Exact Lagrangians in A_n-surface singularities

In this paper we classify Lagrangian spheres in $A_n$ -surface singularities up to Hamiltonian isotopy. Combining with a result of Ritter (Geom Funct Anal 20(3):779–816, 2010), this yields a complete classification of exact Lagrangians in $A_n$ -surface singularities. Our main new tool is the application of a technique which we call ball-swappings and its relative version.     

Duality and Exact Penalization for General Augmented Lagrangians

We consider a problem of minimizing an extended real-valued function defined in a Hausdorff topological space. We study the dual problem induced by a general augmented Lagrangian function. Under a simple set of assumptions on this general augmented Lagrangian function, we obtain strong duality and existence of exact penalty parameter via an abstract convexity approach. We show that every cluster point of a sub-optimal path related to the dual problem is a primal solution. Our assumptions are more general than those recently considered in the related literature.     

Deformations of Symplectic Cohomology and Exact Lagrangians in ALE Spaces

ALE spaces are the simply connected hyperkähler manifolds which at infinity look like ${\mathbb{C}^{2}/G}ALE spaces are the simply connected hyperk?hler manifolds which at infinity look like \mathbbC²/G{\mathbb{C}^{2}/G}, for any finite subgroup G ì SL₂(\mathbbC){G \subset SL_2(\mathbb{C})}. We prove that all exact Lagrangians inside ALE spaces must be spheres. The proof relies on showing the vanishing of a twisted version of symplectic cohomology.     

On null Lagrangians

We consider multiple-integral variational problems where the Lagrangian function, defined on a frame bundle, is homogeneous. We construct, on the corresponding sphere bundle, a canonical Lagrangian form with the property that it is closed exactly when the Lagrangian is null. We also provide a straightforward characterization of null Lagrangians as sums of determinants of total derivatives. We describe the correspondence between Lagrangians on frame bundles and those on jet bundles: under this correspondence, the canonical Lagrangian form becomes the fundamental Lepage equivalent. We also use this correspondence to show that, for a single-determinant null Lagrangian, the fundamental Lepage equivalent and the Carathéodory form are identical.     

Null Lagrangians for nematic elastomers

In this paper, we calculate all possible null Lagrangians (null energies) for the mechanics of a distinguished class of continua, the nematic elastomers. The calculation is done in order to help to relate different physically equivalent theories of nematic elastomers. We discuss both local and global (hence topological) aspects of the problem. __________ Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 10, No. 1, Geometry of Integrable Models, 2004.     

Simple sheaves along dihedral Lagrangians

We prove the unicity of a complex of sheavesF whose microsupport is carried by a “dihedral” Lagrangian Λ ofT ^* X (X=a real manifold) and which is simple with a prescribed shift at a regular point of Λ. Our method consists in reducing Λ, by a real contact transformation, to the conormal bundle to aC ¹-hypersuface, and then in using [K-S 1, Prop. 6.2.1] in the variant of [D'A-Z 1]. This is similar to [Z 2] but more general, since complex contact transformations and calculations of shifts are not required. We then consider the case of a complex manifoldX, and obtain some vanishing theorems for the complex of “microfunctions along Λ” similar to those of [A-G], [A-H], [K-S 1] (cf. also [D'A-Z 3 5], [Z 2]).     

On Natural First order Lagrangians

The natural first order Lagrangian is a function defined onpairs of Riemannian manifolds and which is invariant with respectto immersions and depends continuously on 1-jets of metrics.We prove that there is a canonical bijection between naturalfirst order Lagrangians and functions which are smooth, symmetricand even.     

Generalized Lagrangians and Spinning Particles

We show that the spin structure of elementary particles can be naturally described by the generalized Ostrogradskii Lagrangians depending on higher-order derivatives. One component of a spin is related to the rotation of a particle and the other one, caused by the dependence of a Lagrangian on the acceleration, is known as a zitterbewegung component of spin.     

Augmented Lagrangians which are quadratic in the multiplier

We present a class of new augmented Lagrangian functions with the essential property that each member is concave quadratic when viewed as a function of the multiplier. This leads to an improved duality theory and to a related class of exact penalty functions. In addition, a relationship between Newton steps for the classical Lagrangian and the new Lagrangians is established.This work was supported in part by ARO Grant No. DAAG29-77-G-0125.     

A useful approximation scheme for Lagrangians

We present a new, useful approximation scheme for the integrand of an integral functional, revolving around a generalized bipolarity result. This scheme leads immediately to lower semicontinuity and lower closure results for the integral functional, as well as to other, more general seminormality properties.     

Limiting Lagrangians: A primal approach

In this paper, we consider a convex program with either a finite or an infinite number of constraints and its formal Lagrangian dual. We show that either the primal program satisfies a general condition which implies there is no duality gap or that there is a nonzero vectord with the following properties: First, wheneverd is added to the objective function, where is a positive number not greater than one, the resulting program satisfies the general sufficient condition cited above for no duality gap. Second, the optimal value of this perturbed program is attained and tends to the optimal value of the original program as tends to zero. Third, the optimal solutions of the perturbed programs form a minimizing sequence of the original program. As a consequence of the above, we derive the limiting Lagrangian theory of Borwein, Duffin, and Jeroslow.The authors are indebted to an unknown referee who suggested the very short and elegant proofs of Lemma 2.3 and Theorem 2.3.This work was completed while the first author was a member of the College of Management, Georgia Institute of Technology, Atlanta, Georgia.     

Non-standard power-law Lagrangians in classical and quantum dynamics

Recently, non-standard Lagrangians (NSL) have gained an increasing significance due to their wide implications in the theory of non-linear differential equations, complex dynamical systems and theoretical physics. In this work, we choose the power-law non-standard Lagrangian and we discuss some of its implications in classical and quantum theories.     

Lagrangians for self-adjoint and non-self-adjoint equations

It is relatively easy to obtain a variational formulation for a self-adjoint equation, while it is difficult or impossible to do this for a non-self-adjoint one. This work suggests an alternative approach to the derivation of the Lagrangian for both cases. A trial-Lagrangian is constructed, and the exact variational formulation can be obtained after the identification of the unknown function involved in the trial-Lagrangian.