The integrability conditions in the inverse problem of the calculus of variations for second-order ordinary differential equations (original) (raw)
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We consider the inverse problem of the calculus of variations for any system by writing its differential equations of motion in first-order form. We provide a way of constructing infinitely many Lagrangians for such a system in terms of its constants of motion using a covariant geometrical approach. We present examples of first-order Lagrangians for systems for which no second-order Lagrangians exist. The Hamiltonian theory for first-order (degenerate) Lagrangians is constructed using Dirac's method for singular Lagrangians.
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We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given.
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In this paper we consider nonautonomous differential systems of arbitrary dimension and first find expressions for their inverse Jacobi multipliers and first integrals in some nonautonomous invariant set in terms of the solutions of the differential system. Given an inverse Jacobi multiplier V , we find a relation between the Poincaré translation map Π at time T that extends to arbitrary dimensions the fundamental relation for scalar equations,
The main objective of this paper is to work out a full-scale application of the integrability analysis of the inverse problem of the calculus of variations, as developed in recent papers by Sarlet and Crampin. For this purpose, the celebrated work of Douglas on systems with two degrees of freedom is taken as the reference model. It is shown that the coordinate-free, geometrical calculus used in Sarlet and Crampin's general theoretical developments provides effective tools also to do the practical calculations. The result is not only that all subcases distinguished by Douglas can be given a more intrinsic characterization, but also that in most of the cases, the calculations can be carried out in a more efficient way and often lead to sharper conclusions.
2009
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 demonstrated on several examples. Systems of second-order partial differential equations in two independent and dependent variables are studied and systems which have more than one variational formulation are classified up to contact equivalence.
Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations
Symmetry, Integrability and Geometry: Methods and Applications, 2012
We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler Theorem. We consider a linear partial dierential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: at semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional jet spaces.
The inverse problem of the calculus of variations for discrete systems
Journal of Physics A: Mathematical and Theoretical
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition between the discrete and the continuous problems and propose variationality as an interesting geometric property to take into account in the design and computer simulation of numerical integrators.
Rendiconti del Circolo Matematico di Palermo, 2003
Inverse Jacobi multipliers are a natural generalization of inverse integrating factors to n-dimensional dynamical systems. In this paper, the corresponding theory is developed from its beginning in the formal methods of integration of ordinary differential equations and the "last multiplier" of K. G. Jacobi. We explore to what extent the nice properties of the vanishing set of inverse integrating factors are preserved in the n -dimensional case. In particular, vanishing on limit cycles (in restricted sense) of an inverse Jacobi multiplier is proved by resorting to integral invariants. Extensions of known constructions of inverse integrating factors by means of power series, local Lie Groups and algebraic solutions are provided for inverse Jacobi multipliers as well as a suitable generalization of the concept to systems with discontinuous right-hand side.
Inverse Jacobi Multipliers: Recent Applications in Dynamical Systems
Progress and Challenges in Dynamical Systems, 2013
In this paper we show novel applications of the inverse Jacobi multiplier focusing on questions of bifurcations and existence of periodic solutions admitted by both autonomous and non-autonomous systems of ordinary differential equations. In the autonomous case we focus on dimension n 3 whereas in the non-autonomous we study the cases with n 2. We summarize results already published and additionally we state some recent results to appear. The principal object of this research is two fold: first to prove the existence and smoothness of inverse Jacobi multiplier V in the region of interest in the phase space and second to show that the invariant set under the flow given by the zero-set of an inverse Jacobi multiplier contains under some assumptions orbits which are relevant in its phase portrait such as periodic orbits, limit cycles, stable, unstable and center manifolds and so on. In the non-autonomous T-periodic case we show some relationships between T-periodic orbits and T-periodic inverse Jacobi multipliers.