Noether Theory Research Articles

Lagrange–Noether method for solving second-order differential equations

The purpose of this paper is to provide a new method called the Lagrange–Noether method for solving second-order differential equations. The method is, firstly, to write the second-order differential equations completely or partially in the form of Lagrange equations, and secondly, to obtain the integrals of the equations by using the Noether theory of the Lagrange system. An example is given to illustrate the application of the result.

Multisymplectic structures and the variational bicomplex

AbstractMultisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

Brill–Noether theory and non-special scrolls

In this paper we study the Brill-Noether theory of invertible subsheaves of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with hyperplane sections isomorphic to C. Most of our results are based on degeneration techniques.

Conformal invariance and integration of first-order differential equations

This paper studies a conformal invariance and an integration of first-order differential equations. It obtains the corresponding infinitesimal generators of conformal invariance by using the symmetry of the differential equations, and expresses the differential equations by the equations of a Birkhoff system or a generalized Birkhoff system. If the infinitesimal generators are those of a Noether symmetry, the conserved quantity can be obtained by using the Noether theory of the Birkhoff system or the generalized Birkhoff system.

On coherent systems of type (n, d, n + 1) on Petri curves

We study coherent systems of type (n, d, n + 1) on a Petri curve X of genus g ≥ 2. We describe the geometry of the moduli space of such coherent systems for large values of the parameter α. We determine the top critical value of α and show that the corresponding “flip” has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of α, proving in many cases that the condition for non-emptiness is the same as for large α. We give some detailed results for g ≤ 5 and applications to higher rank Brill–Noether theory and the stability of kernels of evaluation maps, thus proving Butler’s conjecture in some cases in which it was not previously known.

Brill–Noether theory of curves on Enriques surfaces I: the positive cone and gonality

We study the existence of linear series on curves lying on an Enriques surface and general in their complete linear system. Using a method that works also below the Bogomolov–Reider range, we compute, in all cases, the gonality of such curves. We also give a new result about the positive cone of line bundles on an Enriques surface and we show how this relates to the gonality.

Integral of differential equations of three-particle Toda lattice

The differential equations of three-particle Toda lattice are a Hamiltonian system. The integrals of the equations can be obtained by using the Noether theory and the Poisson theory.

SINGULAR CURVES ON A K3 SURFACE AND LINEAR SERIES ON THEIR NORMALIZATIONS

We study the Brill–Noether theory of the normalizations of singular, irreducible curves on a K3 surface. We introduce a singular Brill–Noether number ρsing and show that if Pic (K3) = ℤ[L], there are no [Formula: see text]'s on the normalizations of irreducible curves in |L|, provided that ρsing < 0. We give examples showing the sharpness of this result. We then focus on the case of hyperelliptic normalizations, and classify linear systems |L| containing irreducible nodal curves with hyperelliptic normalizations, for ρsing < 0, without any assumption on the Picard group.

A Birkhoff-Noether method of solving differential equations

In this paper, a Birkhoff–Noether method of solving ordinary differential equations is presented. The differential equations can be expressed in terms of Birkhoff's equations. The first integrals for differential equations can be found by using the Noether theory for Birkhoffian systems. Two examples are given to illustrate the application of the method.

First integrals and stability of second-order differential equations

The stability of second-order differential equations is studied by using their integrals. A system of second-order differential equations can be considered as a mechanical system with holonomic constraints. A conserved quantity of the mechanical system or a part of the system is obtained by using the Noether theory. It is possible that the conserved quantity becomes a Liapunov function and the stability of the system is proved by the Liapunov theorem.

Classical Algebraic Geometry

The worshop on Classical Algebraic Geometry, organized by David Eisenbud (Berkeley), Joe Harris (Harvard) and Frank-Olaf Schreyer (Saarbrücken) was well attended by important senior researchers and many gifted young mathematicians. Classical Algebraic Geometry is characterized by having very basic and concrete problems. However development of the abstract, sophisticated tools of algebraic geometry has often lead to remarkable progress on these problems. It was our intention to make a conference emphasizing progress on the classical problems, and featuring the new tools and their applications. We emphasize a few results: Geometry of the Moduli Spaces of Curves. Based partly on motivations from mathematical physics, the enumerative geometry of the moduli spaces of curves has been an extremely active area for some time. Several of the major conjectures in the area have seen major progress, and may be close to resolution. The extension of Brill–Noether theory for line bundles to the case of bundles of higher rank is a long-standing problem (Mukai's conjecture), and this plays now a novel role in the search for interesting divisors on the moduli space in the work of Gavril Farkas. In the talk of Brendon Hassett we learned how other birational models of the Moduli space can be interpreted in terms of curves with nodes, cusps and tacnodes. Derived Categories. Originally introduced in the late fifties as a tool for generalizing duality theory, derived categories have arisen in a several striking new contexts, including the characterization of birational transformations in the minimal model program, the analysis of the Bernstein–Gel'fand–Gel'fand and Fourier–Mukai transforms. One of the speakers of the conference (Mihnea Popa) suggested an amazing series of parallels between Castelnuovo theory and the Schottky problem, based on these last two contexts. Enumerative Geometry. The high-tech tools of modern geometry have led to a better understanding of degenerations, which has in turn led to the solution of many classical enumerative problems. We heard talks on a solution via specialization of the long-standing problem of intersection theory on flag manifolds, and on the maximal degneration of complex structures through amoebas and tropical geometry. We were particularly pleased by the number of extremely strong young (i.e. untenured) participants, among them for example Gavril Farkas and Mihnea Popa.

Cartan forms for first order constrained variational problems

Given a constrained variational problem on the 1-jet extension J 1 Y of a fibre bundle p : Y → X , under certain conditions on the constraint submanifold S ⊂ J 1 Y , we characterize the space of admissible infinitesimal variations of an admissible section s : X → Y as the image by a certain first order differential operator, P s , of the space of sections Γ ( X , s ∗ V Y ) . In this way we obtain a constrained first variation formula for the Lagrangian density L ω on J 1 Y , which allows us to characterize critical sections of the problem as admissible sections s such that P s + E L ω ( s ) = 0 , where P s + is the adjoint operator of P s and E L ω ( s ) is the Euler–Lagrange operator of the Lagrangian density L ω as an unconstrained variational problem. We introduce a Cartan form on J 2 Y which we use to generalize the Cartan formalism and Noether theory of infinitesimal symmetries to the constrained variational problems under consideration. We study the relation of this theory with the Lagrange multiplier rule as well as the question of regularity in this framework. The theory is illustrated with several examples of geometrical and physical interest.

New method of deriving local energy- and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: basic theory

In this paper we describe a relatively simple and transparent method of obtaining collisionless drift-kinetic, gyrokinetic and more general theories, including local charge, energy and momentum conservation laws. An important feature of the new formalism is, contrary to present-day theories, the exact gauge invariance, thus avoiding certain inconsistencies. The present paper starts with the introduction and proof of the correctness of a Lagrangian for combined Maxwell-kinetic theories in general coordinates as concerns the particle motion. The kinetic part of it is formulated in Eulerian form by means of the equations of motion in the form of Hamilton–Jacobi's equation, used only as a tool, and Dirac's constraint theory. Charge and current densities automatically distinguish between ‘particle-like’ (guiding-centre), polarization and magnetization contributions. This formalism is applied to averaging coordinates derived by a method similar to Kruskal's. Certain properties of the averaging coordinates, according to the basic requirements imposed on them, can be used to obtain a gyroangle independent Lagrangian, from which one can obtain a Lagrangian for the combined Maxwell-kinetic theories in a reduced phase space that is applicable to situations in which one is not interested in the dependence on some kind of gyroangle describing the gyromotion, whose treatment can, however, easily be added. The basic perturbation theory, which aims at obtaining averaging phase-space coordinates, is done solely within the Kruskal formalism in which only the electric and magnetic fields appear, but not the corresponding potentials. This formalism provides, in particular, information about the allowed amplitudes of fluctuations. The results are later used to obtain approximate expressions for the Lagrange functions in the drift-kinetic and the gyrokinetic ordering. For the definition of certain approximations to the exact Lagrangian the central principle is the exact gauge invariance. The terms of the zeroth and first orders needed for such an approximation are given. For the drift-kinetic ordering Littlejohn's Lagrangian is readily rederived and hence the drift-kinetic theory as obtained and investigated by the present authors in some previous work. Conservation laws and their mathematical structures corresponding to the gyroangle-independent Lagrangian are obtained in a following paper. There, in particular, a detailed description of how to obtain variations of gyroangle averaged quantities in the reduced phase space is given, and it is explained how these variations are used in a modified form of Noether's theory which observes exact gauge invariance.

Noether formalism with gauge-invariant variations

The energy–momentum tensor is usually obtained by symmetrizing the canonical tensor, which is sometimes difficult. The fact that the energy–momentum tensor must always be gauge invariant, and insight gained by investigating the Maxwell kinetic equations, lead one to conclude that this tensor should be readily obtainable for a certain class of Lagrange densities. The appropriate method to achieve this consists in performing gauge-invariant variations within the framework of Noether's theory. These variations are equivalent to shifts in space and time, and are considered here instead of the usual shift variations, which are not gauge invariant. These features indicate the existence of a certain underlying structure which is made evident by the proposed method. The Lagrange densities of the class in question are characterized by having a gauge-invariant contribution additional to the Maxwell part. For vanishing potentials, such Lagrange densities lead directly to a symmetric canonical tensor; this implies that there are only scalar fields. Maxwell's equations, being the basic part of such theories, are treated first. Then, the coupled system of electromagnetic fields and the usual scalar fields describing charged matter is considered. Finally, a different kind of scalar field which occurs in a structural non-standard form in collisionless phase-space theories (Vlasov or collisionless Boltzmann-like equations) is treated.

Generally covariant theories: the Noether obstruction for realizing certain space–time diffeomorphisms in phase space

Relying on known results of the Noether theory of symmetries extended to constrained systems, it is shown that there exists an obstruction that prevents certain tangent-space diffeomorphisms being projectable to phase space, for generally covariant theories. This main result throws new light on the old fact that the algebra of gauge generators in the phase space of general relativity, or other generally covariant theories, only closes as a soft algebra and not as a Lie algebra. The deep relationship between these two issues is clarified. In particular, we see that the second one may be understood as a side effect of the procedure to solve the first. It is explicitly shown how the adoption of specific metric-dependent diffeomorphisms, as a way to achieve projectability, causes the algebra of gauge generators (constraints) in phase space not to be a Lie algebra—with structure constants—but a soft algebra—with structure functions.

On conservation laws in generalized dynamic thermoelasticity

The differential equations of generalized dynamic thermoelasticity constitute a coupled non-self-adjoint system of PDEs, which in its variational form does not admit a Lagrangian. However following exterior calculus methodology, we augment the space of dependent variables, accompanying the initial differential equations with suitable adjoint equations and additional auxiliary “fields”. So the construction of a Lagrangian is now possible along with the realization of Noether's theory for finding conservation laws. Finally, in our case, it is possible to characterize the additional dependent variables in terms of the physical fields themselves and obtain then conservation laws expressed via the original known thermoelastic fields.

Noether's theorem of a rotational relativistic variable mass system

Noether's theory of a rotational relativistic variable mass system is studied. Firstly, Jourdain's principle of the rotational relativistic variable mass system is given. Secondly, on the basis of the invariance of the Jourdain's principle under the infinitesimal transformations of groups, Noether's theorem and its inverse theorem of the rotational relativistic variable mass system are presented. Finally, an example is given to illustrate the application of the result.

Singular bielliptic curves and special linear systems

Here we study the Brill–Noether theory of special linear systems on a singular Gorenstein bielliptic curve Y. Any such linear system is associated to a spanned rank 1 torsion-free sheaf, L. We obtain the same results as in the case of a smooth curve if either L∈Pic( Y) or Y has only ordinary nodes or ordinary cusps as singularities.

Noether's theorem of nonholonomic systems of non-Chetaev's type with unilateral constraints

In this paper, we present Noether's theorem and its inverse theorem for nonholonomic systems of non-Chetaev's type with unilateral constraints. We present first the principle of Jourdain for the system and, on the basis of the invariance of the differential variational principle under the infinitesimal transformations of groups, we have established Noether's theory for the above systems. An example is given to illustrate the application of the result.

BRILL–NOETHER THEORY OF CURVES WITH Ak SINGULARITIES AND HYPERPLANE SECTIONS OF K3 SURFACES

Let Y be an integral projective curve whose singularities are of type Ak, i.e. with only tacnodes and planar (perhaps non-ordinary) cusps. Set g:= pa(Y). Here we study the Brill - Noether theory of spanned line bundles on Y. If the singularities are bad enough, we show the existence of spanned degree d line bundles, L, with h0(Y, L) ≥ r + 1 even if the Brill - Noether number ρ(g, d, r) < 0. We apply this result to prove that genus g curves with certain singularities cannot be hyperplane section of a simple K3 surface S ⊂ P g.