Darboux chart on projective limit of weak symplectic Banach manifold
10 Jul 2015-International Journal of Geometric Methods in Modern Physics (World Scientific Publishing Company)-Vol. 12, Iss: 7, pp 1550072
TL;DR: In this article, the projective limit of weak symplectic Banach manifolds is defined and a Frechet space Hx is associated to each point x ∈ M, and it is shown that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak-symmetric structure.
Abstract: Suppose M be the projective limit of weak symplectic Banach manifolds {(Mi, ϕij)}i, j∈ℕ, where Mi are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Frechet space Hx. We prove that if Hx are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.
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21 Sep 2019
TL;DR: In this paper, the existence of Darboux charts on weakly symplectic bounded Frechet manifolds was proved by using the Moser's trick, and sufficient conditions for their existence were provided.
Abstract: We provide sufficient conditions for the existence of Darboux charts on weakly symplectic bounded Frechet manifolds by using the Moser's trick.
4 citations
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TL;DR: In this paper, the Darboux theorem is also true on the direct limit of weak symplectic Banach manifolds without very strong conditions, but not on the upper limit of the manifold.
Abstract: Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show in general, without very strong conditions, the answer is negative. In particular we give an example of an ascending weak symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In a second part, we illustrate this discussion in the context of an ascending sequences of Sobolev manifolds of loops in symplectic finite dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux theorem is true around any point.
3 citations
TL;DR: In this paper, the existence of Darboux charts on weakly symplectic bounded Frechet manifolds was proved by using the Moser's trick, and sufficient conditions for their existence were provided.
Abstract: We provide sufficient conditions for the existence of Darboux charts on weakly symplectic bounded Frechet manifolds by using the Moser's trick.
3 citations
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TL;DR: In this article, the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories) are reviewed and the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation.
Abstract: We review in simple terms the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories) and we discuss the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation. Particular attention is paid to conservation laws related to Poincare invariance within the different approaches. To make the text accessible to a wider audience, we have included an outline of Poisson and symplectic geometry for both classical mechanics and field theory.
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TL;DR: In this paper, a local version of the Darboux theorem was adapted to prove a similar result for Banach manifolds, where the problem is a local one and the corresponding manifold is a 2-form manifold.
Abstract: 1. Normal form. Let M be a Banach manifold. A symplectic structure on M is a closed 2-form Q such that the associated mapping S: T(M)->T*(M) defined by Q(X) = X _ ] 0 is a bundle isomorphism. If M is finite dimensional, Darboux's theorem states that every point in M has a coordinate neighborhood N with coordinate functions (xi, • • • , xn, yi, • • • , yn) such that Q= ]C?=i dxiAdy% on iV. Standard proofs of this theorem (e.g. [4]) use induction on w, so they do not apply to the infinite-dimensional case. It happens, however, that an idea of J. Moser [3] may be adapted to prove a similar result for Banach manifolds. Since the problem is a local one, it suffices to consider a symplectic structure 0 on a neighborhood of 0 in a Banach space B.
30 citations
TL;DR: In this paper, it was shown that if a commutative Frechet-Lie group G has an exponential map, which is a local diffeomorphism, then G is the limit of a projective system of Banach-Lie groups.
Abstract: In this paper we characterize commutative Frechet-Lie groups using the exponential map. In particular we prove that if a commutative Frechet-Lie groupG has an exponential map, which is a local diffeomorphism, thenG is the limit of a projective system of Banach-Lie groups.
29 citations