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Hamiltonian Structure of
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The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian are given in this article for a principal bundle on the principal bundle of a principal manifold.Abstract:
Let C ! M be the bundle of connections of a principal bundle on M. The solutions to Hamilton-Cartan equations for a gauge-invariant Lagrangian �read more
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Coupling between internal and surface waves
TL;DR: In this article, the authors study the regime where long waves propagate in the interfacial mode, which are coupled to a modulational regime for the free surface mode, and the perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler's equations.
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Nearly-Hamiltonian Structure for Water Waves with Constant Vorticity
TL;DR: In this article, the governing equations for two-dimensional gravity water waves with constant nonzero vorticity have a nearly Hamiltonian structure, which becomes Hamiltonian for steady waves.
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Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations
TL;DR: In this paper, the Serre system of equations of water wave theory was derived from a generalized variational principle, and a robust and accurate finite volume scheme was proposed to solve these equations in one horizontal dimension.
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Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions
Darryl D. Holm,Cesare Tronci +1 more
TL;DR: In this article, the collective Hamiltonians are shown to fit into the Kaluza-Klein theory of particles in a Yang-Mills field and these formulations were shown to apply also at the continuum PDE level.
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On a shallow-water approximation to the Green–Naghdi equations with the Coriolis effect
TL;DR: In this article, an asymptotic 1D (in space) rotation-Camassa-Holm (R-CH) model was used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with allowance for the weak Coriolis effect due to the Earth's rotation.
References
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The Hamilton-Cartan formalism in the calculus of variations
TL;DR: In this paper, Caratheodory, Cartan, and De Donder give an exposition of the geometry of the calculus of variations in several variables, and the main emphasis is on the Hamiltonian formalism via the use of a linear differential form studied in detail by Cartan.
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On the bundle of connections and the gauge orbit manifold in Yang-Mills theory
P. K. Mitter,C. M. Viallet +1 more
TL;DR: In this paper, it was shown that the quotienting of the space of connections by the group of gauge transformations in Yang-Mills theory is aC fixme∞ principal fibration, and that the underlying quotient space, the gauge orbit space, is explicitly aC¯¯¯¯∞ manifold modelled on a Hilbert space.
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The geometry of the bundle of connections
TL;DR: In this paper, a generalized symplectic structure on the bundle of connections of an arbitrary principal G-bundle is defined by means of a ÃÂÃÂÃÂÃÂÃÂÃÂÃÂÃÂ$p^{\ast}\mathrm{ad}P$¯¯¯¯ -valued differential 2-form, which is related to the generalized contact structure on ��$J^{1}(P)$¯¯¯¯.