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12.802 Wave Motions in the Ocean and Atmosphere (MIT) 12.802 Wave Motions in the Ocean and Atmosphere (MIT)

Description

This course is an introduction to basic ideas of geophysical wave motion in rotating, stratified, and rotating-stratified fluids. Subject begins with general wave concepts of phase and group velocity. It also covers the dynamics and kinematics of gravity waves with a focus on dispersion, energy flux, initial value problems, etc. Also addressed are subject foundation used to study internal and inertial waves, Kelvin, Poincare, and Rossby waves in homogeneous and stratified fluids. Laplace tidal equations are applied to equatorial waves. Other topics include: resonant interactions, potential vorticity, wave-mean flow interactions, and instability. This course is an introduction to basic ideas of geophysical wave motion in rotating, stratified, and rotating-stratified fluids. Subject begins with general wave concepts of phase and group velocity. It also covers the dynamics and kinematics of gravity waves with a focus on dispersion, energy flux, initial value problems, etc. Also addressed are subject foundation used to study internal and inertial waves, Kelvin, Poincare, and Rossby waves in homogeneous and stratified fluids. Laplace tidal equations are applied to equatorial waves. Other topics include: resonant interactions, potential vorticity, wave-mean flow interactions, and instability.

Subjects

geophysical wave motion | geophysical wave motion | rotating | stratified | and rotating-stratified fluids | rotating | stratified | and rotating-stratified fluids | general wave concepts | general wave concepts | phase | phase | group velocity | group velocity | dynamics and kinematics of gravity waves | dynamics and kinematics of gravity waves | dispersion | dispersion | energy flux | energy flux | initial value problems | initial value problems | internal and inertial waves | internal and inertial waves | Kelvin | Kelvin | Poincare | Poincare | and Rossby waves | and Rossby waves | homogeneous and stratified fluids | homogeneous and stratified fluids | Laplace tidal equations | Laplace tidal equations | equatorial waves | equatorial waves | resonant interactions | resonant interactions | potential vorticity | potential vorticity | wave-mean flow interactions | wave-mean flow interactions | instability | instability | 12. Kelvin | Poincare | and Rossby waves | 12. Kelvin | Poincare | and Rossby waves | Kelvin | Poincare | and Rossby waves | Kelvin | Poincare | and Rossby waves | internal gravity waves | internal gravity waves | surface gravity waves | surface gravity waves | rotation | rotation | large-scale hydrostatic motions | large-scale hydrostatic motions | vertical structure equation | vertical structure equation | equatorial ?-plane | equatorial ?-plane | Stratified Quasi-Geostrophic Motion | Stratified Quasi-Geostrophic Motion

License

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18.385J Nonlinear Dynamics and Chaos (MIT) 18.385J Nonlinear Dynamics and Chaos (MIT)

Description

This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software. This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.

Subjects

chaos | chaos | Floquet theory | Floquet theory | Poincare-Bendixson theory | Poincare-Bendixson theory | phase plane | phase plane | limit cycles | limit cycles | time-dependent systems | time-dependent systems | Poincare maps | Poincare maps | stability of equilibria | stability of equilibria | near-equilibrium dynamics | near-equilibrium dynamics | center manifolds | center manifolds | elementary bifurcations | elementary bifurcations | normal forms | normal forms

License

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18.385J Nonlinear Dynamics and Chaos (MIT) 18.385J Nonlinear Dynamics and Chaos (MIT)

Description

This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software. This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.

Subjects

Phase plane | Phase plane | limit cycles | limit cycles | Poincare-Bendixson theory | Poincare-Bendixson theory | Time-dependent systems | Time-dependent systems | Floquet theory | Floquet theory | Poincare maps | Poincare maps | averaging | averaging | Stability of equilibria | Stability of equilibria | near-equilibrium dynamics | near-equilibrium dynamics | Center manifolds | Center manifolds | elementary bifurcations | elementary bifurcations | normal forms | normal forms | chaos | chaos | 18.385 | 18.385 | 2.036 | 2.036

License

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18.385 Nonlinear Dynamics and Chaos (MIT) 18.385 Nonlinear Dynamics and Chaos (MIT)

Description

Nonlinear dynamics with applications. Intuitive approach with emphasis on geometric thinking, computational and analytical methods. Extensive use of demonstration software. Topics: Bifurcations. Phase plane. Nonlinear coupled oscillators in biology and physics. Perturbation, averaging theory. Parametric resonances, Floquet theory. Relaxation oscillations. Hysterises. Phase locking. Chaos: Lorenz model, iterated mappings, period doubling, renormalization. Fractals. Hamiltonian systems, area preserving maps; KAM theory.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.MATLAB® is a trademark of The MathWorks, Inc. Nonlinear dynamics with applications. Intuitive approach with emphasis on geometric thinking, computational and analytical methods. Extensive use of demonstration software. Topics: Bifurcations. Phase plane. Nonlinear coupled oscillators in biology and physics. Perturbation, averaging theory. Parametric resonances, Floquet theory. Relaxation oscillations. Hysterises. Phase locking. Chaos: Lorenz model, iterated mappings, period doubling, renormalization. Fractals. Hamiltonian systems, area preserving maps; KAM theory.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.MATLAB® is a trademark of The MathWorks, Inc.

Subjects

Phase plane | Phase plane | limit cycles | limit cycles | Poincare-Bendixson theory | Poincare-Bendixson theory | Time-dependent systems | Time-dependent systems | Floquet theory | Floquet theory | Poincare maps | Poincare maps | averaging | averaging | Stability of equilibria | Stability of equilibria | near-equilibrium dynamics | near-equilibrium dynamics | Center manifolds | Center manifolds | elementary bifurcations | elementary bifurcations | normal forms | normal forms | chaos | chaos

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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12.620J Classical Mechanics: A Computational Approach (MIT) 12.620J Classical Mechanics: A Computational Approach (MIT)

Description

12.620J covers the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. The course uses computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.The following topics are covered: the Lagrangian formulation, action, variational principles, and equations of motion, Hamilton's principle, conserved quantities, rigid bodies and tops, Hamiltonian formulation and canonical equations, surfaces of section, chaos, canonical transformations and generating functions, Liouville's theorem and Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems, invariant curves and cantori, nonlinear resonances, resonance ov 12.620J covers the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. The course uses computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.The following topics are covered: the Lagrangian formulation, action, variational principles, and equations of motion, Hamilton's principle, conserved quantities, rigid bodies and tops, Hamiltonian formulation and canonical equations, surfaces of section, chaos, canonical transformations and generating functions, Liouville's theorem and Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems, invariant curves and cantori, nonlinear resonances, resonance ov

Subjects

classical mechanics | classical mechanics | phase space | phase space | computation | computation | Lagrangian formulation | Lagrangian formulation | action | action | variational principles | variational principles | equations of motion | equations of motion | Hamilton's principle | Hamilton's principle | conserved quantities | conserved quantities | rigid bodies and tops | rigid bodies and tops | Hamiltonian formulation | Hamiltonian formulation | canonical equations | canonical equations | surfaces of section | surfaces of section | chaos | chaos | canonical transformations | canonical transformations | generating functions | generating functions | Liouville's theorem | Liouville's theorem | Poincar? integral invariants | Poincar? integral invariants | Poincar?-Birkhoff | Poincar?-Birkhoff | KAM theorem | KAM theorem | invariant curves | invariant curves | cantori | cantori | nonlinear resonances | nonlinear resonances | resonance overlap | resonance overlap | transition to chaos | transition to chaos | chaotic motion | chaotic motion | 12.620 | 12.620 | 6.946 | 6.946 | 8.351 | 8.351

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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12.802 Wave Motions in the Ocean and Atmosphere (MIT)

Description

This course is an introduction to basic ideas of geophysical wave motion in rotating, stratified, and rotating-stratified fluids. Subject begins with general wave concepts of phase and group velocity. It also covers the dynamics and kinematics of gravity waves with a focus on dispersion, energy flux, initial value problems, etc. Also addressed are subject foundation used to study internal and inertial waves, Kelvin, Poincare, and Rossby waves in homogeneous and stratified fluids. Laplace tidal equations are applied to equatorial waves. Other topics include: resonant interactions, potential vorticity, wave-mean flow interactions, and instability.

Subjects

geophysical wave motion | rotating | stratified | and rotating-stratified fluids | general wave concepts | phase | group velocity | dynamics and kinematics of gravity waves | dispersion | energy flux | initial value problems | internal and inertial waves | Kelvin | Poincare | and Rossby waves | homogeneous and stratified fluids | Laplace tidal equations | equatorial waves | resonant interactions | potential vorticity | wave-mean flow interactions | instability | 12. Kelvin | Poincare | and Rossby waves | Kelvin | Poincare | and Rossby waves | internal gravity waves | surface gravity waves | rotation | large-scale hydrostatic motions | vertical structure equation | equatorial ?-plane | Stratified Quasi-Geostrophic Motion

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.385 Nonlinear Dynamics and Chaos (MIT)

Description

Nonlinear dynamics with applications. Intuitive approach with emphasis on geometric thinking, computational and analytical methods. Extensive use of demonstration software. Topics: Bifurcations. Phase plane. Nonlinear coupled oscillators in biology and physics. Perturbation, averaging theory. Parametric resonances, Floquet theory. Relaxation oscillations. Hysterises. Phase locking. Chaos: Lorenz model, iterated mappings, period doubling, renormalization. Fractals. Hamiltonian systems, area preserving maps; KAM theory.Technical RequirementsMATLAB® software is required to run the .m files found on this course site.MATLAB® is a trademark of The MathWorks, Inc.

Subjects

Phase plane | limit cycles | Poincare-Bendixson theory | Time-dependent systems | Floquet theory | Poincare maps | averaging | Stability of equilibria | near-equilibrium dynamics | Center manifolds | elementary bifurcations | normal forms | chaos

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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12.620J Classical Mechanics: A Computational Approach (MIT)

Description

12.620J covers the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. The course uses computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration.The following topics are covered: the Lagrangian formulation, action, variational principles, and equations of motion, Hamilton's principle, conserved quantities, rigid bodies and tops, Hamiltonian formulation and canonical equations, surfaces of section, chaos, canonical transformations and generating functions, Liouville's theorem and Poincaré integral invariants, Poincaré-Birkhoff and KAM theorems, invariant curves and cantori, nonlinear resonances, resonance ov

Subjects

classical mechanics | phase space | computation | Lagrangian formulation | action | variational principles | equations of motion | Hamilton's principle | conserved quantities | rigid bodies and tops | Hamiltonian formulation | canonical equations | surfaces of section | chaos | canonical transformations | generating functions | Liouville's theorem | Poincar? integral invariants | Poincar?-Birkhoff | KAM theorem | invariant curves | cantori | nonlinear resonances | resonance overlap | transition to chaos | chaotic motion | 12.620 | 6.946 | 8.351

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.385J Nonlinear Dynamics and Chaos (MIT)

Description

This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.

Subjects

chaos | Floquet theory | Poincare-Bendixson theory | phase plane | limit cycles | time-dependent systems | Poincare maps | stability of equilibria | near-equilibrium dynamics | center manifolds | elementary bifurcations | normal forms

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.152 Introduction to Partial Differential Equations (MIT) 18.152 Introduction to Partial Differential Equations (MIT)

Description

This course provides a solid introduction to Partial Differential Equations for advanced undergraduate students. The focus is on linear second order uniformly elliptic and parabolic equations. This course provides a solid introduction to Partial Differential Equations for advanced undergraduate students. The focus is on linear second order uniformly elliptic and parabolic equations.

Subjects

Harmonic functions | Harmonic functions | Harnack inequality | Harnack inequality | gradient estimate | gradient estimate | Hopf Maximum Principle | Hopf Maximum Principle | Poincare Inequalities | Poincare Inequalities | Cacciopolli Inequality | Cacciopolli Inequality | Dirichlet problem | Dirichlet problem | Campanato's lemma | Campanato's lemma | Morrey's lemma | Morrey's lemma | Moser's Approach | Moser's Approach

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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12.006J Nonlinear Dynamics I: Chaos (MIT) 12.006J Nonlinear Dynamics I: Chaos (MIT)

Description

This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering. This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering.

Subjects

Forced and parametric oscillators | Forced and parametric oscillators | Phase space | Phase space | Periodic | quasiperiodic | and aperiodic flows | Periodic | quasiperiodic | and aperiodic flows | Sensitivity to initial conditions and strange attractors | Sensitivity to initial conditions and strange attractors | Lorenz attractor | Lorenz attractor | Period doubling | intermittency | and quasiperiodicity | Period doubling | intermittency | and quasiperiodicity | Scaling and universality | Scaling and universality | Analysis of experimental data: Fourier transforms | Analysis of experimental data: Fourier transforms | Poincar? sections | Poincar? sections | fractal dimension | fractal dimension | Lyaponov exponents | Lyaponov exponents

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.385J Nonlinear Dynamics and Chaos (MIT)

Description

This graduate level course focuses on nonlinear dynamics with applications. It takes an intuitive approach with emphasis on geometric thinking, computational and analytical methods and makes extensive use of demonstration software.

Subjects

Phase plane | limit cycles | Poincare-Bendixson theory | Time-dependent systems | Floquet theory | Poincare maps | averaging | Stability of equilibria | near-equilibrium dynamics | Center manifolds | elementary bifurcations | normal forms | chaos | 18.385 | 2.036

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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STS.042J Einstein, Oppenheimer, Feynman: Physics in the 20th Century (MIT) STS.042J Einstein, Oppenheimer, Feynman: Physics in the 20th Century (MIT)

Description

This class will study some of the changing ideas within modern physics, ranging from relativity theory and quantum mechanics to solid-state physics, nuclear and elementary particles, and cosmology. These ideas will be situated within shifting institutional, cultural, and political contexts. The overall aim is to understand the changing roles of physics and of physicists over the course of the twentieth century. This class will study some of the changing ideas within modern physics, ranging from relativity theory and quantum mechanics to solid-state physics, nuclear and elementary particles, and cosmology. These ideas will be situated within shifting institutional, cultural, and political contexts. The overall aim is to understand the changing roles of physics and of physicists over the course of the twentieth century.

Subjects

relativity theory | relativity theory | quantum mechanics | quantum mechanics | solid-state physics | solid-state physics | elementary particles | elementary particles | quarks | quarks | cosmology | cosmology | nuclear weapons | nuclear weapons | Maxwell | Maxwell | Mach | Mach | Bohr | Bohr | Heisenberg | Heisenberg | McCarthyism | McCarthyism | Poincar? | Poincar? | Schr?dinger | Schr?dinger | nuclear particles | nuclear particles | physics | physics | 20th century | 20th century | twentieth century | twentieth century | physicists | physicists | institutional | political | cultural context | institutional | political | cultural context | STS.042 | STS.042 | 8.225 | 8.225

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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12.006J Nonlinear Dynamics I: Chaos (MIT) 12.006J Nonlinear Dynamics I: Chaos (MIT)

Description

This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering. This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering.

Subjects

Forced and parametric oscillators | Forced and parametric oscillators | Phase space | Phase space | Periodic | quasiperiodic | and aperiodic flows | Periodic | quasiperiodic | and aperiodic flows | Sensitivity to initial conditions and strange attractors | Sensitivity to initial conditions and strange attractors | Lorenz attractor | Lorenz attractor | Period doubling | intermittency | and quasiperiodicity | Period doubling | intermittency | and quasiperiodicity | Scaling and universality | Scaling and universality | Analysis of experimental data: Fourier transforms | Analysis of experimental data: Fourier transforms | Poincar? sections | Poincar? sections | fractal dimension | fractal dimension | Lyaponov exponents | Lyaponov exponents

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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12.620J Classical Mechanics: A Computational Approach (MIT) 12.620J Classical Mechanics: A Computational Approach (MIT)

Description

We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration. We will consider the following topics: the Lagrangian formulation; action, variational principles, and equations of motion; Hamilton's principle; conserved quantities; rigid bodies and tops; Hamiltonian formulation and canonical equations; surfaces of section; chaos; canonical transformations and generating functions; Liouville's theorem and Poincaré integral invariants; Poincaré-Birkhoff and KAM theorems; invariant curves and cantori; nonlinear resonances; resonance overl We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration. We will consider the following topics: the Lagrangian formulation; action, variational principles, and equations of motion; Hamilton's principle; conserved quantities; rigid bodies and tops; Hamiltonian formulation and canonical equations; surfaces of section; chaos; canonical transformations and generating functions; Liouville's theorem and Poincaré integral invariants; Poincaré-Birkhoff and KAM theorems; invariant curves and cantori; nonlinear resonances; resonance overl

Subjects

classical mechanics | classical mechanics | computational classical mechanics | computational classical mechanics | structure and interpretation of classical mechanics | structure and interpretation of classical mechanics | phase space | phase space | lagrangian | lagrangian | action | action | variational principles | variational principles | equation of motion | equation of motion | hamilton principle | hamilton principle | rigid bodies | rigid bodies | Hamiltonian | Hamiltonian | canonical equations | canonical equations | surfaces of section | surfaces of section | canonical transformations | canonical transformations | liouville | liouville | Poincare | Poincare | birkhoff | birkhoff | kam theorem | kam theorem | invariant curves | invariant curves | resonance | resonance | chaos | chaos

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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STS.042J Einstein, Oppenheimer, Feynman: Physics in the 20th Century (MIT) STS.042J Einstein, Oppenheimer, Feynman: Physics in the 20th Century (MIT)

Description

This class explores the changing roles of physics and physicists during the 20th century. Topics range from relativity theory and quantum mechanics to high-energy physics and cosmology. The course also examines the development of modern physics within shifting institutional, cultural, and political contexts, such as physics in Imperial Britain, Nazi Germany, U.S. efforts during World War II, and physicists' roles during the Cold War. This class explores the changing roles of physics and physicists during the 20th century. Topics range from relativity theory and quantum mechanics to high-energy physics and cosmology. The course also examines the development of modern physics within shifting institutional, cultural, and political contexts, such as physics in Imperial Britain, Nazi Germany, U.S. efforts during World War II, and physicists' roles during the Cold War.

Subjects

relativity theory | relativity theory | quantum mechanics | quantum mechanics | solid-state physics | solid-state physics | elementary particles | elementary particles | quarks | quarks | cosmology | cosmology | nuclear weapons | nuclear weapons | Maxwell | Maxwell | Mach | Mach | Poincar? | Poincar? | Bohr | Bohr | Heisenberg | Heisenberg | Schr?dinger | Schr?dinger | McCarthyism | McCarthyism | Einstein | Einstein | Planck | Planck | Feynman | Feynman | scientific frontiers | scientific frontiers

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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12.620J Classical Mechanics: A Computational Approach (MIT)

Description

We will study the fundamental principles of classical mechanics, with a modern emphasis on the qualitative structure of phase space. We will use computational ideas to formulate the principles of mechanics precisely. Expression in a computational framework encourages clear thinking and active exploration. We will consider the following topics: the Lagrangian formulation; action, variational principles, and equations of motion; Hamilton's principle; conserved quantities; rigid bodies and tops; Hamiltonian formulation and canonical equations; surfaces of section; chaos; canonical transformations and generating functions; Liouville's theorem and Poincaré integral invariants; Poincaré-Birkhoff and KAM theorems; invariant curves and cantori; nonlinear resonances; resonance overl

Subjects

classical mechanics | computational classical mechanics | structure and interpretation of classical mechanics | phase space | lagrangian | action | variational principles | equation of motion | hamilton principle | rigid bodies | Hamiltonian | canonical equations | surfaces of section | canonical transformations | liouville | Poincare | birkhoff | kam theorem | invariant curves | resonance | chaos

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.152 Introduction to Partial Differential Equations (MIT)

Description

This course provides a solid introduction to Partial Differential Equations for advanced undergraduate students. The focus is on linear second order uniformly elliptic and parabolic equations.

Subjects

Harmonic functions | Harnack inequality | gradient estimate | Hopf Maximum Principle | Poincare Inequalities | Cacciopolli Inequality | Dirichlet problem | Campanato's lemma | Morrey's lemma | Moser's Approach

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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STS.042J Einstein, Oppenheimer, Feynman: Physics in the 20th Century (MIT)

Description

This class explores the changing roles of physics and physicists during the 20th century. Topics range from relativity theory and quantum mechanics to high-energy physics and cosmology. The course also examines the development of modern physics within shifting institutional, cultural, and political contexts, such as physics in Imperial Britain, Nazi Germany, U.S. efforts during World War II, and physicists' roles during the Cold War.

Subjects

relativity theory | quantum mechanics | solid-state physics | elementary particles | quarks | cosmology | nuclear weapons | Maxwell | Mach | Poincar? | Bohr | Heisenberg | Schr?dinger | McCarthyism | Einstein | Planck | Feynman | scientific frontiers

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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12.006J Nonlinear Dynamics I: Chaos (MIT)

Description

This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering.

Subjects

Forced and parametric oscillators | Phase space | Periodic | quasiperiodic | and aperiodic flows | Sensitivity to initial conditions and strange attractors | Lorenz attractor | Period doubling | intermittency | and quasiperiodicity | Scaling and universality | Analysis of experimental data: Fourier transforms | Poincar? sections | fractal dimension | Lyaponov exponents

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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12.006J Nonlinear Dynamics I: Chaos (MIT)

Description

This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering.

Subjects

Forced and parametric oscillators | Phase space | Periodic | quasiperiodic | and aperiodic flows | Sensitivity to initial conditions and strange attractors | Lorenz attractor | Period doubling | intermittency | and quasiperiodicity | Scaling and universality | Analysis of experimental data: Fourier transforms | Poincar? sections | fractal dimension | Lyaponov exponents

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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12.006J Nonlinear Dynamics I: Chaos (MIT)

Description

This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering.

Subjects

Forced and parametric oscillators | Phase space | Periodic | quasiperiodic | and aperiodic flows | Sensitivity to initial conditions and strange attractors | Lorenz attractor | Period doubling | intermittency | and quasiperiodicity | Scaling and universality | Analysis of experimental data: Fourier transforms | Poincar? sections | fractal dimension | Lyaponov exponents

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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STS.042J Einstein, Oppenheimer, Feynman: Physics in the 20th Century (MIT)

Description

This class will study some of the changing ideas within modern physics, ranging from relativity theory and quantum mechanics to solid-state physics, nuclear and elementary particles, and cosmology. These ideas will be situated within shifting institutional, cultural, and political contexts. The overall aim is to understand the changing roles of physics and of physicists over the course of the twentieth century.

Subjects

relativity theory | quantum mechanics | solid-state physics | elementary particles | quarks | cosmology | nuclear weapons | Maxwell | Mach | Bohr | Heisenberg | McCarthyism | Poincar? | Schr?dinger | nuclear particles | physics | 20th century | twentieth century | physicists | institutional | political | cultural context | STS.042 | 8.225

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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