Related papers: Second-order integrable Lagrangians and WDVV equat…

We discuss an elementary derivation of variational symmetries and corresponding integrals of motion for the Lagrangian systems depending on acceleration. Providing several examples, we make the manuscript accessible to a wide range of…

For the linearized setting of the dynamics of complex bodies we construct variational integrators and prove their convergence by making use of BV estimates on the rate fields. We allow for peculiar substructural inertia and internal…

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and non-local) and the…

We find general non-linear lagrangians of a U(1) field invariant under electric-magnetic duality. They are characterized by an arbitrary function and go to the Maxwell theory in the weak field limit. We give some explicit examples which are…

This paper aims at the most comprehensive and systematic construction and tabulation of mechanical systems that admit a second invariant, quadratic in velocities, other than the Hamiltonian. The configuration space is in general a 2D…

We construct, for a second-order homogeneous Lagrangian in two independent variables, a differential 2-form with the property that it is closed precisely when the Lagrangian is null. This is similar to the property of the 'fundamental…

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…

Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering…

We calculate the effective electromagnetic Lagrangian up to the lowest-order corrections in the derivatives for two fermionic systems of interest in condensed matter physics in the linearized approximation of the tight-binding Hamiltonian…

Based on previous work we consturct an equation (Lagrange equation) and relate it with a system of generalized integrals and differential equations in such a way to provide useful evaluations and connections between them.

The Noether-like operators that play an essential role in writing down the invariants for systems of two ordinary differential equations (ODEs) are constructed. The classification of such operators is carried out with the help of analytic…

We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the…

We solve the classification problem for integrable lattices of the form $u_{,t}=f(u_{-2},\dots,u_2)$ under the additional assumption of invariance with respect to the group of linear-fractional transformations. The obtained list contains 5…

In the context of classical mechanics, we study the conditions under which higher-order derivative theories can evade the so-called Ostrogradsky instability. More precisely, we consider general Lagrangians with second order time…

The fundamental problem of calculus of variations is considered when solutions are differentiable curves on locally convex spaces. Such problems admit an extension of the Euler-Lagrange equations [Orlov 2002] for continuously normally…

A link between first-order ordinary differential equations (ODEs) and 2-dimensional Riemannian manifolds is explored. Given a first-order ODE, an associated Riemannian metric on the variable space is defined, and some properties of the…

In this study, linear second-order conformable differential equations using a proportional derivative are shown to be formally self-adjoint equations with respect to a certain inner product and the associated self-adjoint boundary…

The integrable systems associated with Seiberg-Witten geometry are considered both from the Hitchin-Donagi-Witten gauge model and in terms of intermediate Jacobians of Calabi-Yau threefolds. Dual pairs and enhancement of gauge symmetries…

We propose a notion of a pluri-Lagrangian problem, which should be understood as an analog of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems…

In this paper, we will give a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. Moreover, we work in the setting of Lie groupoids and Lie algebroids which is enough general to…