相关论文: A Lagrangian Description of the Higher-Order Painl…
We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to the same…
We show that given an ordinary differential equation of order four, it may be possible to determine a Lagrangian if the third derivative is absent (or eliminated) from the equation. This represents a subcase of Fels'conditions [M. E. Fels,…
Starting from the second Painlev\'{e} equation, we obtain Painlev\'{e} type equations of higher order by using the singular point analysis.
In a recent paper by Ibragimov [N. H. Ibragimov, Invariant Lagrangians and a new method of integration of nonlinear equations, J. Math. Anal. Appl. 304 (2005) 212--235] a method was presented in order to find Lagrangians of certain…
Differential equations with the Painlev\'e property have been studied extensively due to their appearance in many branches of mathematics and their applicability in physics. Although a modern, differential algebraic treatment of the order…
We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to several…
The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with Euler-Poincar\'e and Lagrange Poincar\'e type…
We use a recently found method to characterise all the invertible fourth-order difference equations linear in the extremal values based on the existence of a discrete Lagrangian. We also give some result on the integrability properties of…
If a Lagrangian defining a variational problem has order $k$ then its Euler-Lagrange equations generically have order $2k$. This paper considers the case where the Euler-Lagrange equations have order strictly less than $2k$, and shows that…
We prove a necessary optimality condition of Euler-Lagrange type for quantum variational problems involving Hahn's derivatives of higher-order.
A geometric approach is used to study a family of higher-order nonlinear Abel equations. The inverse problem of the Lagrangian dynamics is studied in the particular case of the second-order Abel equation and the existence of two alternative…
A geometric approach is used to study the Abel first order differential equation of the first kind. The approach is based on the recently developed theory of quasi-Lie systems which allows us to characterise some particular examples of…
We show that a method presented in [S.L. Trubatch and A. Franco, Canonical Procedures for Population Dynamics, J. Theor. Biol. 48 (1974), 299-324] and later in [G.H. Paine, The development of Lagrangians for biological models, Bull. Math.…
We formulate higher order variations of a Lagrangian in the geometric framework of jet prolongations of fibered manifolds. Our formalism applies to Lagrangians which depend on an arbitrary number of independent and dependent variables,…
We obtain new regularity conditions for problems of calculus of variations with higher-order derivatives. As a corollary, we get non-occurrence of the Lavrentiev phenomenon. Our main regularity result asserts that autonomous integral…
The problem of Painleve classification of ordinary differential equations lasting since the end of XIX century saw significant advances for the limited equation order, however not that much for the equations of higher orders. In this work…
Searching for a Lagrangian may seem either a trivial endeavour or an impossible task. In this paper we show that the Jacobi last multiplier associated with the Lie symmetries admitted by simple models of classical mechanics produces (too?)…
Using supervector fields and graded forms along a morphism, we study the geometry of ordinary differential superequations, extend the formalism of higher order Lagrangian mechanics to the graded context and prove a generalization of…
We study higher--order variational derivatives of a generic second--order Lagrangian ${\cal L}={\cal L}(x,\phi,\partial\phi,\partial^2\phi)$ and in this context we discuss the Jacobi equation ensuing from the second variation of the action.…
This work presents higher order Lagrangian dynamics possessing locally conformal character. More concretely, locally conformal higher order Euler-Lagrange equations are written with particular focus on the second- and the third-order cases.