A Symplectic Slice Theorem

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model, the so-called Chu map, can be used instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.     

Revisiting Noether's Theorem on constants of motion

In this paper we revisit Noether's theorem on the constants of motion for Lagrangian mechanical systems in the ODE case, with some new perspectives on both the theoretical and the applied side. We make full use of invariance up to a divergence, or, as we call it here, Bessel-Hagen (BH) invariance. By recognizing that the Bessel-Hagen (BH) function need not be a total time derivative, we can easily deduce nonlocal constants of motion. We prove that we can always trivialize either the time change or the BH-function, so that, in particular, BH-invariance turns out not to be more general than Noether's original invariance. We also propose a version of time change that simplifies some key formulas. Applications include Lane-Emden equation, dissipative systems, homogeneous potentials and superintegrable systems. Most notably, we give two derivations of the Laplace-Runge-Lenz vector for Kepler's problem that require space and time change only, without BH invariance, one with and one without use of the Lagrange equation.     

A geometric approach to Noether's Second Theorem in time-dependent lagrangian mechanics

We analyse Noether's Second Theorem from a geometric viewpoint using the concepts of vector fields and forms along tangent bundle projections.     

Noether's Theorem and the Conservation Laws of the KORTEWEG-DE VRIES Equation

A method developed recently by the author to derive a continuum of conservation laws by Noether's theorem from the so-called extended Bäcklund transformations is applied to the KORTEWEG -DE VRIES equation that describes various nonlinear dispersive wave phenomena in hydrodynamics, plasma physics and solid state physics. Further applications of Noether's theorem concerning this equation are given. It is shown that the Galilean transformation in the present case has an analogous function as Lie's transformation has with respect to the sine-Gordon equation.     

Symplectic Structures on Moduli Spaces of Parabolic Higgs Bundles and Hilbert Scheme

Parabolic triples of the form (E_*,,) are considered, where (E_*,) is a parabolic Higgs bundle on a given compact Riemann surface X with parabolic structure on a fixed divisor S, and is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle (E_*,) a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by d. On the other hand, there is a map from the moduli space of stable parabolic triples to a Hilbert scheme Hilb(Z), where Z denotes the total space of the line bundle K_{X X}(S), that sends a triple (E_*,,) to the divisor defined by the section on the spectral curve corresponding to the parabolic Higgs bundle (E_*,). Using this map and a meromorphic one–form on Hilb(Z), a natural two–form on the moduli space of stable parabolic triples is constructed. It is shown here that this form coincides with the above mentioned form d.     

On Noether's theorem in quantum field theory

Extending the construction of local generators of symmetries in (S. Doplicher, Commun. Math. Phys.85 (1982), 73; S. Doplicher and R. Longo, Commun. Math. Phys.88 (1983), 399) to space-time and supersymmetries, we establish a weak form of Noether's theorem in quantum field theory. We also comment on the physical significance of the “split property,” underlying our analysis, and discuss some local aspects of superselection rules following from our results.     

Noether's theory of Lagrange systems in discrete case

In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the well-known Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether's theorem is presented. An example was discussed to illustrate these results.     

Noether's theorem for singular Lagrangians

The correspondence between constants of motion and symmetries of a singular Langrangian system is studied. It is shown to be a one-to-one correspondence after an appropriate definition of both concepts. The theory is illustrated with an example.     

Conventional formulations of Noether's theorem in classical field theory

The role of invariance considerations in conventional formulations of Noether's theorem in classical field theory is investigated and found weaker than is usually supposed. It is shown how nonfulfilment of the conventional assumptions going into Noether's theorem brings about nonconservation.     

Coalgebra Bundles

We develop a generalised theory of bundles and connections on them in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory, embeddable quantum homogeneous spaces and braided group gauge theory, the latter being introduced now by these means. Examples include ones in which the gauge groups are the braided line and the quantum plane. Received: 22 February 1996 / Accepted: 29 May 1997     

事件空间中Birkhoff系统的Noether理论

研究事件空间中Birkhoff系统的Noether对称性与Noether守恒量.首先,建立了事件空间中Birkhoff系统的参数方程;其次,基于Pfaff作用量在无限小变换下的不变性,给出了事件空间中Birkhoff系统的Noether定理及其逆定理;最后,举例说明结果的应用. 关键词： 事件空间 Birkhoff系统 Pfaff作用量 Noether对称性     

Generalization of the Noether's Identities and Application

Starting from the transformation property of the action integral of a system under the local and non-local transformation, we derive the generalized Noether's identities connecting with non-local transformation. The applications of the theory to the Yang-Mills field with high-order derivatives are presented. A new conservative PBRS charge is found which differs from BRS conservative charge. The other conservative charge connecting with non-local transformation is also obtained.     

Symplectic Structure in Brane Mechanics

This paper treats the generalization to brane dynamics of the covariant canonical variational procedure leading to the construction of a conserved bilinear symplectic current in the manner originally developed by Witten, Zuckerman, and others in the context of field theory. After a general presentation, including a review of the relationships between the various (Lagrangian, Eulerian, and other) relevant kinds of variation, the procedure is illustrated by application to the particularly simple case of branes of the Dirac—Goto—Nambu type, in which internal fields are absent.     

Symplectic quantum mechanics

Using the notion of symplectic structure and Weyl (or star) product of non-commutative geometry, we construct unitary representations for the Galilei group and show how to rewrite the Schrödinger equation in phase space. This approach gives rise to a new procedure to derive Wigner functions without the use of the Liouville-von Neumann equation. Applications are presented by deriving the states of linear and nonlinear oscillators in terms of amplitudes of probability in phase space. The notion of coherent states is also discussed in this context.     

Formal Symplectic Groupoid

The multiplicative structure of the trivial symplectic groupoid over ^d associated to the zero Poisson structure can be expressed in terms of a generating function. We address the problem of deforming such a generating function in the direction of a non-trivial Poisson structure so that the multiplication remains associative. We prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it. This formula turns out to be the semi-classical approximation of Kontsevichs deformation formula. For the case of a linear Poisson structure, the deformed generating function reduces exactly to the CBH formula of the associated Lie algebra. The methods used to prove existence are interesting in their own right as they come from an at first sight unrelated domain of mathematics: the Runge–Kutta theory of the numeric integration of ODEs.A.S.C. acknowledges partial support of SNF Grant No. 20-100029/1.B.D. and G.F. acknowledge partial support of SNF Grant No. 21-65213.01.Acknowledgement The second author thanks Ernst Hairer for useful discussions, and suggestions.     

Symplectic twistor spaces

The paper describes the geometry of the bundle (M, ω) of the compatible complex structures of the tangent spaces of an (almost) symplectic manifold (M, ω), from the viewpoint of general twistor spaces [3], [9], [1]. It is shown that M has an either complex or almost Kaehler twistor space iff it has a flat symplectic connection. Applications of the twistor space to the study of the differential forms of M, and to the study of mappings : N → M, where N is a Kaehler manifold are indicated.     

Soldered Bundles in Particle Physics

The use in gauge theories of soldered fiber bundles with Cartan connection is reviewed with particular emphasis on geometrically motivated symmetry breaking.