Maximum Principles for the Hill's Equation

By Alberto Cabada, José Ángel Cid and Lucía López-Somoza

Maximum Principles for the Hill's Equation focuses on the application of these methods to nonlinear equations with singularities (e.g. Brillouin-bem focusing equation, Ermakov-Pinney,…) and for problems with parametric dependence. The authors discuss the properties of the related Green’s functions coupled with different boundary value conditions. In addition, they establish the equations’ relationship with the spectral theory developed for the homogeneous case, and discuss stability and constant sign solutions. Finally, reviews of present classical and recent results made by the authors and by other key authors are included.

• Evaluates classical topics in the Hill’s equation that are crucial for understanding modern physical models and non-linear applications
• Describes explicit and effective conditions on maximum and anti-maximum principles
• Collates information from disparate sources in one self-contained volume, with extensive referencing throughout

• Language: English
• Publisher: Academic Press
• Release date: Oct 27, 2017
• ISBN: 9780128041260

Alberto Cabada

Alberto Cabada is Professor at the University of Santiago de Compostela (Spain). His line of research is devoted to the existence and multiplicity of solutions of nonlinear differential equations, both ordinary and partial, as well as difference and fractional ones. He is the author of more than one hundred forty research articles indexed in the Citation Index Report and has authored two monographs.

Book preview

2017

Chapter 1

Introduction

Abstract

In Section 1.1 we briefly introduce Hill's equation pointing out some applications. In Section 1.2 we deal with the concept of Lyapunov stability for Hill's equation that will be used along the book. Finally, at Section 1.3 we present the characterization of the stability by means of Floquet's theory, giving an explicit condition in terms of the discriminant of the Hill's equation.

Keywords

Hill's equation; Lyapunov stability; Floquet theory; Lyapunov criterium

Chapter Outline

1.1  Hill's Equation

1.2  Stability in the Sense of Lyapunov

1.3  Floquet's Theorem for the Hill's Equation

References

1.1 Hill's Equation

The Hill's equation,

(1.1)

has numerous applications in engineering and physics. Among them we can find some problems in mechanics, astronomy, circuits, electric conductivity of metals and cyclotrons. Hill's equation is named after the pioneering work of the mathematical astronomer George William Hill (1838–1914), see [6]. He also made contributions to the three and the four body problems.

Moreover, the theory related to the Hill's equation can be extended to every differential equation in the form

(1.2)

such that the coefficients p and q have enough regularity. This is due to the fact that, with a suitable change of variable, the previous equation transforms in one of the type of (1.1) (see the details in Section 2.2 of Chapter 2).

As a first example we could consider a mass-spring system, that is, a spring with a mass m the position of the mass at the instant t and assuming absence of friction, the previous model can be expressed as

the elastic constant of the spring.

However, in a real physical system, there exists a friction force which opposes the movement and is proportional to the object's speed. In this case the situation can be modelled by the equation

with μ the so-called friction coefficient. The value of such coefficient is characteristic of the environment where the object oscillates, and depends, among other variables, on the density, temperature, and pressure of the environment. However, it could be considered a situation in which the spring moves between two different environments, each one with its particular friction coefficient. Also, the environment could have strong variations of density or temperature that could cause changes in the friction coefficient depending on time. This could be modelled by substituting the friction coefficient μ

Another possible situation would be that one in which there exists another external force acting periodically on the mass in such a way that it tends to move the mass back into its position of equilibrium, acting in proportion to the distance to that position. Including this new term in the previous model we have

In any of the two cases, we obtain an equation in the form has enough regularity, we could do the following change of variable

and transform the equation in one in the form (1.1).

A second example studied in [4,8] is the inverted pendulum. A mathematical pendulum consists of a particle of mass m connected to a base through a string (which is supposed to be rigid and of despicable weight) in such a way that the mass moves in a fixed vertical plane. If the particle moves by the force of gravity, then the movement of the pendulum is given by the equation

where g denotes the gravity, l the length of the string, and θ represents the angle between the string and the perpendicular line to the base.

so the equation of movement could be rewritten as

. Then, as it is proved in [4], the equation of movement would change into

which is of the form (1.1).

Other equations which fit into the framework of the Hill's equation are the following ones:

•  Airy's equation: (see [13])

This equation appears in the study of the diffraction of light, the diffraction of radio waves around the Earth's surface, in aerodynamics and in the swing of a uniform vertical column which bounds under its own weight.

•  Mathieu's equation: (see [2,14,15])

It is the result of the analysis of the phenomenon of parametric resonance associated with an oscillator whose parameters change with time. It appears in problems related to periodic movements, as the trajectory of an electron in a periodic arrange of atoms.

When studying oscillation phenomena of the solutions of (1.1), it is observed that these are determined by the potential alarge enough. Moreover, the larger the potential a is, the faster the solutions of (1.1) oscillate.

By simply considering that every integrable function can be rewritten as

, it is obvious that studying the potentials a for which the solutions of Eq. for which the equation

(1.3)

, has no trivial solution.

If we consider Eq. (1.3) submitted to boundary value conditions, we have a spectral problem.

First studies about the Hill's equation were focused on the homogeneous case, from the point of view of the classical oscillation theory of Sturm-Liouville ([5,9,13]). In particular, from the study of Eq. (1.3) under periodic boundary conditions, important results related to the stability of solutions were obtained.

Afterwards the nonhomogeneous periodic problem,

. In this case it results specially interesting the study of constant sign solutions when σ does not change sign. This situation could be interpreted in a physical way by considering σ as an external force acting over the system; then constant sign solutions would mean that a positive perturbation maintains oscillations above or below the equilibrium point. The study of constant sign solutions carried to consider comparison principles (MP and AMP) which, later, were related to the constant sign of the Green's function.

1.2 Stability in the Sense of Lyapunov

In this section we will restrict ourselves to the concept of stability in the case of the Hill's equation (1.1), although this concept can be stated for the solution of any differential equation, see [3]. Firstly, consider the potential a . We will denote the unique solution of

(1.4)

in order to stress its dependence on the initial conditions.

Definition 1

Stability in the sense of Lyapunov

of such that if

then

.

Definition 2

of such that

.

are stable but unbounded. However, in the linear case, and in particular for problem (1.4), we have the following useful result.

Theorem 1

The following claims are equivalent:

1.  The solution of (1.4) is stable for each .

2.  The solution of (1.4) is stable.

3.  The solution of (1.4) is bounded for each .

Proof

Firstly, we are going to prove that 1 and 2 are equivalent. Taking into account that, by linearity,

we have that

.

Now, we will show that 2 and 3 are equivalent. Again by linearity we have

. □

Definition 3

In view of .

Otherwise, we will say that Eq. (1.1), or the trivial solution of (1.1), is unstable.

Floquet's theory for first order linear differential systems with T-periodic coefficients, see , where

(1.5)

are the solutions of (1.1) satisfying the initial conditions

system

(1.6)

of (1.6) is a nonsingular matrix C satisfying