Numerical Hamiltonian Problems

J.M. Sanz-Serna, M.P. Calvo

• Publisher: Dover Publications
• Year: 2018
• Print ISBN: 9780486824109
• eBook ISBN: 9780486831527
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

This advanced text explores a category of mathematical problems that occur frequently in physics and other sciences. Five preliminary chapters make the book accessible to students without extensive background in this area. Topics include Hamiltonian systems, symplecticness, numerical methods, order conditions, and implementation.
The heart of the book, chapters 6 through 10, explores symplectic integration, symplectic order conditions, available symplectic methods, numerical experiments, and properties of symplectic integrators. The final four chapters contain more advanced material: generating functions, Lie formalism, high-order methods, and extensions. Many numerical examples appear throughout the text.

CHAPTER 1

Hamiltonian systems

1.1 Hamiltonian systems

This chapter and the next are a first introduction to Hamiltonian problems: more advanced material is presented later as required. A good starting point for the mathematical theory of Hamiltonian systems is the textbook by Arnold (1989). MacKay and Meiss (1987) have compiled an excellent collection of important papers in Hamiltonian dynamics. The article by Berry in this collection is particularly recommended. For an introduction to the more geometric modern approach the book by Marsden (1992) is an advisable choice.

We start by describing the class of problems we shall be concerned with and by presenting some notation. Let Ω be a domain (i.e., a nonempty, open, connected subset) in the oriented Euclidean space

of the points (p, q) = (p₁, . . . p_d, q₁, . . ., q_d). We denote by I an open interval of the real line

of the variable t (time); I may be bounded, I = (a, b), or unbounded, I = (−∞, b), I = (a, ∞), I = (−∞, ∞). If H = H(p, q, t) is a sufficiently smooth real function defined in the product Ω × I, then the Hamiltonian system of differential equations with Hamiltonian H is, by definition, given by

The integer d is called the number of degrees of freedom and Ω is the phase space. The product Ω × I is the extended phase space. The exact amount of smoothness required of H will vary from place to place and will not be explicitly stated, but throughout we assume at least C² continuity, so that the right-hand side of the system (1.1) is C¹ and the standard existence and uniqueness theorems apply to the corresponding initial value problem. Sometimes, the symbol S_H will be used to refer to the system (1.1).

Usually, in applications to mechanics (Arnold (1989)), the q variables are generalized coordinates, the p variables the conjugated generalized momenta and H corresponds to the total mechanical energy.

In many Hamiltonian systems of interest, the Hamiltonian H does not explicitly depend on t; then (1.1) is an autonomous system of differential equations. For autonomous problems we shall consider H as a function defined in the phase space Ω, rather than as a function defined in

and independent of the last variable.

It is sometimes useful to combine all the dependent variables in (1.1) in a 2d-dimensional vector y = (p, q). Then (1.1) takes the simple form

where ∇ is the gradient operator

and J is the 2d × 2d skew-symmetric matrix

(I and 0 respectively represent the unit and zero d × d matrices).

Upon differentiation of H with respect to t along a solution of (1.1), we find

so that, in view of (1.2) and of the skew-symmetry of J⁻¹,

In particular, if H is autonomous, dH/dt = 0. Then H is a conserved quantity that remains constant along solutions of the system. In the applications, this usually corresponds to conservation of energy.

We now turn to some concrete examples of Hamiltonian systems. These examples have been chosen for their simplicity. More realistic examples from celestial mechanics, plasma physics, molecular dynamics etc. can be found in the literature of the corresponding fields.

1.2 Examples of Hamiltonian systems

1.2.1 The harmonic oscillator

This is the well-known system with d = 1 (one degree of freedom)

Here, m and k are positive constants that, for the familiar case of a material point attached to a spring, respectively correspond to mass and spring constant. Of course...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface
• 1. Hamiltonian systems
• 2. Symplecticness
• 3. Numerical methods
• 4. Order conditions
• 5. Implementation
• 6. Symplectic integration
• 7. Symplectic order conditions
• 8. Available symplectic methods
• 9. Numerical experiments
• 10. Properties of symplectic integrators
• 11. Generating functions
• 12. Lie formalism
• 13. High-order methods
• 14. Extensions
• References
• Symbol index
• Index