We consider the variational complex on infinite jet space and the complex of variational derivatives for Lagrangians of multidimensional paths and study relations between them. The discussion of the variational (bi)complex is set up in terms of a flat connection in the jet bundle. We extend it to supercase using a particular new class of forms. We establish relation of the complex of variational derivatives and the variational complex. Certain calculus of Lagrangians of multidimensional paths is developed. It is shown how covariant Lagrangians of higher order can be used to represent characteristic classes.
In this paper we present a new theory of calculus over $k$-dimensional domains in a smooth $n$-manifold, unifying the discrete, exterior, and continuum theories. The calculus begins at a single point and is extended to chains of finitely many points by linearity, or superposition. It converges to the smooth continuum with respect to a norm on the space of ``pointed chains,'' culminating in the chainlet complex. Through this complex, we discover a broad theory of coordinate free, multivector analysis in smooth manifolds for which both the classical Newtonian calculus and the Cartan exterior cal…
Basic elements of integral calculus over algebras of iterated differential forms, are presented. In particular, defining complexes for modules of integral forms are described and the corresponding berezinians and complexes of integral forms are computed. Various applications and the integral calculus over the algebra $Λ_{\infty}$ will be discussed in subsequent notes.
On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic form, we find a new complex. In particular, on complex projective space with its Fubini-Study form and connection, we can build a series of differential complexes akin to the Bernstein-Gelfand-Gelfand complexes from parabolic differential geometry.
We connect the well-known theory of functional forms of variational bicomplex with the theory of antiexact differential forms. We identify antiexact functional forms as an obstruction to the variationality of differential equations. The most prominent result of this observation is the formulation of the variational problem for some differential equations that are not variational and neither have a variational multiplier. Heat, Navier-Stokes, and KdV equations illustrate this new variational principle.
The calculus of variations for lagrangians which are not functions on the tangent bundle, but sections certain affine bundles is developed. We follow a general approach to variational principles which admits boundary terms of variations.
For a compact, oriented, hyperbolic $n$-manifold $(M,g)$, realised as $M= Γ\backslash \mathbb{H}^{n}$ where $Γ$ is a torsion-free cocompact subgroup of $SO(n,1)$, we establish and study a relationship between differential geometric cohomology on $M$ and algebraic invariants of the group $Γ$. In particular for $\mathbb{F}$ an irreducible $SO(n,1)$-module, we show that the group cohomology with coefficients $H^{\bullet}(Γ, \mathbb{F})$ is isomorphic to the cohomology of an appropriate projective BGG complex on $M$. This yields the geometric interpretation that $H^{\bullet}(Γ, \mathbb{F})$ parame…
We present a geometric interpretation of the integration-by-parts formula on an arbitrary vector bundle. As an application we give a new geometric formulation of higher-order variational calculus.
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propose variationality as an interesting geometric property to take into account in the design and computer simulation of numerical integrators.
Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for analogs of holonomic, vakonomic and nonholonomic constraints. This general model covers majority of first-order Lagrangian systems which are present in the literature and reduces to the standard variational calculus and the Euler-Lagrange equations in Classical Mechanics for E=TM.
We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number of independent variables increased by one), and explain its relation with the classical Helmholtz criterion using the de Rham complex on an infinite-dimensional space of fields.
We present a family of complexes playing the same role, for homogeneous variational problems, that the horizontal parts of the variational bicomplex play for variational problems on a fibred manifold. We show that, modulo certain pullbacks, each of these complexes (apart from the first one) is globally exact. All the complexes may be embedded in bicomplexes, and we show that, again modulo pullbacks, the latter are locally exact. The edge sequence is an important part of such a bicomplex, and may be used for the study of homogeneous variational problems.
In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to "telescope" to accomodate a level of detail appropriate to a set of c…
It is shown that the Euler-Lagrange equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of reduction and the relation with Lagrange multiplier method are also studied.
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a differential relation of special kind (a Zakrzewski morphism). In the paper we investigate basic properties of higher algebroids and show some applications. Namely, we develop a geometric framework for variational calculus on higher algebroids (including both forces and momenta). Suc…
The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulation for a subsystem of the given PDE. No constraints on the differential order or number of dependent or independent variables are assumed. The proof follows a recent observation of Bridges, Hydon and Lawson and generalizes an older result of Henneaux from ordinary differential equ…
We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) such as the Schouten algebra - these give a convenient language for describing deformation problems; 2) the Deligne 2-groupoid associated to such a dg Lie algebra, which represents the moduli of formal deformations; 3) a refined version of the Schouten bracket in the formal calculus…
The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of the first--order variational integrals related to the second--order Euler--Lagrange systems.
We propose a unified approach to the theory of connections in the geometry of sprays and Finsler metrics which, in particular, gives a simple explanation of the well-known fact that all the classical Finslerian connections provide exactly the same formulas appearing in the calculus of variations.
In this paper we continue the investigation of Loday's Leibniz cohomology as a new invariant for differentiable manifolds. In particular the Leibniz coboundary of a k-tensor (in the sense of differential geometry) is computed in a local coordinate chart and then interpreted in terms of the calculus of variations. For example, the Leibniz coboundary of the metric two-tensor reduces to the first variation of arc length.
We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms between our bundles of smooth maps, we introduce the concept of fiberwise $(k,r)$-jet. The relations to the Euler-Lagrange morphism of the variational calculus are described.
We develop a frame and dyad gauge-independent formalism for the calculus of variations of functionals involving spinorial objects. As part of this formalism we define a modified variation operator which absorbs frame and spin dyad gauge terms. This formalism is applicable to both the standard spacetime (i.e. SL(2,C)) 2-spinors as well as to space (i.e. SU(2,C)) 2-spinors. We compute expressions for the variations of the connection and the curvature spinors.
The paper studies the harmonic maps on a direction between a Riemannian space and a generalized Lagrange space. Also, it is proved there that the solutions of C^2 class of certain ODEs or PDEs are harmonic maps, in the sense of this paper.
We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler-Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi's theorem on conjugate points and Noether's theorem. We close with an application to low regularity Riemannian geometry.
In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory and so on. This generalized variational calculus is based on two main notions: the tangent lift of curves and the notion of complete lift of a vector field. Both concepts are also adapted for the case of skew-symmetric algebroids, therefore, our formalism easily extends to the case of Lie algebroids and nonholonomic systems. …
A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and sufficient algebraic and differential conditions for the existence of a variational multiplier are derived. It is shown that the number of independent variational multipliers is determined by the nullity of a completely algebraic system of equations associated to the given system of partial differential equations. An algorithm for solving the inverse problem is…
Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. 'Integration by parts' procedure for this variation is then described in purely formal language and is shown to consist in application of nonlinear Green formula to the vertical differential of the Lagrangian. Euler-Lagrange expressions and the Green operator are calculated by simple pull-backs of certain vector bundle valued differential forms associated with the given va…
