Numerical Hamiltonian Problems

Numerical Hamiltonian Problems

by J.M. Sanz-Serna, M.P. Calvo
Numerical Hamiltonian Problems

Numerical Hamiltonian Problems

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

eBook

$11.49  $14.95 Save 23% Current price is $11.49, Original price is $14.95. You Save 23%.

Available on Compatible NOOK Devices and the free NOOK Apps.
WANT A NOOK?  Explore Now

Related collections and offers

LEND ME® See Details

Overview

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.

Product Details

ISBN-13: 9780486831527
Publisher: Dover Publications
Publication date: 06/13/2018
Series: Dover Books on Mathematics
Sold by: Barnes & Noble
Format: eBook
Pages: 224
File size: 10 MB

About the Author



J. M. Sanz-Serna is on the faculty of Charles III University of Madrid.
M. P. Calvo teaches at the University of Valladolid.

Read an Excerpt

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 R2d of the points (p, q) = (p1, ..., pd, q1, ..., qd). We denote by I an open interval of the real line R 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 sm∞th real function defined in the product Ω x I, then the Hamiltonian system of differential equations with Hamiltonian H is, by definition, given by

[MATHEMATICAL EXPRESSION OMITTED] (1.1)

The integer d is called the number of degrees of freedom and Ω is the phase space. The product Ω x 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 SH 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 Ω x R 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

dy/dt = J-1[nabla]H, (1.2)

where [nabla] is the gradient operator

[MATHEMATICAL EXPRESSION OMITTED] (1.3)

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

[MATHEMATICAL EXPRESSION OMITTED] (1.4)

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

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

[MATHEMATICAL EXPRESSION OMITTED]

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

[MATHEMATICAL EXPRESSION OMITTED]

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)

H = T + V, T = p21(2m), V = kq21/2.

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 T and V are the kinetic and potential energies.

In situations with d = 1, it is clearly convenient to use the notation p and q for the dependent variables (rather than p1, q1). With this notation, the equations (1.1) for the harmonic oscillator read

[MATHEMATICAL EXPRESSION OMITTED] (1.5)

Here and elsewhere dots represent differentiation with respect to t. The general solution for q is an oscillation

q(t) = C1sin(ωt + C2), ω = [square root of ((k/m))],

with angular frequency ω (period T = 2π/ω, frequency v = 1/T = ω/(2π)); C1 and C2 are integration constants. Similarly p is given by

p(t) = mωC1 cos(ωt + C2).

The particular solution that takes the initial value (p(O), q(O)) at t = 0 is easily written in matrix form:

[MATHEMATICAL EXPRESSION OMITTED] (1.6)

When plotted in the phase (p, q)-plane, the parametric curves (p(t), q(t)) correspond to the ellipses

H(p, q) = p2/(2m) + kq2/2 = constant.

These are circles when mk = 1 (or, equivalently, when mω = 1).

1.2.2 The pendulum

If the units are chosen in such a way that the mass of the blob, the length of the rod and the acceleration of gravity are all unity, then

H = T + V, T = p2/2, V = -cos q,

where q is the angle between the rod and a vertical, downward oriented axis. The equations of motion are then

p = -sin q, q = p.

In the phase (p, q)-plane depicted in Fig. 1.1 the solutions lie in the level curves H = constant. We only consider the situation near q = O; the phase portrait repeats itself periodically along the q-axis because H is a periodic function of q. There is a stable equilibrium at p = 0, q = 0 (the pendulum rests in its lowest position), surrounded by libration solutions where q varies periodically between values qmax > 0 and qmin = -qmax. The libration trajectories fill the region -1 < H 1. For H > 1 we find rotation solutions, where q varies monotonically. The level set H = 1 is composed of the unstable equilibria at q = π, q = -π (pendulum resting in the highest position) and of the separatrices connecting them.

To integrate the equations of motion, we substitute p by q in the energy equation H(p, q) = h, with h a constant. This yields a differential equation

[MATHEMATICAL EXPRESSION OMITTED] (1.7)

that is readily integrated in terms of quadratures of elementary functions.

In particular, we see that the period of a libration solution is given by

[MATHEMATICAL EXPRESSION OMITTED]

Note that h = H (0, qmax) = -cos qmax. For amplitudes qmax close to 0, the period T is close to 2π (the period of the linearized equations p = -q, q = p). As qmax approaches π, the trajectory approaches the separatrix and the period approaches ∞. The dependence of the period on the amplitude is typical of nonlinear oscillations; for the (linear) harmonic oscillator all solutions have of course the same period.

1.2.3 The double harmonic oscillator

This has two degrees of freedom and

[MATHEMATICAL EXPRESSION OMITTED]

In the equations of motion p1 = -ω1q1, q1 = ω1p1, p2 = -ω2q2, q2 = ω2p2, the (p1, q1) variables are not coupled to the variables (p2, q2); we are considering two uncoupled harmonic oscillators. According to our previous discussion of the harmonic oscillator, the projections of the solutions of the double harmonic oscillator onto the (pi, qi)-plane correspond to circles and possess angular frequency ωi > 0. If ω1/ω2 is a rational number r/s, then the solutions of the double harmonic oscillator are periodic, with period T = 2πr/ω1 = 2πs/ω2; the trajectory returns to its initial position in the four-dimensional phase space after having completed r cycles of the (p1, q1) variables and s cycles of the (p2, q2) variables. If ω1/ω2 is irrational, the trajectory never returns to its initial location.

A geometric picture is useful. The system has two conserved quantities

[MATHEMATICAL EXPRESSION OMITTED]

that represent the energies in each of the two uncoupled oscillators. In the phase space (p1, p2, q1, q2), the level sets H1 = constant1, H2 = constant2 represent 2-dimensional tori. These tori are invariant: if a trajectory is at time t = 0 on one of the tori it is on that torus for all times t. The phase φ1 = arctan(p1/q1) of the first oscillation represents the longitude in the torus and the phase φ2 of the second oscillation represents the latitude in the torus. The longitude and latitude along a trajectory vary periodically with angular frequencies ω1 and ω2. For ω1/ω2 = r/s the trajectory returns to its initial location after winding itself on the torus r times in the direction of the parallels and s times in the direction of the meridians. For ω1/ω2 irrational, the trajectory never returns to its initial position. It can be shown that it is actually dense on the torus surface, and even ergodic, i.e., the trajectory stays in each domain D on the torus surface an amount of time proportional to the area of D (Arnold (1989), Section 51).

In the case with ω1/ω2 irrational the vector-valued function (p1(t), p2(t), q1(t), q2(t)) is quasiperiodic (Siegel and Moser (1971), Section 36). In general, a function F(t) is said to be quasiperiodic, with frequencies ω1, ω2, if it can be expanded in a series of the form

[MATHEMATICAL EXPRESSION OMITTED]

If ω2 is an integer multiple of ω1 (or, more generally, if ω1/ω2 is rational), then this series reduces to a Fourier series for a periodic function. All quantities mω1 + nω2 are then integer multiples of a single value ω.

Quasiperiodic functions with k > 2 frequencies ωi can be defined in an obvious way and would appear in the study of k uncoupled harmonic oscillators.

1.2.4 Kepler's problem

Kepler's problem describes the motion in a plane (the configuration plane) of a material point that is attracted towards the origin with a force inversely proportional to the distance squared. In nondimensional form,

[MATHEMATICAL EXPRESSION OMITTED]

The equations of motion are then

[MATHEMATICAL EXPRESSION OMITTED]

Since the problem is autonomous, the Hamiltonian (energy) H is a conserved quantity. Furthermore, due to the central character of the force (Arnold (1989), Sections 6-7), there is a second conserved quantity: the angular momentum

M = q1p2 - q2p1. (1.8)

For the analysis, it is best to employ polar coordinates (r, θ) in the configuration (q1, q2)-plane. Then, the corresponding momenta are pr = r and pθ = r2θ and the Hamiltonian becomes

[MATHEMATICAL EXPRESSION OMITTED]

The equations (1.1) are then

[MATHEMATICAL EXPRESSION OMITTED] (1.9)

pθ = 0, (1.10)

r = pr, (1.11)

θ = pθ/r2. (1.12)

From (1.10) we see that pθ is a constant of motion; in fact an easy computation shows that pθ is in fact the polar coordinate expression of the angular momentum M whose cartesian expression is (1.8). Upon replacing pθ by a constant M and pr by r in the equation H = h = constant, that expresses conservation of energy, we obtain a first-order differential equation for r

[MATHEMATICAL EXPRESSION OMITTED] (1.13)

that can be solved by quadratures, cf. (1.7). If the constant H is negative and M ≠ 0, then r = r(t) librates periodically between a minimum rmin > 0 and a maximum rmax. The points in the configuration plane where r is minimum are called pericentres; those corresponding to maximum r are called apocentres. The period of r is found to be (Arnold (1989), Section 8)

[MATHEMATICAL EXPRESSION OMITTED] (1.14)

Once r = r(t) is known, a quadrature in (1.12) (with pθ = M) yields θ = θ(t). It turns out that the polar angle θ between a pericentre and the next pericentre is exactly 2π. Hence after time T, not only r reassumes its initial value, but the moving material point returns to its initial position in the configuration plane.

Hence the trajectory in this plane is a closed curve; this trajectory is an ellipse, of course. All four functions (pr, pθ, r, θ) (or the cartesian (p1, p2, q1, q2)) are periodic with period (1.14).

Example 1.1 Consider the initial conditions

[MATHEMATICAL EXPRESSION OMITTED] (1.15)

where e is a parameter (0 ≤ e ≤ 1). The period (1.14) of the solution is readily found to be 2π. The values rmax and rmin can be computed by setting r = 0 in the equation of conservation of energy. It turns out that rmax = 1 + e and rmin = 1 - e. Hence the initial condition (1.15) corresponds to the pericentre and the major semiaxis of the ellipse is 1. Furthermore the distance from the centre of the ellipse to the origin (focus of the ellipse) equals e, so that the parameter e represents the eccentricity.

(Continues…)


Excerpted from "Numerical Hamiltonian Problems"
by .
Copyright © 1994 J. M. Sanz-Serna and M. P. Calvo.
Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Table of Contents

Preface1. Hamiltonian systems2. Symplecticness3. Numerical methods4. Order conditions5. Implementation6. Symplectic integration7. Symplectic order conditions8. Available symplectic methods9. Numerical experiments10. Properties of symplectic integrators11. Generating functions12. Lie formalism13. High-order methods14. ExtensionsReferencesSymbol indexIndex
From the B&N Reads Blog

Customer Reviews