Nav: Home

Problems in mechanics open the door to the orderly world of chaos

August 22, 2016

The word "chaos" has two meanings that are almost exact opposites. In general usage, it means "wild unpredictable confusion". In physics and mathematics it often refers to the behavior of systems for which the tools of traditional mathematics fail and one is forced to think in a rigorous new semi-quantitative way.

The first person to realize this was the French mathematician Henri Poincaré. In the late 19th century he was challenged to prove that the solar system is stable, but instead raised the possibility that it might not be! If one ignores the gravitational attraction between the planets one is left with a solvable problem that appears in all undergraduate mechanics texts. Including these small perturbations leads to a problem that can be described with partial differential equations, but these equations have no proper solutions except possibly in terms of infinite series. These equations can be solved numerically of course, but the results show an odd combination of order and unpredictability. Poincaré went on to lay the foundation for the study of such systems. He identified criteria that make a system inescapably chaotic and showed how a kind of order can emerge from the chaos. In this century, the Russian mathematician Kolomogrov developed what has come to be called the KAM theorem; one of the crowning works of modern mathematics. He showed that there is an odd fractal-like landscape of infinite series solutions in otherwise unsolvable problems.

Lectures in Nonlinear Mechanics and Chaos Theory begins by reviewing the tools of traditional classical mechanics--the Hamiltonian formulation, abstract transformation theory, and perturbation theory--and shows how they ultimately fail. It then moves on to the landmarks of chaos theory, the Poincaré-Hopf or "hairy ball" theorem, followed by the Poincaré-Birkoff theorem for rational winding numbers, and finally, the KAM theorem. These are discussed in terms of rigorous mathematics and illustrated with numerous examples of computer-drawn solutions. It finishes with a discussion of the relevance of the KAM theorem and measure theory to the ergodic hypothesis.

This book is based on a one-quarter course in graduate mechanics that has been given in the Physics Department of Oregon State University. It is intended to be used as a textbook to review conventional mechanics and introduce students to more recent developments in chaos theory.

This book retails for US$35 / £29 (paperback) and US$70 / £58 (hardback), and is also available on Amazon, Barnes and Noble, and other major online booksellers. To know more about the book or to purchase a copy, visit http://www.worldscientific.com/worldscibooks/10.1142/10070
-end-
About the Author

Albert W Stetz is Professor Emeritus in the Physics Department of Oregon State University, USA, with a career in nuclear and elementary particle physics. He currently teaches courses for non-majors in topics such as quantum mechanics, cosmology, and the physics and philosophy of time.

About World Scientific Publishing Co.

World Scientific Publishing is a leading independent publisher of books and journals for the scholarly, research, professional and educational communities. The company publishes about 600 books annually and about 130 journals in various fields. World Scientific collaborates with prestigious organizations like the Nobel Foundation, US National Academies Press, as well as its subsidiary, the Imperial College Press, amongst others, to bring high quality academic and professional content to researchers and academics worldwide. To find out more about World Scientific, please visit http://www.worldscientific.com. For more information, contact Amanda Yun at heyun@wspc.com

World Scientific

Related Mathematics Articles:

People can see beauty in complex mathematics, study shows
Ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a beautiful landscape painting or a piano sonata.
Improving geothermal HVAC systems with mathematics
Sustainable heating, ventilation, and air conditioning systems, such as those that harness low-enthalpy geothermal energy, are needed to reduce collective energy use and mitigate the continued effects of a warming climate.
How the power of mathematics can help assess lung function
Researchers at the University of Southampton have developed a new computational way of analyzing X-ray images of lungs, which could herald a breakthrough in the diagnosis and assessment of chronic obstructive pulmonary disease (COPD) and other lung diseases.
Mathematics pushes innovation in 4-D printing
New mathematical results will provide a potential breakthrough in the design and the fabrication of the next generation of morphable materials.
More democracy through mathematics
For democratic elections to be fair, voting districts must have similar sizes.
More Mathematics News and Mathematics Current Events

Best Science Podcasts 2019

We have hand picked the best science podcasts for 2019. Sit back and enjoy new science podcasts updated daily from your favorite science news services and scientists.
Now Playing: TED Radio Hour

Rethinking Anger
Anger is universal and complex: it can be quiet, festering, justified, vengeful, and destructive. This hour, TED speakers explore the many sides of anger, why we need it, and who's allowed to feel it. Guests include psychologists Ryan Martin and Russell Kolts, writer Soraya Chemaly, former talk radio host Lisa Fritsch, and business professor Dan Moshavi.
Now Playing: Science for the People

#537 Science Journalism, Hold the Hype
Everyone's seen a piece of science getting over-exaggerated in the media. Most people would be quick to blame journalists and big media for getting in wrong. In many cases, you'd be right. But there's other sources of hype in science journalism. and one of them can be found in the humble, and little-known press release. We're talking with Chris Chambers about doing science about science journalism, and where the hype creeps in. Related links: The association between exaggeration in health related science news and academic press releases: retrospective observational study Claims of causality in health news: a randomised trial This...