Oana Constantinescu | Universitatea Alexandru Ioan Cuza Iasi (original) (raw)

Papers by Oana Constantinescu

We present a reformulation of the inverse problem of the calculus of variations for time dependen... more We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, J 1 π. We prove that a system of time dependent SODE, identified with a semispray S, is Lagrangian if and only if a special class, Λ 1 S (J 1 π), of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class Λ 1 S (J 1 π) of semi-basic 1-forms. Each such class contains the Poincaré-Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1-form in Λ 1 S (J 1 π), which is not a Poincaré-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.

International Journal of Geometric Methods in Modern Physics, 2012

The aim of this paper is to study from the point of view of linear connections the data (M, D, g,... more The aim of this paper is to study from the point of view of linear connections the data (M, D, g, W ) with M a smooth (n + p) dimensional real manifold, (D, g) a n-dimensional semi-Riemannian distribution on M, G the conformal structure generated by g and W a Weyl substructure: a map W :

Symmetry, Integrability and Geometry: Methods and Applications, 2012

We address the integrability conditions of the inverse problem of the calculus of variations for ... more 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.

International Journal of Geometric Methods in Modern Physics, 2011

To a system of second order ordinary differential equations (SODE) one can assign a canonical non... more To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical nonlinear connection also to a system of higher order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third or fourth order ordinary differential equations.

We present a reformulation of the inverse problem of the calculus of variations for time dependen... more We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, J 1 π. We prove that a system of time dependent SODE, identified with a semispray S, is Lagrangian if and only if a special class, Λ 1 S (J 1 π), of semi-basic 1-forms is not empty. We provide global Helmholtz conditions to characterize the class Λ 1 S (J 1 π) of semi-basic 1-forms. Each such class contains the Poincaré-Cartan 1-form of some Lagrangian function. We prove that if there exists a semi-basic 1-form in Λ 1 S (J 1 π), which is not a Poincaré-Cartan 1-form, then it determines a dual symmetry and a first integral of the given system of SODE.

International Journal of Geometric Methods in Modern Physics, 2012

The aim of this paper is to study from the point of view of linear connections the data (M, D, g,... more The aim of this paper is to study from the point of view of linear connections the data (M, D, g, W ) with M a smooth (n + p) dimensional real manifold, (D, g) a n-dimensional semi-Riemannian distribution on M, G the conformal structure generated by g and W a Weyl substructure: a map W :

Symmetry, Integrability and Geometry: Methods and Applications, 2012

We address the integrability conditions of the inverse problem of the calculus of variations for ... more 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.

International Journal of Geometric Methods in Modern Physics, 2011

To a system of second order ordinary differential equations (SODE) one can assign a canonical non... more To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical nonlinear connection also to a system of higher order ordinary differential equations (HODE). For this nonlinear connection we develop its geometry, and explicitly compute all curvature components of the corresponding Jacobi endomorphism. Using these curvature components we derive a Jacobi equation that describes the behavior of nearby geodesics to a HODE. We motivate the applicability of this nonlinear connection using examples from the equivalence problem, the inverse problem of the calculus of variations, and biharmonicity. For example, using components of the Jacobi endomorphism we express two Wuenschmann-type invariants that appear in the study of scalar third or fourth order ordinary differential equations.