Darboux Theorem Research Articles

An exactly soluble Schrödinger equation with a bistable potential

For a bistable potential which is the sum of two hyperbolic cosine functions, the Schrödinger equation for the low-lying states of a homonuclear diatomic molecule can be solved analytically. In this model the potential and the energy eigenvalues depend on three parameters, and the resulting wave functions are continuous and have continuous derivatives everywhere. This last property of the wave functions enables one to generate a family of soluble bistable potentials by applying the theorem of Darboux to the discrete eigenfunctions of the Schrödinger equation.

Generalization of Darboux's theorem

Generalization of Darboux's theorem

Aspects of cauchy elastic materials

The Darboux theorem on the classification and canonical forms of first degree exterior differential forms is used to obtain a canonical representation of Cauehy elastic materials. It is shown that the balance of moment of momentum and the principle of material indifference are independent for materials whose Darboux class is greater than 1. By means of the Caratheodory's theorem on inaccessibility it is shown that any homogeneous deformation state can be reached from any other homogeneous deformation state by a path of zero work whenever the Darboux class of the material is 3 or greater.

A guiding center Hamiltonian: A new approach

A Hamiltonian treatment of the guiding center problem is given which employs noncanonical coordinates in phase space. Separation of the unperturbed system from the perturbation is achieved by using a coordinate transformation suggested by a theorem of Darboux. As a model to illustrate the method, motion in the magnetic field B=B (x,y) ? is studied. Lie transforms are used to carry out the perturbation expansion.

Transformation methods in the analysis of series for critical properties

Abstract During the past twenty years, considerable use has been made of conformal transformations as an aid to series analysis in the study of critical phenomena; however, there has been no evident systematic approach to the task of choosing the most suitable transformation for a given series. In this review we discuss the theory of transformations and provide such a rational and systematic method of approach. The purpose of transformation is to map the singularity of interest (usually the critical point) significantly closer to the origin than any other singularity, so that it dominates the later coefficients of the series; extrapolation methods based on Darboux's theorems may then be employed. We show that, for series likely to arise in thermodynamics, it is always possible to find transformations which achieve this purpose. We provide a set of conditions, which should be satisfied by any transformation function, to ensure straightforward and valid analysis, and we discuss the basic types of transformation and their selection in practice. In the final sections, we illustrate the approach with some important examples, and show that it leads to transformed series which are much smoother than those previously obtained.

Darboux classification of cauchy elastic bodies and some of its consequences

The Darboux theorem on the classification and canonical forms of first degree differential forms is used to obtain a classification and canonical representation of Cauchy elastic solids. If the stress is assumed to be determined uniquely by the Green deformation tensor C, the double Piola-Kirchhoff stress tensor is shown to belong to one of six possible classes, each of which possesses a unique representation in terms of linear combinations of gradients with respect to C of scalar-valued functions with scalar-valued coefficients. There are three distinct classes for isotropic materials, each of which has a unique representation. It is shown that these three classes are distinguishable in terms of the existence or nonexistence of cycles with nonzero total work and by the accessibility or inaccessibility of any deformation state from any neighboring deformation state by a path of zero work.

Darboux's Theorem and Points of Inflection

Darboux's Theorem and Points of Inflection

Almost-Riemannian Structures on Banach Manifolds: The Morse Lemma and the Darboux Theorem

In this paper we introduce the notion of an almost Riemannian manifold. Briefly speaking, an almost-Riemannian structure on a Banach manifold is a generalization of the notion of a Riemannian structure on a Hilbert manifold, common examples would be manifolds of maps modelled on the Sobolev spaces Lkp. The successful use of weak Riemannian structures in hard problems has been given by Ebin [2] and Ebin and Marsden [10].

Darboux's Theorem Fails for Weak Symplectic Forms

Darboux's Theorem Fails for Weak Symplectic Forms

Lectures on Differential Geometry.

Algebraic Preliminaries: 1. Tensor products of vector spaces 2. The tensor algebra of a vector space 3. The contravariant and symmetric algebras 4. Exterior algebra 5. Exterior equations Differentiable Manifolds: 1. Definitions 2. Differential maps 3. Sard's theorem 4. Partitions of unity, approximation theorems 5. The tangent space 6. The principal bundle 7. The tensor bundles 8. Vector fields and Lie derivatives Integral Calculus on Manifolds: 1. The operator $d$ 2. Chains and integration 3. Integration of densities 4. $0$ and $n$-dimensional cohomology, degree 5. Frobenius' theorem 6. Darboux's theorem 7. Hamiltonian structures The Calculus of Variations: 1. Legendre transformations 2. Necessary conditions 3. Conservation laws 4. Sufficient conditions 5. Conjugate and focal points, Jacobi's condition 6. The Riemannian case 7. Completeness 8. Isometries Lie Groups: 1. Definitions 2. The invariant forms and the Lie algebra 3. Normal coordinates, exponential map 4. Closed subgroups 5. Invariant metrics 6. Forms with values in a vector space Differential Geometry of Euclidean Space: 1. The equations of structure of Euclidean space 2. The equations of structure of a submanifold 3. The equations of structure of a Riemann manifold 4. Curves in Euclidean space 5. The second fundamental form 6. Surfaces The Geometry of $G$-Structures: 1. Principal and associated bundles, connections 2. $G$-structures 3. Prolongations 4. Structures of finite type 5. Connections on $G$-structures 6. The spray of a linear connection Appendix I: Two existence theorems Appendix II: Outline of theory of integration on $E^n$ Appendix III: An algebraic model of transitive differential geometry Appendix IV: The integrability problem for geometrical structures References Index.

Locally one-to-one mappings and a classical theorem on schlicht functions

Abstract : Some topological theorems are given which are similar to Darboux's Theorem and from one of which (Theorem 1) Darboux's Theorem is easily deduced. Our theorems are like Darboux's Theorem in the sense that we assume a certain mapping to be one-to-one on the boundary of its domain and conclude that it is one-to-one everywhere on its domain. (Author)