Related papers: An old Method of Jacobi to find Lagrangians
It is shown that a given non-autonomous system of two first-order ordinary differential equations can be expressed in Hamiltonian form. The derivation presented here allow us to obtain previously known results such as the infinite number of…
In an attempt to look for the root of nonstandard Lagrangians in the theories of the inverse variational problem we introduce a logarithmic Lagrangian (LL) in addition to the so-called reciprocal Lagrangian (RL) that exists in the…
We investigate integrability of Euler-Lagrange equations associated with 2D second-order Lagrangians of the form \begin{equation*} \int f(u_{xx},u_{xy},u_{yy})\ dxdy. \end{equation*} By deriving integrability conditions for the Lagrangian…
In this work we present a formal generalization of the Hamilton-Jacobi formalism, recently developed for singular systems, to include the case of Lagrangians containing variables which are elements of Berezin algebra. We derive the…
A set of linear second-order differential equations is converted into a semigroup, whose algebraic structure is used to generate many novel equations. Two independent methods that can be used to derive the equations of the semigroup are…
A new method for finding first integrals of discrete equations is presented. It can be used for discrete equations which do not possess a variational (Lagrangian or Hamiltonian) formulation. The method is based on a newly established…
We show that $\lambda$-symmetries can be algorithmically obtained by using the Jacobi last multiplier. Several examples are provided.
We consider a class of Lagrangians that depend not only on some configurational variables and their first time derivatives, but also on second time derivatives, thereby leading to fourth-order evolution equations. The proposed higher-order…
In addition to standard and non-standard Lagrangians of classical mechanics, we consider, in this work, null Lagrangians that (i) identically satisfy the Euler-Lagrange equation and at the same time can be expressed as (ii) the total…
The paper presents a new method for finding first integrals of ordinary difference equations which do not possess Lagrangians, nor Hamiltonians. As an example we solve a third order nonlinear ordinary differential equation and its invariant…
A Lagrangian method is introduced recently for deriving indefinite integrals of special functions that satisfy homogeneous (nonhomogeneous) second-order linear differential equations. This paper extends this method to include indefinite…
We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with point masses…
We study a class of weakly coupled Hamilton-Jacobi systems with a specific aim to perform a qualitative analysis in the spirit of weak KAM theory. Our main achievement is the definition of a family of related action functionals containing…
A new method to find first integrals of nonlinear differential equations in Jacobi-type form is presented. The basic idea of our approach is to use one-parameter perturbed motions to find well-conceived nonlocal constants that are conserved…
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral…
We discuss several numerical methods for calculating Lyapunov exponents (a quantitative measure of chaos) in systems of ordinary differential equations. We pay particular attention to constrained systems, and we introduce a variety of…
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…
An equation is obtained to find the Lagrangian for a one-dimensional autonomous system. The continuity of the first derivative of its constant of motion is assumed. This equation is solved for a generic nonconservative autonomous system…
We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new…
The problems that are connected with Lagrangians which depend on higher order derivatives (namely additional degrees of freedom, unbound energy from below, etc.) are absent if effective Lagrangians are considered because the equations of…