Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems

This paper develops the notion of implicit Lagrangian systems and presents some of their basic properties in the context of Dirac structures. This setting includes degenerate Lagrangian systems and systems with both holonomic and nonholonomic constraints, as well as networks of Lagrangian mechanical systems. The definition of implicit Lagrangian systems with a configuration space $Q$ makes use of Dirac structures on $T^{1}Q$ that are induced from a constraint distribution on $Q$ as well as natural symplectomorphisms between the spaces $T^{1}TQ$, $T{T}^{1}Q$, and $T^1{T}^{1}Q$. Two illustrative examples are presented; the first is a nonholonomic system, namely a vertical disk rolling on a plane, and the second is an L–C circuit, a degenerate Lagrangian system with holonomic constraints.     

Discrete Lagrangian Reduction,Discrete Euler–Poincaré Equations,and Semidirect Products

A discrete version of Lagrangian reduction is developed within the context of discrete time Lagrangian systems on G × G, where G is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of an isotropy subgroup of a fixed element in the representation space of G. Within this context, the reduction of the discrete Euler–Lagrange equations is shown to lead to the so-called discrete Euler–Poincaré equations. A constrained variational principle is derived. The Legendre transformation of the discrete Euler–Poincaré equations leads to discrete Hamiltonian (Lie–Poisson) systems on a dual space to a semiproduct Lie algebra.     

On Mpc-structures and symplectic Dirac operators

We prove that the kernels of the restrictions of the symplectic Dirac operator and one of the two symplectic Dirac–Dolbeault operators on natural sub-bundles of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute these kernels explicitly for complex projective spaces and show that the remaining Dirac–Dolbeault operator has infinite dimensional kernels on these finite rank sub-bundles. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabiliser of a Lagrangian subspace) in the $M{p}^c$ group and classify $G$-invariant $M{p}^c$-structures on symplectic manifolds with a $G$-action. We prove a variant of Parthasarathy’s formula for the commutator of two symplectic Dirac-type operators on general symmetric symplectic spaces.     

Omni-Lie 2-algebras and their Dirac structures

We introduce the notion of omni-Lie 2-algebra, which is a categorification of Weinstein’s omni-Lie algebras. We prove that there is a one-to-one correspondence between strict Lie 2-algebra structures on 2-sub-vector spaces of a 2-vector space V$\mathbb{V}$ and Dirac structures on the omni-Lie 2-algebra gl(V)⊕V$\mathfrak{g}\mathfrak{l}\left(\mathbb{V}\right)\oplus \mathbb{V}$. In particular, strict Lie 2-algebra structures on V$\mathbb{V}$ itself one-to-one correspond to Dirac structures of the form of graphs. Finally, we introduce the notion of twisted omni-Lie 2-algebra to describe (non-strict) Lie 2-algebra structures. Dirac structures of a twisted omni-Lie 2-algebra correspond to certain (non-strict) Lie 2-algebra structures, which include string Lie 2-algebra structures.     

Legendre Transformation for Regularizable Lagrangians in Field Theory

Hamilton equations based not only upon the Poincaré–Cartan equivalent of a first-order Lagrangian, but also upon its Lepagean equivalent are investigated. Lagrangians which are singular within the Hamilton–De Donder theory, but regularizable in this generalized sense are studied. Legendre transformation for regularizable Lagrangians is proposed and Hamilton equations, equivalent with the Euler–Lagrange equations, are found. It is shown that all Lagrangians affine or quadratic in the first derivatives of the field variables are regularizable. The Dirac field and the electromagnetic field are discussed in detail.     

W-algebras and the equivalence of bihamiltonian,Drinfeld–Sokolov and Dirac reductions

We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld–Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic subspace. In addition, using this result we prove again that the transverse Poisson structure to a nilpotent orbit is polynomial and we better clarify the relation between classical and finite $W$-algebras.     

Complex structures adapted to magnetic flows

Let $M$ be a compact real-analytic manifold, equipped with a real-analytic Riemannian metric $g$, and let $\beta$ be a closed real-analytic 2-form on $M$, interpreted as a magnetic field. Consider the Hamiltonian flow on $T^{1}M$ that describes a charged particle moving in the magnetic field $\beta$. Following an idea of T. Thiemann, we construct a complex structure on a tube inside $T^{1}M$ by pushing forward the vertical polarization by the Hamiltonian flow “evaluated at time $i$”. This complex structure fits together with $\omega -{\pi }^1\beta$ to give a Kähler structure on a tube inside $T^{1}M$. When $\beta =0$, our magnetic complex structure is the adapted complex structure of Lempert–Szőke and Guillemin–Stenzel.We describe the magnetic complex structure in terms of its $(1,0)$-tangent bundle, at the level of holomorphic functions, and via a construction using the embeddings of Whitney–Bruhat and Grauert. We describe an antiholomorphic intertwiner between this complex structure and the complex structure induced by $-\beta$, and we give two formulas for local Kähler potentials, which depend on a local choice of vector potential $1$-form for $\beta$. Finally, we compute the magnetic complex structure explicitly for constant magnetic fields on ${\mathbb{R}}^2$ and $S^2$.     

Surface plasmons in a semi-bounded massless Dirac plasma

The collective excitation of surface plasmons in a massless Dirac plasma (e.g., graphene) half-space (bounded by air) is investigated using a relativistic quantum fluid model. The unique features of such surface waves are discussed and compared with those in a Fermi plasma. It is found that in contrast to Fermi plasmas, the long-wavelength surface plasmon frequency (ω) in massless Dirac plasmas is explicitly nonclassical, i.e., $\omega \propto 1/\sqrt{?}$, where $h=2\pi ?$ is the Planck's constant. Besides some apparent similarities between the surface plasmon frequencies in massless Dirac plasmas and Fermi plasmas, several notable differences are also found and discussed. Our findings elucidate the properties of surface plasmons that may propagate in degenerate plasmas where the relativistic and quantum effects play a vital role.     

Affine Hamiltonians in Higher Order Geometry

Affine Hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of Lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine Hamiltonians and Lagrangians of order k≥2 are studied, relating them to a Hamilton equation. An Ostrogradski type theorem is proved: the Hamilton equation of an affine Hamiltonian h is equivalent with Euler–Lagrange equation of its dual Lagrangian L. Zermelo condition is also studied and some non-trivial examples are given. The authors were partially supported by the CNCSIS grant A No. 81/2005.     

LEPAGE EQUIVALENTS OF SECOND-ORDER EULER–LAGRANGE FORMS AND THE INVERSE PROBLEM OF THE CALCULUS OF VARIATIONS

In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler–Lagrange equations. They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton–Jacobi theory and theory of integration of variational equations. In this paper, Lepage equivalents of second-order Euler–Lagrange quasi-linear PDE's are characterised explicitly. A closed (n + 1)-form uniquely determined by the Euler–Lagrange form is constructed, and used to find a geometric solution of the inverse problem of the calculus of variations.