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Geometric Hamiltonian Formalism for Reparametrization Invariant Theories with Higher Derivatives  [PDF]
Petr Dunin-Barkowski,Alexei Sleptsov
Physics , 2008, DOI: 10.1007/s11232-009-0005-7
Abstract: Reparametrization invariant Lagrangian theories with higher derivatives are considered. We investigate the geometric structures behind these theories and construct the Hamiltonian formalism in a geometric way. The Legendre transformation which corresponds to the transition from the Lagrangian formalism to the Hamiltonian formalism is non-trivial in this case. The resulting phase bundle, i.e. the image of the Legendre transformation, is a submanifold of some cotangent bundle. We show that in our construction it is always odd-dimensional. Therefore the canonical symplectic two-form from the ambient cotangent bundle generates on the phase bundle a field of the null-directions of its restriction. It is shown that the integral lines of this field project directly to the extremals of the action on the configuration manifold. Therefore this naturally arising field is what is called the Hamilton field. We also express the corresponding Hamilton equations through the generilized Nambu bracket.
Geometric Polynomial Constraints in Higher-Order Graph Matching  [PDF]
Mayank Bansal,Kostas Daniilidis
Computer Science , 2014,
Abstract: Correspondence is a ubiquitous problem in computer vision and graph matching has been a natural way to formalize correspondence as an optimization problem. Recently, graph matching solvers have included higher-order terms representing affinities beyond the unary and pairwise level. Such higher-order terms have a particular appeal for geometric constraints that include three or more correspondences like the PnP 2D-3D pose problems. In this paper, we address the problem of finding correspondences in the absence of unary or pairwise constraints as it emerges in problems where unary appearance similarity like SIFT matches is not available. Current higher order matching approaches have targeted problems where higher order affinity can simply be formulated as a difference of invariances such as lengths, angles, or cross-ratios. In this paper, we present a method of how to apply geometric constraints modeled as polynomial equation systems. As opposed to RANSAC where such systems have to be solved and then tested for inlier hypotheses, our constraints are derived as a single affinity weight based on $n>2$ hypothesized correspondences without solving the polynomial system. Since the result is directly a correspondence without a transformation model, our approach supports correspondence matching in the presence of multiple geometric transforms like articulated motions.
Geometric Momentum for a Particle on a Curved Surface  [PDF]
Q. H. Liu
Physics , 2011,
Abstract: When a two-dimensional curved surface is conceived as a limiting case of a curved shell of equal thickness d, where the limit d\rightarrow0 is then taken, the well-known geometric potential is induced by the kinetic energy operator, in fact by the second order partial derivatives. Applying this confining procedure to the momentum operator, in fact to the first order partial derivatives, we find the so-called geometric momentum instead. This momentum is compatible with the Dirac's canonical quantization theory on system with second-class constraints. The distribution amplitudes of the geometric momentum on the spherical harmonics are analytically determined, and they are experimentally testable for rotational states of spherical molecules such as C_{60}.
Nonholonomic Constraints with Fractional Derivatives  [PDF]
Vasily E. Tarasov,George M. Zaslavsky
Physics , 2006, DOI: 10.1088/0305-4470/39/31/010
Abstract: We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle.
Estimation in high dimensions: a geometric perspective  [PDF]
Roman Vershynin
Statistics , 2014,
Abstract: This tutorial provides an exposition of a flexible geometric framework for high dimensional estimation problems with constraints. The tutorial develops geometric intuition about high dimensional sets, justifies it with some results of asymptotic convex geometry, and demonstrates connections between geometric results and estimation problems. The theory is illustrated with applications to sparse recovery, matrix completion, quantization, linear and logistic regression and generalized linear models.
Asymptotic vacua with higher derivatives  [PDF]
Spiros Cotsakis,Georgios Kolionis,Antonios Tsokaros
Physics , 2013,
Abstract: We study limits of vacuum, isotropic universes in the full, effective, four-dimensional theory with higher derivatives. We show that all flat vacua as well as general curved ones are globally attracted by the standard, square root scaling solution at early times. Open vacua asymptote to horizon-free, Milne states in both directions while closed universes exhibit more complex logarithmic singularities, starting from initial data sets of a smaller dimension.
Stability of equilibrium under constraints: Role of second-order constrained derivatives  [PDF]
Tamas Gal
Physics , 2007, DOI: 10.1088/1751-8113/43/42/425208
Abstract: In the stability analysis of an equilibrium, given by a stationary point of a functional F[n] (free energy functional, e.g.), the second derivative of F[n] plays the essential role. If the system in equilibrium is subject to the conservation constraint of some extensive property (e.g. volume, material, or energy conservation), the Euler equation determining the stationary point corresponding to the equilibrium alters according to the method of Lagrange multipliers. Here, the question as to how the effects of constraints can be taken into account in a stability analysis based on second functional derivatives is examined. It is shown that the concept of constrained second derivatives incorporates all the effects due to constraints; therefore constrained second derivatives provide the proper tool for the stability analysis of equilibria under constraints. For a physically important type of constraints, it is demonstrated how the presented theory works. Further, the rigorous derivation of a recently obtained stability condition for a special case of equilibrium of ultrathin-film binary mixtures is given, presenting a guide for similar analyses. [For details on constrained derivatives, see also math-ph/0603027, physics/0603129, physics/0701145.]
Gauging Higher Derivatives  [PDF]
Physics , 1995, DOI: 10.1143/PTP.94.105
Abstract: The usual prescription for constructing gauge-invariant Lagrangian is generalized to the case where a Lagrangian contains second derivatives of fields as well as first derivatives. Symmetric tensor fields in addition to the usual vector fields are introduced as gauge fields. Covariant derivatives and gauge-field strengths are determined.
Higher order fractional derivatives  [PDF]
Richard Herrmann
Mathematics , 2009,
Abstract: Based on the Liouville-Weyl definition of the fractional derivative, a new direct fractional generalization of higher order derivatives is presented. It is shown, that the Riesz and Feller derivatives are special cases of this approach.
Higher geometric prequantum theory  [PDF]
Domenico Fiorenza,Christopher L. Rogers,Urs Schreiber
Mathematics , 2013,
Abstract: We develop the refinement of geometric prequantum theory to higher geometry (higher stacks), where a prequantization is given by a higher principal connection (a higher gerbe with connection). We show fairly generally how there is canonically a tower of higher gauge groupoids and Courant groupoids assigned to a higher prequantization, and establish the corresponding Atiyah sequence as an integrated Kostant-Souriau infinity-group extensions of higher Hamiltonian symplectomorphisms by higher quantomorphisms. We also exhibit the infinity-group cocycle which classifies this extension and discuss how its restrictions along Hamiltonian infinity-actions yield higher Heisenberg cocycles. In the special case of higher differential geometry over smooth manifolds we find the L-infinity-algebra extension of Hamiltonian vector fields which is the higher Poisson bracket of local observables and show that it is equivalent to the construction proposed by the second author in n-plectic geometry. Finally we indicate a list of examples of applications of higher prequantum theory in the extended geometric quantization of local quantum field theories and specifically in string geometry.
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