Coordonnées canoniques

Résumé

En mathématiques et en mécanique classique, les coordonnées canoniques sont des ensembles de coordonnées sur l'espace des phases qui peuvent être utilisées pour décrire un système physique à un moment donné dans le temps. Les coordonnées canoniques sont utilisées dans la formulation hamiltonienne de la mécanique classique. Un concept étroitement lié apparaît également en mécanique quantique ; voir le théorème de Stone-von Neumann et les relations de commutation canoniques pour plus de détails. Comme la mécanique hamiltonienne est généralisée par la géométrie symplectique et que les transformations canoniques sont généralisées par les transformations de contact, la définition du des coordonnées canoniques en mécanique classique peut être généralisée à une définition plus abstraite du des coordonnées sur le faisceau cotangent d'une variété (la notion mathématique d'espace des phases). Définition en mécanique classique En mécanique classique, les coordonnées canoniques so

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https://fr.wikipedia.org/wiki/Coordonn%C3%A9es_canoniques

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Présentation des méthodes de la mécanique analytique (équations de Lagrange et de Hamilton) et introduction aux notions de modes normaux et de stabilité.

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Séances de cours associées (31)

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Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo p(2) of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse-Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration. We also extend Nygaard's approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.

WALTER DE GRUYTER GMBH2021

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Single speed solutions of the Vlasov-Poisson equations for Coulombian and Newtonian systems in nD

Philippe Choquard

Results of theoretical and numerical investigations concerning the space-time evolution in nD of Coulombian systems known in Plasma Physics and Newtonian systems known in Cosmology are reported here. A generic, canonical Hamiltonian formulation is given (1) for studying single speed solutions of their Coulomb- or Jeans-Vlasov-Poisson descriptions. (2) It is shown that their equations of motion are integrable for vorticity free planar motions in nD (3) and that their characteristics are solutions of explicit non-linear ODES in general (4). Some simple examples of trajectories are illustrated and the problem of their crossing is evoked. (c) 2007 Elsevier B.V. All rights reserved.

2008

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The method of symmetrical components is not effective for fault location in the case of untransposed lines, due to potential couplings between the sequence circuits. This paper proposes a non-iterative algorithm in the phase-coordinates for wide-area fault location on untransposed transmission networks. In doing so, first, an improved two-terminal method is suggested to accurately locate faults on untransposed lines. Next, an algorithm is proposed to infer voltage and current phasors at the faulted line ends without direct measurements, by taking advantage of the data provided by phasor measurement units (PMUs). Accordingly, the adverse effect of close instrument transformers transients on the estimation accuracy is minimized. Being highly nonlinear in terms of fault distance and impedance, the fault equations are derived and made linear in this paper by defining six suitable auxiliary variables. The resulting system of equations is solved using the least-squares method to obtain three-phase voltages and currents at the faulted line ends. A main feature of the proposed algorithm is that it only requires a limited number of current and voltage synchrophasors. An additional advantage of the proposed algorithm is that the faulted line is not required to be known a-priori. The proposed algorithm is validated using extensive simulation studies on the New England 39-bus test system, accounting for different fault locations, types and resistances.

Institute of Electrical and Electronics Engineers2016

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