S. G. Rajeev
- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199670857
- eISBN:
- 9780191775154
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199670857.003.0011
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter derives the consequences of scale invariance in the three body problem. Jacobi co-ordinates are introduced. The orbits are geodesics in a certain metric. The three body problem with ...
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This chapter derives the consequences of scale invariance in the three body problem. Jacobi co-ordinates are introduced. The orbits are geodesics in a certain metric. The three body problem with inverse cube force is shown to be more symmetric. Montgomery's ‘pair of pants’ metric is derived. The orbits are geodesics in a metric of negative curvature on the plane with three points removed. The Virial Theorem is suggested as an exercise.Less
This chapter derives the consequences of scale invariance in the three body problem. Jacobi co-ordinates are introduced. The orbits are geodesics in a certain metric. The three body problem with inverse cube force is shown to be more symmetric. Montgomery's ‘pair of pants’ metric is derived. The orbits are geodesics in a metric of negative curvature on the plane with three points removed. The Virial Theorem is suggested as an exercise.
S. G. Rajeev
- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199670857
- eISBN:
- 9780191775154
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199670857.003.0012
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter includes a detailed study of the three body problem where one of the bodies is infinitesimal and the other two are in circular motion. This applies to the Sun-Jupiter-Asteroid, ...
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This chapter includes a detailed study of the three body problem where one of the bodies is infinitesimal and the other two are in circular motion. This applies to the Sun-Jupiter-Asteroid, Earth-Moon-Satellite systems. Lagrange's surprising discovery of stable equilibrium points and orbits around them is explained. Jacobi's integral and Hill's regions are derived. A spin-off is a theory of stability in the presence of velocity dependent (Coriolis) forces: sometimes a maximum of the potential can be a stable equilibrium.Less
This chapter includes a detailed study of the three body problem where one of the bodies is infinitesimal and the other two are in circular motion. This applies to the Sun-Jupiter-Asteroid, Earth-Moon-Satellite systems. Lagrange's surprising discovery of stable equilibrium points and orbits around them is explained. Jacobi's integral and Hill's regions are derived. A spin-off is a theory of stability in the presence of velocity dependent (Coriolis) forces: sometimes a maximum of the potential can be a stable equilibrium.
S. G. Rajeev
- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199670857
- eISBN:
- 9780191775154
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199670857.001.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This book begins with the ancient parts of classical mechanics: the variational principle, Lagrangian and Hamiltonian formalisms, and Poisson brackets. The simple pendulum provides a glimpse of the ...
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This book begins with the ancient parts of classical mechanics: the variational principle, Lagrangian and Hamiltonian formalisms, and Poisson brackets. The simple pendulum provides a glimpse of the beauty of elliptic curves, which will also appear later in rigid body mechanics. Geodesics in Riemannian geometry are presented as an example of a Hamiltonian system. Conversely, the path of a non-relativistic particle is a geodesic in a metric that depends on the potential. Orbits around a black hole are found. Hamilton-Jacobi theory is discussed, showing a path towards quantum mechanics and a connection to the eikonal of optics. The three body problem is studied in detail, including small orbits around the Lagrange points. The dynamics of a charged particle in a magnetic field, especially a magnetic monopole, is studied in the Hamiltonian formalism. Spin is shown to be a classical phenomenon. Symplectic integrators that allow numerical solutions of mechanical systems are derived. A simplified version of Feigenbaum's theory of period doubling introduces chaos. Following a classification of Mobius transformations, this book studies chaos on the complex plane: Julia sets, Fatou sets, and the Mandelblot are explained. Newton's method for solution of non-linear equations is viewed as a dynamical system, allowing a novel approach to the reduction of matrices to canonical form. This is used as a stepping stone to the KAM theory of maps of a circle to itself, unravelling a connection to the Diophantine problem of number theory. KAM theory of the solution of the Hamilton-Jacobi equation using Newton's iteration concludes the book.Less
This book begins with the ancient parts of classical mechanics: the variational principle, Lagrangian and Hamiltonian formalisms, and Poisson brackets. The simple pendulum provides a glimpse of the beauty of elliptic curves, which will also appear later in rigid body mechanics. Geodesics in Riemannian geometry are presented as an example of a Hamiltonian system. Conversely, the path of a non-relativistic particle is a geodesic in a metric that depends on the potential. Orbits around a black hole are found. Hamilton-Jacobi theory is discussed, showing a path towards quantum mechanics and a connection to the eikonal of optics. The three body problem is studied in detail, including small orbits around the Lagrange points. The dynamics of a charged particle in a magnetic field, especially a magnetic monopole, is studied in the Hamiltonian formalism. Spin is shown to be a classical phenomenon. Symplectic integrators that allow numerical solutions of mechanical systems are derived. A simplified version of Feigenbaum's theory of period doubling introduces chaos. Following a classification of Mobius transformations, this book studies chaos on the complex plane: Julia sets, Fatou sets, and the Mandelblot are explained. Newton's method for solution of non-linear equations is viewed as a dynamical system, allowing a novel approach to the reduction of matrices to canonical form. This is used as a stepping stone to the KAM theory of maps of a circle to itself, unravelling a connection to the Diophantine problem of number theory. KAM theory of the solution of the Hamilton-Jacobi equation using Newton's iteration concludes the book.
Mingwei Song
- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780252041754
- eISBN:
- 9780252050428
- Item type:
- chapter
- Publisher:
- University of Illinois Press
- DOI:
- 10.5622/illinois/9780252041754.003.0007
- Subject:
- Literature, 20th-century and Contemporary Literature
This chapter introduces the life and work of Liu Cixin, a Chinese science-fiction writer who has played a major role in reviving the genre in twenty-first-century China. The chapter discusses Liu’s ...
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This chapter introduces the life and work of Liu Cixin, a Chinese science-fiction writer who has played a major role in reviving the genre in twenty-first-century China. The chapter discusses Liu’s work in the context of the genre’s history in China. Liu and other writers belonging to the same generation have created a new wave, in which the genre has gained unprecedented popularity in China. The main part of the chapter analyzes several major works by Liu and attempts to theorize the aesthetics and politics of the new wave as represented in Liu’s stories and novels. The new wave makes visible the hidden dimensions of Chinese science fiction, together with the darker side of reality that it speaks to.Less
This chapter introduces the life and work of Liu Cixin, a Chinese science-fiction writer who has played a major role in reviving the genre in twenty-first-century China. The chapter discusses Liu’s work in the context of the genre’s history in China. Liu and other writers belonging to the same generation have created a new wave, in which the genre has gained unprecedented popularity in China. The main part of the chapter analyzes several major works by Liu and attempts to theorize the aesthetics and politics of the new wave as represented in Liu’s stories and novels. The new wave makes visible the hidden dimensions of Chinese science fiction, together with the darker side of reality that it speaks to.
Veronica Hollinger
- Published in print:
- 2017
- Published Online:
- May 2019
- ISBN:
- 9781496811523
- eISBN:
- 9781496811561
- Item type:
- chapter
- Publisher:
- University Press of Mississippi
- DOI:
- 10.14325/mississippi/9781496811523.003.0002
- Subject:
- Literature, Film, Media, and Cultural Studies
In “‘Great Wall Planet’: Estrangements of Chinese Science Fiction,” Veronica Hollinger offers a few observations about Chinese SF, specifically in terms of its potential to defamiliarize what ...
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In “‘Great Wall Planet’: Estrangements of Chinese Science Fiction,” Veronica Hollinger offers a few observations about Chinese SF, specifically in terms of its potential to defamiliarize what scholars have begun to refer to as “global science fiction.” She suggests five ways in which Chinese SF can estrange a taken-for-granted Anglo-American mainstream: 1) as an “alien” cultural product; 2) as a product of China’s “rise” as a global superpower; 3) as the product of an “alternate” cultural history; 4) as representative of something called “global science fiction”; and 5) as a kind of “second-language” version of the discourse of Anglo-American globalization. She uses fictional works by Liu Cixin and Han Song to emphasize her points.Less
In “‘Great Wall Planet’: Estrangements of Chinese Science Fiction,” Veronica Hollinger offers a few observations about Chinese SF, specifically in terms of its potential to defamiliarize what scholars have begun to refer to as “global science fiction.” She suggests five ways in which Chinese SF can estrange a taken-for-granted Anglo-American mainstream: 1) as an “alien” cultural product; 2) as a product of China’s “rise” as a global superpower; 3) as the product of an “alternate” cultural history; 4) as representative of something called “global science fiction”; and 5) as a kind of “second-language” version of the discourse of Anglo-American globalization. She uses fictional works by Liu Cixin and Han Song to emphasize her points.
David D. Nolte
- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198805847
- eISBN:
- 9780191843808
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805847.003.0009
- Subject:
- Physics, History of Physics
Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are ...
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Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable. In the hands of Vladimir Arnold and Jürgen Moser, this became the Kolmo–Arnol–Mos (KAM) theory of Hamiltonian chaos. This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory. Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor. Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.Less
Half a century after Poincaré first glimpsed chaos in the three-body problem, the great Russian mathematician Andrey Kolmogorov presented a sketch of a theorem that could prove that orbits are stable. In the hands of Vladimir Arnold and Jürgen Moser, this became the Kolmo–Arnol–Mos (KAM) theory of Hamiltonian chaos. This chapter shows how KAM theory fed into topology in the hands of Stephen Smale and helped launch the new field of chaos theory. Edward Lorenz discovered chaos in numerical models of atmospheric weather and discovered the eponymous strange attractor. Mathematical aspects of chaos were further developed by Mitchell Feigenbaum studying bifurcations in the logistic map that describes population dynamics.
Nathalie Deruelle and Jean-Philippe Uzan
- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198786399
- eISBN:
- 9780191828669
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198786399.003.0013
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter discusses the N-body problem. In 1886, Karl Weierstrass submitted the following question to the scientific community on the occasion of a mathematical competition to mark the 60th ...
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This chapter discusses the N-body problem. In 1886, Karl Weierstrass submitted the following question to the scientific community on the occasion of a mathematical competition to mark the 60th birthday of King Oscar II of Sweden. Weierstrass asked that, ‘given a system of arbitrarily many mass points that attract each other according to Newton’s laws, try to find, under the assumption that no two points ever collide, a representation of the coordinates of each point as a series in a variable which is some known function of time and for all of whose values the series converges uniformly’. Henri Poincaré showed that the equations of motion for more than two gravitational bodies are not in general integrable and won the competition. However, the jury awarded the prize to Poincaré not for solving the problem, but for coming up with the first ideas of what later became known as chaos theory.Less
This chapter discusses the N-body problem. In 1886, Karl Weierstrass submitted the following question to the scientific community on the occasion of a mathematical competition to mark the 60th birthday of King Oscar II of Sweden. Weierstrass asked that, ‘given a system of arbitrarily many mass points that attract each other according to Newton’s laws, try to find, under the assumption that no two points ever collide, a representation of the coordinates of each point as a series in a variable which is some known function of time and for all of whose values the series converges uniformly’. Henri Poincaré showed that the equations of motion for more than two gravitational bodies are not in general integrable and won the competition. However, the jury awarded the prize to Poincaré not for solving the problem, but for coming up with the first ideas of what later became known as chaos theory.
