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Meyer Kenneth R. - Introduction to Hamiltonian Dynamical Systems and the N-Body Problem

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Meyer Kenneth R. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem
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This third edition text provides expanded material on the restricted three body problem and celestial mechanics. With each chapter containing new content, readers are provided with new material on reduction, orbifolds, and the regularization of the Kepler problem, all of which are provided with applications. The previous editions grew out of graduate level courses in mathematics, engineering, and physics given at several different universities. The courses took students who had some background in differential equations and lead them through a systematic grounding in the theory of Hamiltonian mechanics from a dynamical systems point of view. This text provides a mathematical structure of celestial mechanics ideal for beginners, and will be useful to graduate students and researchers alike. Reviews of the second edition: The primary subject here is the basic theory of Hamiltonian differential equations studied from the perspective of differential dynamical systems. The N-body problem is used as the primary example of a Hamiltonian system, a touchstone for the theory as the authors develop it. This book is intended to support a first course at the graduate level for mathematics and engineering students. ... It is a well-organized and accessible introduction to the subject ... This is an attractive book ... (William J. Satzer, The Mathematical Association of America, March, 2009) The second edition of this text infuses new mathematical substance and relevance into an already modern classic ... and is sure to excite future generations of readers. ... This outstanding book can be used not only as an introductory course at the graduate level in mathematics, but also as course material for engineering graduate students. ... it is an elegant and invaluable reference for mathematicians and scientists with an interest in classical and celestial mechanics, astrodynamics, physics, biology, and related fields. (Marian Gidea, Mathematical Reviews, Issue 2010 d).;Beginnings -- Hamiltonian Systems -- Celestial Mechanics -- The Restricted Problem -- Topics in Linear Theory -- Local Geometric Theory -- Symplectic Geometry -- Special Coordinates -- Poincars Continuation Method -- Normal Forms -- Bifurcations of Periodic Orbits -- Stability and KAM Theory -- Variational Techniques.

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Meyer Kenneth R. - Introduction to Hamiltonian Dynamical Systems and the N-Body Problem
Introduction to Hamiltonian Dynamical Systems and the N-Body Problem
Meyer Kenneth R.
Introduction to Hamiltonian Dynamical Systems and the N-Body Problem

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Springer International Publishing AG 2017
Kenneth R. Meyer and Daniel C. Offin Introduction to Hamiltonian Dynamical Systems and the N-Body Problem Applied Mathematical Sciences 10.1007/978-3-319-53691-0_1
1. Beginnings
Kenneth R. Meyer 1 and Daniel C. Offin 2
(1)
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA
(2)
Department of Mathematic and Statistics, Queens University, Kingston, ON, Canada
Newtons second law of motion gives rise to a system of second-order differential equations in Picture 1 and so to a system of first-order equations in Picture 2 , an even-dimensional space. If the forces are derived from a potential function, the equations of motion of the mechanical system have many special properties, most of which follow from the fact that the equations of motion can be written as a Hamiltonian system. The Hamiltonian formalism is the natural mathematical structure to develop the theory of conservative mechanical systems such as the equations of celestial mechanics.
This chapter defines a Hamiltonian system of ordinary differential equations, gives some basic results about such systems, and presents several classical examples. This discussion is informal. Some of the concepts introduced in the setting of these examples are fully developed later.
1.1 Hamiltonian Equations
A Hamiltonian system is a system of 2 n ordinary differential equations of the form
Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 3
(1.1)
where H = H ( t , q , p ), the Hamiltonian, is a smooth real-valued function defined for Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 4 , an open set in Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 5 . The vectors.
The variable t is called time, and the dot represents differentiation with respect to t, i.e., Picture 6 . This notation goes back to Newton who created a fluxion from a fluent by placing a dot over it.
The variables q and p are said to be conjugate variables: p is conjugate to q . The concept of conjugate variable grows in importance as the theory develops. The integer n is the number of degrees of freedom of the system.
In general, introduce the 2 n vector z , the 2 n 2 n skew-symmetric matrix J , and the gradient by
Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 7
where 0 is the n n zero matrix and I is the n n identity matrix. In this notation () becomes
Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 8
(1.2)
One of the basic results from the general theory of ordinary differential equations is the existence and uniqueness theorem. This theorem states that for each Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 9 , there is a unique solution z =( t , t 0, z 0) of ( ) for details of the theory of ordinary differential equations.
In the special case when H is independent of t so that Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 10 where Picture 11 is some open set in Picture 12 , the differential equations () satisfying (0, z 0)= z 0. The solutions are pictured as parameterized curves called orbits in Picture 13 , and the set Picture 14 is called the phase space . By the existence and uniqueness theorem, there is a unique curve through each point in Picture 15 ; and by the uniqueness theorem, two such solution curves cannot cross in Picture 16 . The space obtained by identifying an orbit to a point is called the orbit space . As we shall see in this chapter, orbit spaces can be very nice or very bad.
An integral for ( We say such a system in completely integrable .
A classical example is angular momentum for the central force problem. The equation Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 17 where Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 18 can be written as a Hamiltonian system
where The angular momentum A q p is a vector of integrals since 12 The - photo 19
where
The angular momentum A q p is a vector of integrals since 12 The - photo 20
The angular momentum A = q p is a vector of integrals since
12 The Poisson Bracket Many of the special properties of Hamiltonian systems - photo 21
1.2 The Poisson Bracket
Many of the special properties of Hamiltonian systems are formulated in terms of the Poisson bracket operator, so this operator plays a central role in the theory developed here. Let H , F , and G be smooth functions from Introduction to Hamiltonian Dynamical Systems and the N-Body Problem - image 22 into and define the Poisson bracket of F and G by 13 Clearly F G is - photo 23 , and define the Poisson bracket of F and G by
13 Clearly F G is a smooth map from to as well and one can easily - photo 24
(1.3)
Clearly { F , G } is a smooth map from Picture 25 to as well and one can easily verify that is skew-symmetric and bilinear A - photo 26 as well, and one can easily verify that {,} is skew-symmetric and bilinear. A little tedious calculation verifies Jacobis identity :
14 By a common abuse of notation let H t H t t t 0 z 0 - photo 27
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