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Methods Of Modern Mathematical Physics

Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

http://www.math.ecnu.edu.cn/~zmwang/teaching/Advanced_ODE-2014-2015/1-Textbook_and_Reference_book/2-Ordinary%20Differential%20Equations%20with%20Applications.pdf

Ordinary Differential Equations with Applications

Publisher's description: The use of geometric methods in classical mechanics has proven fruitful, with wide applications in physics and engineering. In this book, Professor Marsden concentrates on these geometric aspects, especially on symmetry techniques. The main points he covers are: the stability of relative equilibria, which is analyzed using the block diagonalization technique; geometric phases, studied using the reduction and reconstruction technique; and bifurcation of relative equilibria and chaos in mechanical systems. A unifying theme for these points is provided by reduction theory, the associated mechanical connection and techniques from dynamical systems. These methods can be applied to many control and stabilization situations, and this is illustrated using rigid bodies with internal rotors, and the use of geometric phases in mechanical systems. To illustrate the above ideas and the power of geometric arguments, the author studies a variety of specific systems, including the double spherical pendulum and the classical rotating water molecule.

/pdf/lectures-on-mechanics-4k4jcuwf13.pdf

Lectures on Mechanics

Can one hear the shape of a drum

Reviews - Eigenfunction Expansions Associated with Second-Order Differential Equations. By E. C. Titchmarsh Pp. 175. 20s. 1946. (Oxford University)

*The first author's work was partially supported by FONDECYT (Chile), project 0132-88, and by a Summer Research Fellowship provided by the Research Council of the University of Missouri-Columbia. He would like to thank the Physics Department and others at the Universidad de Chile for their hospitality during his visit in April, 1990, when much of this research was completed. The second author was supported in part by FONDECYT (Chile), projects 0132-88 and 1238-90. Both authors also thank Fritz Gesztesy for general comments and encouragement.

A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions

This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H.C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S n or H n . Introduction The Eigenvalue Problems The first eigenvalue problem we shall introduce is that of the fixed membrane, or Dirichlet Laplacian . We consider the eigenvalues and eigenfunctions of –Δ on a bounded domain (=connected open set) Ω in Euclidean space R n , i.e., the problem It is well-known that this problem has a real and purely discrete spectrum where Here each eigenvalue is repeated according to its multiplicity. An associated orthonormal basis of real eigenfunctions will be denoted u 1 , u 2 , u 3 , …. In fact, throughout this paper we will assume that all functions we consider are real. This entails no loss of generality in the present context. The next problem we introduce is that of the free membrane, or Neumann Laplacian .

Isoperimetric and universal inequalities for eigenvalues

In 1955 and 1956 Payne, Pólya, and Weinberger considered the problem of bounding ratios of eigenvalues for homogeneous membranes of arbitrary shape [PPW1, PPW2]. Among other things, they showed that the ratio k2IK °f * ^ r s t t w o eigenvalues was less than or equal to 3 and went on to conjecture that the optimal upper bound for A2/Aj was its value for the disk, approximately 2.539. It is this conjecture which we establish below. Since 1956 various authors have attempted to prove the conjecture of Payne, Pólya, and Weinberger and some have been able to improve upon the constant 3. Specifically, Brands [Br] in 1964 obtained the value 2.686, de Vries [dV] in 1967 obtained 2.658, and Chiti [Ch2] in 1983 obtained 2.586. In addition, Thompson [Th] gave the natural extension of the PPW argument to dimension n, obtaining

/pdf/proof-of-the-payne-polya-weinberger-conjecture-1nghp1xg04.pdf

Proof of the Payne-Pólya-Weinberger conjecture

On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions

On demontre la borne inferieure optimale λ 2 −λ 1 ≥3Π 2 /d 2 pour la difference des deux premieres valeurs propres d'un operateur de Schrodinger a une dimension −d 2 /dx 2 +V(x) avec un potentiel symetrique a un seul puits sur un intervalle de longueur d et avec des conditions aux limites de Dirichlet

https://www.ams.org/proc/1989-105-02/S0002-9939-1989-0942630-X/S0002-9939-1989-0942630-X.pdf

Mark S. Ashbaugh

Papers

A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions

Isoperimetric and universal inequalities for eigenvalues

Proof of the Payne-Pólya-Weinberger conjecture

On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions

Optimal lower bound for the gap between the first two eigenvalues of one-dimensional Schrödinger operators with symmetric single-well potentials