Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

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

This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control theoretical applications. The basic methodology is that of geometric mechanics applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a ``body reference frame'' relates part of the momentum equation to the components of the Euler-Poincar\'{e} equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory. September 1994 Revised, March 1995 Revised, June 1995

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Nonholonomic Mechanical Systems with Symmetry

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2F60590931937BF33789B5A986069C6D/S0025557200107697a.pdf/div-class-title-span-class-italic-curves-and-singularities-span-by-j-w-bruce-and-p-j-giblin-pp-222-1984-isbn-0-521-24945-7-hardback-25-27091-x-paperback-8-95-cambridge-university-press-div.pdf

Curves and singularities, by J. W. Bruce and P. J. Giblin. Pp 222. 1984. ISBN 0-521-24945-7 (Hardback) £25, 27091-X (Paperback) £8·95 (Cambridge University Press)

Introduction Part 1. Cobordism: Hamiltonian cobordism Abstract moment maps The linearization theorem Reduction and applications Part 2. Quantization: Geometric quantization The quantum version of the linearization theorem Quantization commutes with reduction Part 3. Appendices: Signs and normalization conventions Proper actions of Lie groups Equivariant cohomology Stable complex and Spin$^{\mathrm{c}}$structures Assignments and abstract moment maps Assignment cohomology Non-degenerate abstract moment maps Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization The Kawasaki Riemann-Roch formula Cobordism invariance of the index of a transversally elliptic operator Bibliography Index.

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Moment Maps, Cobordisms, and Hamiltonian Group Actions

I. The harmonic oscillator.- 1. Hamilton's equations and Sl symmetry.- 2. S1 energy momentum mapping.- 3. U(2) momentum mapping.- 4. The Hopf fibration.- 5. Invariant theory and reduction.- 6. Exercises.- II. Geodesics on S3.- 1. The geodesic and Delaunay vector fields.- 2. The SO(4) momentum mapping.- 3. The Kepler problem.- 3.1 The Kepler vector field.- 3.2 The so(4) momentum map.- 3.3 Kepler's equation.- 3.4 Regularization of the Kepler vector field.- 4. Exercises.- III The Euler top.- 1. Facts about SO(3).- 1.1 The standard model.- 1.2 The exponential map.- 1.3 The solid ball model.- 1.4 The sphere bundle model.- 2. Left invariant geodesics.- 2.1 Euler-Arnol'd equations on SO(3).- 2.2 Euler-Arnol'd equations on T1S2 x R3.- 3. Symmetry and reduction.- 3.1 Construction of the reduced phase space.- 3.2 Geometry of the reduction map.- 3.3 Euler's equations.- 4. Qualitative behavior of the reduced system.- 5. Analysis of the energy momentum map.- 6. Integration of the Euler-Arnol'd equations.- 7. The rotation number.- 7.1 An analytic formula.- 7.2 Poinsot's construction.- 8. A twisting phenomenon.- 9. Exercises.- IV. The spherical pendulum.- 1. Liouville integrability.- 2. Reduction of the Sl symmetry.- 3. The energy momentum mapping.- 4. Rotation number and first return time.- 5. Monodromy.- 6. Exercises.- V. The Lagrange top.- 1. The basic model.- 2. Liouville integrability.- 3. Reduction of the right Sl action.- 3.1 Reduction to the Euler-Poisson equations.- 3.2 The magnetic spherical pendulum.- 4. Reduction of the left S1 action.- 5. The Poisson structure.- 6. The Euler-Poisson equations.- 6.1 The Poisson structure.- 6.2 The energy momentum mapping.- 6.3 Motion of the tip of the figure axis.- 7. The energy momemtum mapping.- 7.1 Topology of ???1(h,a,b) and H?1(h).- 7.2 The discriminant locus.- 7.3 The period lattice and monodromy.- 8. The Hamiltonian Hopf bifurcation.- 8.1 The linear case.- 8.2 The nonlinear case.- 9. Exercises.- Appendix A. Fundamental concepts.- 1. Symplectic linear algebra.- 2. Symplectic manifolds.- 3. Hamilton's equations.- 4. Poisson algebras and manifolds.- 5. Exercises.- Appendix B. Systems with symmetry.- 1. Smooth group actions.- 2. Orbit spaces.- 2.1 Orbit space of a proper action.- 2.2 Orbit space of a free action.- 2.3 Orbit space of a locally free action.- 3. Momentum mappings.- 3.1 General properties.- 3.2 Normal form.- 4. Reduction: the regular case.- 5. Reduction: the singular case.- 6. Exercises.- Appendix C. Ehresmann connections.- 1. Basic properties.- 2. The Ehresmann theorems.- 3. Exercises.- Appendix D. Action angle coordinates.- 1. Local action angle coordinates.- 2. Monodromy.- 3. Exercises.- Appendix E. Basic Morse theory.- 1. Preliminaries.- 2. The Morse lemma.- 3. The Morse isotopy lemma.- 4. Exercises.- Notes.- References.- Acknowledgements.

Global Aspects of Classical Integrable Systems

Let $(M,\omega)$ be a Hamiltonian $G$-space with a momentum map $F:M \to {\frak g}^*$. It is well-known that if $\alpha$ is a regular value of $F$ and $G$ acts freely and properly on the level set $F^{-1}(G\cdot \alpha)$, then the reduced space $M_\alpha :=F^{-1}(G\cdot \alpha)/G$ is a symplectic manifold. We show that if the regularity assumptions are dropped the space $M_\alpha $ is a union of symplectic manifolds, and that the symplectic manifolds fit together in a nice way. In other words the reduced space is a {\em symplectic stratified space}. This extends results known for the Hamiltonian action of compact groups.

Proper group actions and symplectic stratified spaces

Let (M;!) be a Hamiltonian G-space with a momentum map F : M ! g: It is well-known that if is a regular value of F and G acts freely and properly on the level set F 1 (G); then the reduced space M := F 1 (G )=G is a symplectic manifold. We show that if the regularity assumptions are dropped the space M is a union of symplectic manifolds, and that the symplectic manifolds t together in a nice way. In other words the reduced space is a symplectic stratied space. This extends results known for the Hamiltonian action of compact groups.

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We show that the Hamiltonian description of a particle moving in a potential field shaped like the punt of a champagne bottle (more properly anS
1 symmetric double well) has monodromy, which is a global obstruction to the construction of action-angle variables.

Larry Bates

Papers

Global Aspects of Classical Integrable Systems

Proper group actions and symplectic stratified spaces

Proper group actions and symplectic stratified spaces

Monodromy in the champagne bottle

Geometry of nonholonomic constraints