Related papers: Exterior Differential Systems and Euler-Lagrange P…
Skew-symmetric differential forms play an unique role in mathematics and mathematical physics. This relates to the fact that closed exterior skew-symmetric differential forms are invariants. The concept of "Exterior differential forms" was…
We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of…
We define on-shell symmetries and characterize them for Lagrangian systems. The terms appearing in the variation of the Poincare'-Cartan form, which vanish because of field equations, are found to be strongly constrained if the space of…
This paper presents the Euler-Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses…
In this paper, we review two related aspects of field theory: the modeling of the fields by means of exterior algebra and calculus, and the derivation of the field dynamics, i.e., the Euler-Lagrange equations, by means of the stationary…
We study the geometry of multidimensional scalar $2^{nd}$ order PDEs (i.e. PDEs with $n$ independent variables) with one unknown function, viewed as hypersurfaces $\mathcal{E}$ in the Lagrangian Grassmann bundle $M^{(1)}$ over a…
This paper has several goals. The first idea is to study the geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness in a modern and rigorous way. Although the idea is not new, our main Theorems…
We start discussing basic properties of Lie groupoids and Lie pseudo-groups in view of applying these techniques to the analysis of Jordan-H\"older resolutions and the subsequent integration of partial differential equations which is the…
The Lagrangian formulation of classical mechanics is widely applicable in solving a vast array of physics problems encountered in the undergraduate and graduate physics curriculum. Unfortunately, many treatments of the topic lack…
Functional bases of second-order differential invariants of the Euclid, Poincar\'e, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant…
We present a new class of solutions for the inverse problem in the calculus of variations in arbitrary dimension $n$. This is the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary…
A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants…
We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimising frame, also known as the Normal, Parallel or Bishop frame. Such systems have previously been…
An intrinsic description of the Hamilton-Cartan formalism for first-order Berezinian variational problems determined by a submersion of supermanifolds is given. This is achieved by studying the associated higher-order graded variational…
The inverse problem of the calculus of variations consists in determining if the solutions of a given system of second order differential equations correspond with the solutions of the Euler-Lagrange equations for some regular Lagrangian.…
In this paper we consider multi-dimensional partial differential equations of parabolic type involving divergence form operators that possess a discontinuous coefficient matrix along some smooth interface. The solution of the equation is…
We construct a pseudo-Lagrangian that is invariant under rigid $E_{11}$ and transforms as a density under $E_{11}$ generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on…
Using parametrized curves (Section 1) or parametrized sheets (Section 3), and suitable metrics, we treat the jet bundle of order one as a semi-Riemann manifold. This point of view allows the description of solutions of DEs as pregeodesics…
We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered…
The important unsolved problem in theory of integrable systems is to find conditions guaranteeing existence of a Lax representation for a given PDE. The use of the exotic cohomology of the symmetry algebras opens a way to formulate such…