This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote the configuration fiber bundle, we show that both the multisymplectic structure on $J^3Y$ as well as the Noether theorem arise from the first variation of the action function. We generalize the multisymplectic form formula derived for first order field theories in \cite{MPS}, to the case of second-order field theories, and we apply our theory to the Camassa-Holm (CH) equation in both the continuous and discrete settings. Our discretization produces a multisymplectic-momentum integrator, a generalization of the Moser-Veselov rigid body algorithm to the setting of nonlinear PDEs with second order Lagrangians.
We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler--Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called `back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm, and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with $2$-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.
This paper presents a variational and multisymplectic formulation of both compressible and incompressible models of continuum mechanics on general Riemannian manifolds. A general formalism is developed for non-relativistic first-order multisymplectic field theories with constraints, such as the incompressibility constraint. The results obtained in this paper set the stage for multisymplectic reduction and for the further development of Veselov-type multisymplectic discretizations and numerical algorithms. The latter will be the subject of a companion paper.
We use the kernel of a premultisymplecic form to classify its solutions, inspired by the work of M. Gotay and J. Nester. In the case of variational premultisymplectic forms, there is an equivalence relation which classify the solutions in general distributions called expanded solutions. We also present an equivalence relation for sections and a reduction procedure for the system. We apply this results to mechanics, electromagnetism and Metric-Affine gravity, and compare them with other notions of gauge freedom.
The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric structures of the jet manifold. When a Lagrangian function is given, we find the equations of motion in terms of a Cartan form canonically associated to the Lagrangian. The Hamiltonian formalism is also extended to this setting and we find the relation between the solutions of both formalism. When the first Lie algebroid is a tangent bundle we give a variational description of the equations of motion.…
Field Theories in Physics can be formulated giving a local Lagrangian density. Locality is imposed using the infinite jet bundle. That bundle is viewed as a pro-finite dimensional smooth manifold and that point of view has been compared to different topological and Frechét structures on it. A category of local (insular) manifolds has been constructed. Noether's second theorem is reviewed and the notion of Lie pseudogroups is explored using these concepts. The L-infinity algebra of local observables is defined depending only on the cohomology of the Lagrangian (using a result in multisymplectic…
A new geometric framework is developed to describe non-conservative classical field theories, which is based on multisymplectic and contact geometries. Assuming certain additional conditions and using the forms that define this multicontact structure, as well as other geometric elements that are derived from them, we can introduce variational field equations in the multicontact manifolds. These equations are stated using different geometric tools; namely, sections, multivector fields and Ehresmann connections in fiber bundles. Then, this framework can be adapted to the jet bundle description o…
The Lagrange problem is established in the discrete field theory subject to constraints with values in a Lie group. For the admissible sections that satisfy a certain regularity condition, we prove that the critical sections of such problems are the solutions of a canonically unconstrained variational problem associated with the Lagrange problem (discrete Lagrange multiplier rule). This variational problem has a discrete Cartan 1-form, from which a Noether theory of symmetries and a multisymplectic form formula are established. The whole theory is applied to the Euler-Poincaré reduction in the…
The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Secondly, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures…
We consider the second variational derivative of a given gauge-natural invariant Lagrangian taken with respect to (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. By requiring such a second variational derivative to vanish, {\em via} the Second Noether Theorem we find that a covariant strongly conserved current is canonically associated with the deformed Lagrangian obtained by contracting Euler--Lagrange equations of the original Lagrangian with (prolongations of) vertical parts of gauge-natural lifts of infinitesimal principal automorphisms ly…
The Carathéodory form of the calculus of variations belongs to the class of Lepage equivalents of first-order Lagrangians in field theory. Here, this equivalent is generalized for second- and higher-order Lagrangians by means of intrisic geometric operations applied to the well-known Poincaré--Cartan form and principal component of Lepage forms, respectively. For second-order theory, our definition coincides with the previous result obtained by Crampin and Saunders in a different way. The Carathéodory equivalent of the Hilbert Lagrangian in general relativity is discussed.
It is proved that the set of geodesic circles in two dimensions may be given a variational description and the explicit form of it is presented. In the limit case of the Euclidean geometry a certain claim of uniqueness of such description is proved. A formal notion of 'spin' force is discovered as a by-product of the variation procedure involving the acceleration.
A second order variational description of the autoparallel curves of some differential-geometric connection for the third order Mathisson's 'new mechanics' of a relativistic free spinning particle is suggested starting from general requirements of invariance and 'variationality'.
We study variational problems for integral invariants, which are defined as integrations of invariant functions of the second fundamental form, of a smooth map between pseudo-Riemannian manifolds. We derive the first variational formulae for integral invariants defined from invariant homogeneous polynomials of degree two. Among these integral invariants, we show that the Euler-Lagrange equation of the Chern-Federer energy functional is reduced to a second order PDE. Then we give some examples of Chern-Federer submanifolds in Riemannian space forms.
We establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.
The present paper extends the classical second-order variational problem of Herglotz type to the more general context of the Euclidean sphere S^n following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler-Lagrange equations is established. This problem covers some classical variational problems posed on the Riemannian manifold S^n such as the problem of finding cubic polynomials on S^n. It also finds applicability on the dynamics of the simple pendulum in a resistive medium.
J.Eells and L. Lemaire introduced $k$-harmonic maps, and Wang Shaobo showed the first variation formula. In this paper, we give the second variation formula of $k$-energy, and give a notion of index, nullity and weakly stable. We also study $k$-harmonic maps into the product Riemannian manifold, and $k$-harmonic curves into a Riemannian manifold with constant sectional curvature, and show their non-trivial solutions.
In this note, we compute the second variational formula for the functional $\int_M v^{(6)}(g)dv_g$, which was introduced by Graham-Juhl and the first variational formula was obtained by Chang-Fang. We also prove that Einstein manifolds (with dimension $\ge 7$) with positive scalar curvature is a strict local maximum within its conformal class, unless the manifold is isometric to round sphere with the standard metric up to a multiple of constant. Note that when $(M,g)$ is locally conformally flat, this functional reduces to the well-studied $\int_M σ_3(g)dv_g$. Hence, our result generalize a pr…
J.Eells and L. Lemaire introduced k-harmonic maps, and T. Ichiyama, J. Inoguchi and H.Urakawa showed the first variation formula. In this paper, we give the second variation formula of k-harmonic maps, and show non-existence theorem of proper k-harmonic maps into a Riemannian manifold of non-positive curvature (k >= 2). We also study k-harmonic maps into the product Riemannian manifold, and describe the ordinary differential equations of 3-harmonic curves and 4-harmonic curves into a sphere, and show their non-trivial solutions.
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We discover here that parabolic regularity is the key to derive new calculus rules and computation formulas for major second-order generalized differential constructions of variational analysis in connection with some properties of sets that go back to classical differential geometr…
Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of $L_g$ over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility.
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…
We show how the homogeneous variational bicomplex provides a useful formalism for describing a number of properties of single-integral variational problems, and we introduce a subsequence of one of the rows of the bicomplex which is locally exact with respect to the variational derivative. We are therefore able to recover a Lagrangian from a set of equations given as a variationally-closed differential form. As an example, we show how to recover a first-order Lagrangian from a suitable set of second-order equations.
