Integrability of the Yang-Mills Hamiltonian system

This paper considers the integrability of generalized Yang-Mills system with the HamiltonianH _a (p, q)=1/2(p ₁ ² +p ₂ ² +a ₁ q ₁ ² +a ₂ q ₂ ² )+1/4q ₁ ⁴ +1/4a ₃ q ₂ ⁴ + 1/2a ₄ q ₁ ² q ₂ ² . We prove that the system is integrable for the cases: (A)a ₁=a ₂,a ₃=a ₄=1; (b)a ₁=a ₂,a ₃=1,a ₄=3; (C)a ₁=a ₂/4,a ₃=16,a ₄=6. Our main result is the presentation of these integrals. Only for cases A and B does the Yang-Mills Hamiltonian possess the Painlevé property. Therefore the Painlevé test does not take account of the integrability for the case C.     

Normalization and the detection of integrability: The generalized Van Der Waals potential

Deprit and Miller have conjectured that normalization of integrable Hamiltonians may produce normal forms exhibiting degenerate equilibria to very high order. Several examples in the class of coupled elliptic oscillators are known. In order to test the utility of normalization as a detector of integrability we normalize, to high order, a perturbed Keplerian system known to have several integrable limits; the generalized van der Waals Hamiltonian for a hydrogen atom. While the separable limits give rise to high order degeneracy we find a non-separable, integrable limit for which the normal form does not exhibit degeneracy. We conclude that normalization may, in certain cases, indicate integrability but is not guaranteed to uncover all integrable limits.     

Non-Existence of New Meromorphic First Integrals in the Planar Three-Body Problem

We consider the planar three-body problem and prove that, apart from some exceptional cases, there is no additional first integral meromorphic with respect to positions, mutual distances and momenta.     

从灰度变化获取表面形状的研究

提出了一种基于可信度操作的从灰度变化信息获取表面形状的方法,并结合在频率域的强制可积约束,把对表面梯度空间(p,q)的处理经过高斯球变换到(f,g)空间处理,不仅较好地保证了算法的收敛性和可靠性,而且有利于边界条件的确定。     

On Fröhlich's solution for Boussinesq's problem

The concentration factor introduced by O. K. Fröhlich is visualized as a procedure for examining the pattern of load transfer from surface loads to the interior of a geomaterial. The historical details that led to the introduction of the concentration factor are scant although it is widely used in the area of soil mechanics problems associated with tillage‐induced soil compaction. The purpose of this note is to examine the concentration factor in terms of the geomechanics of an elastic continuum and to identify the precise conditions that are satisfied by the distribution of stresses and strains that accommodate the concentration factor. Copyright © 2013 John Wiley & Sons, Ltd.     

On Integrable Hamiltonians with Velocity Dependent Potentials

We study strongly and weakly integrable 2-dimensional Hamiltonian systems with velocity dependent potentials. We determine the set of conditions which must be satisfied in order to allow the existence of an independent second invariant polynomial in the momenta. We then investigate the linear case for which a complete solution of the problem can be obtained. We recover the classical set of linear strongly integrable systems and provide several new examples of weakly integrable systems whose equations of motion can be explicitly solved at a fixed value of the energy.     

Non-Integrability of the Generalized Two Fixed Centres Problem

We consider two fixed centres attracting a third body. Each centre is the source of a force field with potential V=−ar ⁻²ⁿ , where n is a real number. We prove that this generalization of the classical two fixed centres problem is non-integrable except when n= 0, 1/2, −1 and −2. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Mel’nikov Subharmonic Function and the Non-existence of Analytic Integrals in Non-autonomous Systems

We consider a one-dimensional non-degenerate Hamiltonian system perturbed by a periodically time dependent non-Hamiltonian vector field and show that the non-vanishing of the Melnikov subharmonic function is strongly related to the non-existence of an analytic integral in the perturbed system.     

Generalized Hill's problem: Some cases of complete integrability

In this paper, we are investigating cases of integrability in the planar Hill's problem. The external potential U _extis supposed to be time independent in a given uniformly rotating frame. Cases of integrability of the relative motion of two interacting particles in the vicinity of an equilibrium solution of U _extare found. In all these cases, the form of the second integral is explicitly given, the first being the Jacobian one. Cases in which the interacting potential U between the two particles is of newtonian type are particularized.     

Nonintegrability Detected from Geometrically Similar Orbits

H. Yoshida’s criterion of nonintegrability is restated in the light of the inverse problem of dynamics. It is shown that, given a monoparametric family of geometrically similar orbits, one may be able to assert nonintegrability of all or some homogeneous potentials of integer degreem (≠ 0, ± 2)which can produce this family. This revised version was published online in July 2006 with corrections to the Cover Date.