On Curvatures and Focal Points of Distributions of Dynamical Lagrangian Distributions and their Reductions by First Integrals

Pairs (Hamiltonian system, Lagrangian distribution) called dynamical Lagrangian distributions, appear naturally in differential geometry, calculus of variations, and rational mechanics. The basic differential invariants of a dynamical Lagrangian distribution with respect to the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and curvature form. These invariants can be considered as generalizations of the classical curvature tensor in Riemannian geometry. In particular, in terms of these invariants one can localize the focal points along extremals of the corresponding variational problems. In the present paper we study the behavior of the curvature operator, the curvature form, and the focal points of a dynamical Lagrangian distribution after its reduction by arbitrary first integrals in involution. An interesting phenomenon is that the curvature form of so-called monotone increasing Lagrangian dynamical distributions, which appear naturally in mechanical systems, does not decrease after reduction. It also turns out that the set of focal points to the given point with respect to the monotone increasing dynamical Lagrangian distribution and the corresponding set of focal points with respect to its reduction by one integral are alternating sets on the corresponding integral curve of the Hamiltonian system of the considered dynamical distributions. Moreover, the first focal point corresponding to the reduced Lagrangian distribution comes before any focal point related to the original dynamical distribution. We illustrate our results on the classical N-body problem. 2000 Mathematics Subject Classification. 37J15, 37J05, 53D20.     

3-D NONLINEAR SUPERCAVITATING FLOW IN A GRAVITY FIELD

In this paper,perturbation method and Fourier-cosine-expansion method are used to solve a 3-D nonlinear problem of a supercavitating flow in an inclined field of gravity at large Froude numbers.By expanding the velocity potential into a power series of a small parameter,the original 3-D nonlinear problem is reduced to a number of 2-D ones. The solutions of the first three orders are derived in detail and expressed in terms of the complete elliptic integrals of the first and second kinds.Then the boundary integral equation method is applied to get the numerical solutions for each order.Computational results are provided for supercavitating flows past cones under various flow conditions.     

Stereology of tubular structures

The morphology of a tubular structure can be characterized, in at least some of its important respects, through stereological methods. We study the geometric meaning of standard stereological quantities when applied to tubular structures, with particular regard to their curvature or tortuosity. Measures defined specifically in relation to tubular structures are also introduced for practical use. The ideal smooth bent cylinder, used here, is not realistic but provides general principles from which a more specialized investigation should be developed. The emphasis is placed on simple counting methods of measurement, specifically the tangent counts introduced by DeHoff, and counts of section profiles exhibiting a particular character (such as ‘figure-eights’). These measures convey information about the tortuosity of tubular structures, whereas the standard indices V_v, S_v, M_v give no information about tortuosity. Some data from human testicular tubules are discussed.     

涡流检测中轴对称场的快速计算

本对涡流检测中解析求解轴对称场时遇到的无穷积分给出一种简单有效的数值积分方法,提出一种改进积分收敛性的方法,加速了无穷积分的截断,显提高了轴对称场的计算效率。几组典型数据的计算结果与ＦＥＭ－ＢＥＭ组合法计算结果的比较表明,本的方法是准确、快速的,这一计算方法的高效率使其可应用于涡流无损检测中缺陷的重构。