} ''Symmetry in Mechanics: A Gentle, Modern Introduction'' is an undergraduate textbook on mathematics and

mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

, centered on the use of

symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the ...

to solve the

Kepler problem In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ...

. It was written by Stephanie Singer, and published by

Birkhäuser Birkhäuser was a Swiss publisher founded in 1879 by Emil Birkhäuser. It was acquired by Springer Science+Business Media in 1985. Today it is an imprint used by two companies in unrelated fields: * Springer continues to publish science (particu ...

in 2001.

Topics

The

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...

is a special case of the

two-body problem In classical mechanics, the two-body problem is to predict the motion of two massive objects which are abstractly viewed as point particles. The problem assumes that the two objects interact only with one another; the only force affecting each ...

in which two point masses interact by

Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...

(or by any

central force In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force. : \vec = \mathbf(\mathbf) = \left\vert F( \mathbf ) \right\vert \hat where \vec F is the force, F is a vecto ...

obeying an

inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be unde ...

). The book starts and ends with this problem, the first time in an ad hoc manner that represents the problem using a system of twelve variables for the positions and momentum vectors of the two bodies, uses the conservation laws of physics to set up a system of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...

s obeyed by these variables, and solves these equations. The second time through, it describes the positions and variables of the two bodies as a single point in a 12-dimensional

phase space In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usual ...

, describes the behavior of the bodies as a

Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...

, and uses

symplectic reduction In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the actio ...

s to shrink the phase space to two dimensions before solving it to produce

Kepler's laws of planetary motion In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular or ...

in a more direct and principled way. The middle portion of the book sets up the machinery of symplectic geometry needed to complete this tour. Topics covered in this part include

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

s, vector fields and

differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many application ...

pushforward The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. * Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" op ...

s and pullbacks,

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...

s, Hamiltonian energy functions, the representation of finite and infinitesimal physical symmetries using

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

s and

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

s, and the use of the

moment map In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the actio ...

to relate symmetries to conserved quantities. In these topics, as well, concrete examples are central to the presentation.

Audience and reception

The book is written as a textbook for undergraduate mathematics and physics students, with many exercises, and it assumes that the students are already familiar with

multivariable calculus Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather t ...

and

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matric ...

, a significantly lower level of background material than other books on symplectic geometry in mechanics. It is not comprehensive in its coverage of symplectic geometry and mechanics, but could be used as auxiliary reading in a class that covers that material from other sources, such as Abraham and Marsden's ''Foundations of Mechanics'' or Arnold's ''Mathematical Methods of Classical Mechanics''. Alternatively, on its own, it can provide a more accessible first course in this material, before presenting it more comprehensively in another course. Reviewer William Satzer writes that this book "makes serious efforts to address real students and their potential difficulties" and shifts comfortably between mathematical and physical views of its problem. Similarly, reviewer J. R. Dorfman writes that it "removes some of the language barriers that divide the worlds of mathematics and physics", and reviewer Jiří Vanžura calls it "remarkable" in its dual ability to motivate mathematical methods for physics students and provide applications in physics for mathematics students, adding that "The book is perfectly written and serves very well its purpose." Reviewer Ivailo Mladenov notes with approval the book's attention to example-first exposition, and despite pointing to a minor inaccuracy regarding the nationality of

Sophus Lie Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. Life and career Marius Sophu ...

, recommends it to both undergraduate and graduate students. Reviewer Richard Montgomory writes that the book does "an excellent job of leading the reader from the Kepler problem to a view of the growing field of symplectic geometry".

References

{{reflist, refs= {{citation , last = Abbott , first = Steve , date = November 2001 , doi = 10.2307/3621823 , issue = 504 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, jstor = 3621823 , page = 571 , title = Review of ''Symmetry in Mechanics'' , volume = 85 {{citation , last = Dorfman , first = J. R. , date = January 2002 , doi = 10.1063/1.1457270 , issue = 1 , journal =

Physics Today ''Physics Today'' is the membership magazine of the American Institute of Physics. First published in May 1948, it is issued on a monthly schedule, and is provided to the members of ten physics societies, including the American Physical Society ...

, pages = 57–57 , title = Review of ''Symmetry in Mechanics'' , volume = 55 {{citation , last1 = Jamiołkowski , first1 = A. , last2 = Mrugała , first2 = R. , bibcode = 2002RpMP...49..123J , date = February 2002 , doi = 10.1016/s0034-4877(02)80009-x , issue = 1 , journal =

Reports on Mathematical Physics ''Reports on Mathematical Physics'' () is a peer-reviewed scientific journal, started in 1970, which publishes papers in theoretical physics that present a rigorous mathematical approach to problems of quantum and classical mechanics, field theo ...

, pages = 123–124 , title = Review of ''Symmetry in Mechanics'' , volume = 49 {{citation , last = Mladenov , first = Ivailo , journal =

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ...

, title = Review of ''Symmetry in Mechanics'' , zbl = 0970.70003; see also Mladenov's review in {{MR, 1816059 {{citation , last = Montgomery , first = Richard , date = April 2003 , doi = 10.2307/3647898 , issue = 4 , journal =

American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an ...

, jstor = 3647898 , pages = 348–353 , title = Review of ''Symmetry in Mechanics'' , url = http://montgomery.math.ucsc.edu/papers/Symm_in_Mech_Review.PDF , volume = 110 {{citation , last = Satzer , first = William J. , date = December 2005 , journal = MAA Reviews , publisher = Mathematical Association of America , title = Review of ''Symmetry in Mechanics'' , url = https://www.maa.org/press/maa-reviews/symmetry-in-mechanics-a-gentle-modern-introduction {{citation , last = Vanžura , first = Jiří , issue = 1 , journal = Mathematica Bohemica , page = 112 , title = Review of ''Symmetry in Mechanics'' , url = https://dml.cz/dmlcz/133933 , volume = 128 , year = 2003 Hamiltonian mechanics Mathematics textbooks 2001 non-fiction books