A Cantor set of tori with monodromy near a focus-focus singularity

We write down an asymptotic expression for action coordinates in an integrable Hamiltonian system with a focus-focus equilibrium. From the singularity in the actions we deduce that the Arnol'd determinant grows infinitely large near the pinched torus. Moreover, we prove that it is possible to globally parametrise the Liouville tori by their frequencies. If one perturbs this integrable system, then the KAM tori form a Whitney smooth family: they can be smoothly interpolated by a torus bundle that is diffeomorphic to the bundle of Liouville tori of the unperturbed integrable system. As is well-known, this bundle of Liouville tori is not trivial. Our result implies that the KAM tori have monodromy. In semi-classical quantum mechanics, quantisation rules select sequences of KAM tori that correspond to quantum levels. Hence a global labeling of quantum levels by two quantum numbers is not possible.Comment: 11 pages, 2 figure

Design, fabrication and test of graphite/polyimide composite joints and attachments

The design, analysis, and testing performed to develop four types of graphite/polyimide (Gr/PI) bonded and bolted composite joints for lightly loaded control surfaces on advanced space transportation systems that operate at temperatures up to 561 K (550 F) are summarized. Material properties and small specimen tests were conducted to establish design data and to evaluate specific design details. Static discriminator tests were conducted on preliminary designs to verify structural adequacy. Scaled up specimens of the final joint designs, representative of production size requirements, were subjected to a series of static and fatigue tests to evaluate joint strength. Effects of environmental conditioning were determined by testing aged (125 hours at 589 K (600 F)) and thermal cycled (116 K to 589 K (-250 F to 600 F), 125 times) specimens. It is concluded Gr/PI joints can be designed and fabricated to carry the specified loads. Test results also indicate a possible resin loss or degradation of laminates after exposure to 589 K (600 F) for 125 hours

Zeeman's monotonicity conjecture

In this paper we prove a conjecture of Zeeman about the monotonicity of the rotation number of a family of dieomorphisms φ of the first quadrant Q of R

The Cartan form for constrained Lagrangian systems and the nonholonomic Noether theorem

This paper deals with conservation laws for mechanical systems with nonholonomic constraints. It uses a Lagrangian formulation of nonholonomic systems and a Cartan form approach. We present what we believe to be the most general relations between symmetries and first integrals. We discuss the so-called nonholonomic Noether theorem in terms of our formalism, and we give applications to Riemannian submanifolds, to Lagrangians of mechanical type, and to the determination of quadratic first integrals.Comment: 25 page

Nonlinear stability of two-layer shallow water flows with a free surface

The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two ‘baroclinic’ parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability

Test and analysis of Celion 3000/PMR-15, graphite/polyimide bonded composite joints: Summary

Standard single lap, double lap and symmetric step lap bonded joints of Celion 3000/PMR-15 graphite/polyimide composite were evaluated. Composite to composite and composite to titanium joints were tested at 116K (-250 F), 294K (70 F) and 561K (550 F). Joint parameters evaluated were lap length, adherend thickness, adherend axial stiffness, lamina stacking sequence and adherend tapering. Tests of advanced joint concepts were also conducted to establish the change in performance of preformed adherends, scalloped adherends and hybrid systems. Special tests were conducted to establish material properties of the high temperature adhesive, designated A7F, used for bonding. Most of the bonded joint tests resulted in interlaminar shear or peel failures of the composite. There were very few adhesive failures. Average test results agree with expected performance trends for the various test parameters. Results of finite element analyses and of test/analysis correlations are also presented