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

August 22, 2016The 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:**

More democracy through mathematics

For democratic elections to be fair, voting districts must have similar sizes.

For democratic elections to be fair, voting districts must have similar sizes.

How to color a lizard: From biology to mathematics

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

Mathematics supports a new way to classify viruses based on structure

New research supports a structure-based classification system for viruses which could help in the identification and treatment of emerging viruses.

New research supports a structure-based classification system for viruses which could help in the identification and treatment of emerging viruses.

US educators awarded for exemplary teaching in mathematics

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Authors of year's best books in mathematics honored

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

The mathematics of coffee extraction: Searching for the ideal brew

Composed of over 1,800 chemical components, coffee is one of the most widely-consumed drinks in the world.

Composed of over 1,800 chemical components, coffee is one of the most widely-consumed drinks in the world.

Even physicists are 'afraid' of mathematics

Physicists avoid highly mathematical work despite being trained in advanced mathematics, new research suggests.

Physicists avoid highly mathematical work despite being trained in advanced mathematics, new research suggests.

Mathematics and music: New perspectives on the connections between these ancient arts

World-leading experts on music and mathematics present insights on the connections between these two ancient arts, especially as they relate to composition and performance, as well as creativity, education, and geometry.

World-leading experts on music and mathematics present insights on the connections between these two ancient arts, especially as they relate to composition and performance, as well as creativity, education, and geometry.

Kindergarteners' mathematics success hinges on preschool skills

In a study funded by the National Science Foundation, researchers at the University of Missouri discovered that preschoolers who better process words associated with numbers and understand the quantities associated with these words are more likely to have success with math when they enter kindergarten.

In a study funded by the National Science Foundation, researchers at the University of Missouri discovered that preschoolers who better process words associated with numbers and understand the quantities associated with these words are more likely to have success with math when they enter kindergarten.

First international mathematics research institute launched in Australia

World leaders in the mathematical sciences are visiting Melbourne for a series of research programs at Australia's first international research institute for mathematics and statistics.

World leaders in the mathematical sciences are visiting Melbourne for a series of research programs at Australia's first international research institute for mathematics and statistics.

**Related Mathematics Reading:**

## 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**

**Changing The World**

What does it take to change the world for the better? This hour, TED speakers explore ideas on activismâwhat motivates it, why it matters, and how each of us can make a difference. Guests include civil rights activist Ruby Sales, labor leader and civil rights activist Dolores Huerta, author Jeremy Heimans, "craftivist" Sarah Corbett, and designer and futurist Angela Oguntala.

**Now Playing: Science for the People**

**#520 A Closer Look at Objectivism**

This week we broach the topic of Objectivism. We'll be speaking with Keith Lockitch, senior fellow at the Ayn Rand Institute, about the philosophy of Objectivism as it's taught through Ayn Rand's writings. Then we'll speak with Denise Cummins, cognitive scientist, author and fellow at the Association for Psychological Science, about the impact of Objectivist ideology on society. Related links: This is what happens when you take Ayn Rand seriously Another Critic Who Doesn’t Care What Rand Thought or Why She Thought It, Only That She’s Wrong Quote is from "A Companion to Ayn Rand"