, operators, Dirac notation, representations, measurement theory. Harmonic oscillator: operator algebra, states. Quantum mechanics in three dimensions: central potentials and the radial equation, bound and scattering states, qualitative analysis of wave functions. Angular momentum: operators, commutator algebra, eigenvalues and eigenstates, spherical harmonics. Spin: Stern-Gerlach devices and measurements, nuclear magnetic resonance, spin and statistics. Addition of angular momentum: Clebsch-Gordan series and coefficients, spin systems, and allotropic forms of hydrogen. Limited to 20.B. ZwiebachTextbooks (Spring 2023) 781b155fdc