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Ioan Bucataru

Publications and source records attributed to Ioan Bucataru.

At least 19 recordsLinked to original sources
arXiv2026

Strong Hamel functions and symmetries

For the geodesic spray of a Finsler space, a strong Hamel function is a Hamel function that is the geodesic derivative of a $0$-homogeneous potential function. Similarly, strong dual symmetries and strong dynamical symmetries are geodesically invariant $1$-forms and vector fields, respectively, associated with $0$-homogeneous potential functions. We prove that strong Hamel functions can be characterized in terms of strong dual symmetries and strong dynamical symmetries. We show that projective deformations by strong Hamel functions preserve the $χ$-curvature and analyse the relationship with other classes of functions (Funk and weak Funk functions) that preserve the curvature tensors under projective deformations.

math.DG↗
arXiv2023

Finsler metrizabilities and geodesic invariance

We prove that various Finsler metrizability problems for sprays can be reformulated in terms of the geodesic invariance of two tensors (metric and angular). We show that gyroscopic sprays is the the largest class of sprays with geodesic invariant angular metric. Scalar functions associated to these geodesically invariant tensors will be invariant as well and therefore will provide first integrals for the given spray.

math.DG↗
arXiv2022

First integrals for Finsler metrics with vanishing $χ$-curvature

We prove that in a Finsler manifold with vanishing $χ$-curvature (in particular with constant flag curvature) some non-Riemannian geometric structures are geodesically invariant and hence they induce a set of non-Riemannian first integrals. Two alternative expressions of these first integrals can be obtained either in terms of the mean Berwald curvature, or as functions of the mean Cartan torsion and the mean Landsberg curvature.

math.DG↗
arXiv2022

Invariant volume forms and first integrals for geodesically equivalent Finsler metrics

Two geodesically (projectively) equivalent Finsler metrics determine a set of invariant volume forms on the projective sphere bundle. Their proportionality factors are geodesically invariant functions and hence they are first integrals. Being 0-homogeneous functions, the first integrals are common for the entire projective class. In Theorem 1.1 we provide a practical and easy way of computing these first integrals as the coefficients of a characteristic polynomial.

math.DG↗
arXiv2019

An Algebraic Characterisation for Finsler Metrics of Constant Flag Curvature

In this paper we prove that a Finsler metrics has constant flag curvature if and only if the curvature of the induced nonlinear connection satisfies an algebraic identity with respect to some arbitrary second rank tensors. Such algebraic identity appears as an obstruction to the formal integrability of some operators in Finsler geometry, [4,7]. This algebraic characterisation for Finsler metrics of constant flag curvature allows to provide yet another proof for the Finslerian version of Beltrami's Theorem, [2,3].

math.DG↗
arXiv2018

A characterisation for Finsler metrics of constant curvature and a Finslerian version of Beltrami Theorem

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function. This way we provide a Finslerian version of Beltrami Theorem.

math.DG↗
arXiv2016

Frobenius integrability and Finsler metrizability for $2$-dimensional sprays

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is provided by a closed, homogeneous $1$-form from the annihilator of the Berwald distribution. We discuss both the degenerate and non-degenerate cases using the fact that the regularity of the Finsler function is encoded into a regularity condition of a $2$-form, canonically associated to the given spray. The integrability of the Berwald distribution and the regularity of the $2$-form have simple and useful expressions in terms of the Berwald frame.

math.DG↗
arXiv2016

Invariant metrizability and projective metrizability on Lie groups and homogeneous spaces

In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.

math.DG↗
arXiv2015

Funk functions and projective deformations of sprays and Finsler spaces of scalar flag curvature

In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature tensor, [14, page 184]. In this paper, we provide an answer to this question, within the class of Finsler metrics of scalar flag curvature. In Theorem 3.1, we show that the answer is negative, for non-vanishing scalar flag curvature. The answer is known to be positive when the scalar flag curvature vanishes, [12, 14] and this positive answer is related to the existence of many solutions to Hilbert's Fourth Problem. As a generalisation of this problem, we can ask if it is possible for a given spray, with non-vanishing scalar flag curvature, to represent, after reparametrisation, the geodesic spray of a Finsler metric. In Proposition 3.3, we show how to construct sprays whose projective class does not contain any Finsler metrizable spray with the same Riemann curvature tensor.

math.DG↗
arXiv2015

Generalized Helmholtz conditions for non-conservative Lagrangian systems

In this paper we provide generalized Helmholtz conditions, in terms of a semi-basic 1-form, which characterize when a given system of second order ordinary differential equations is equivalent to the Lagrange equations, for some given arbitrary non-conservative forces. For the particular cases of dissipative or gyroscopic forces, these conditions, when expressed in terms of a multiplier matrix, reduce to those obtained in [18]. When the involved geometric structures are homogeneous with respect to the fibre coordinates, we show how one can further simplify the generalized Helmholtz conditions. We provide examples where the proposed generalized Helmholtz conditions, expressed in terms of a semi-basic 1-form, can be integrated and the corresponding Lagrangian and Lagrange equations can be found.

math.DG↗
arXiv2015

Symmetries in Lagrangian Field Theory

By generalizing the cosymplectic setting for time-dependent Lagrangian mechanics, we propose a geometric framework for the Lagrangian formulation of classical field theories with a Lagrangian depending on the independent variables. For that purpose we consider the first order jet bundles $J^1π$ of a fiber bundle $π:E\to {\mathbb R}^k$ where ${\mathbb R}^k$ is the space of independent variables. Generalized symmetries of the Lagrangian are introduced and the corresponding Noether Theorem is proved.

math-ph↗
arXiv2013

A setting for higher order differential equations fields and higher order Lagrange and Finsler spaces

We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss the projective metrizability problem for higher order differential equation fields. We provide necessary and sufficient conditions for higher order projectivpre-e metrizability in terms of homogeneous semi-basic 1-forms. Such a semi-basic 1-form is the Poincaré-Cartan 1-form of a higher order Finsler function, while the potential of such semi-basic 1-form is a higher order Finsler function.

math.DG↗
arXiv2013

Metrizable isotropic second-order differential equations and Hilbert's fourth problem

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. The proof of Theorem 3.1 provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. One condition of Theorem 3.1, regarding the regularity of the sought after Finsler function, can be relaxed. By relaxing this condition, we provide examples of sprays that are metrizable by conic pseudo-Finsler functions as well as degenerate Finsler functions. Hilbert's fourth problem asks to determine the Finsler functions with rectilinear geodesics. A Finsler function that is a solution to Hilbert's fourth problem is necessarily of constant or scalar flag curvature. Therefore, we can use the conditions of [11, Theorem 4.1] and Theorem 3.1 to test when the projective deformations of a flat spray, which are isotropic, are metrizable by Finsler functions of constant or scalar flag curvature. We show how to use the algorithms provided by the proofs of [11, Theorem 4.1] and Theorem 3.1 to construct solutions to Hilbert's fourth problem.

math.DG↗
arXiv2012

Sprays metrizable by Finsler functions of constant flag curvature

In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary differential equations represents the Euler-Lagrange equations of a Finsler function of constant flag curvature. The conditions we provide are tensorial equations on the Jacobi endomorphism. We identify the class of homogeneous SODE where the Finsler metrizability is equivalent with the metrizability by a Finsler function of constant flag curvature.

math.DG↗
arXiv2012

Symmetries, Newtonoids vector fields and conservation laws in the Lagrangian $k$-symplectic formalism

In this paper we study symmetries, Newtonoid vector fields, conservation laws, Noether's Theorem and its converse, in the framework of the $k$-symplectic formalism, using the Frölicher-Nijenhuis formalism on the space of $k^1$-velocities of the configuration manifold. For the case $k=1$, it is well known that Cartan symmetries induce and are induced by constants of motions, and these results are known as Noether's Theorem and its converse. For the case $k>1$, we provide a new proof for Noether's Theorem, which shows that, in the $k$-symplectic formalism, each Cartan symmetry induces a conservation law. We prove that, under some assumptions, the converse of Noether's Theorem is also true and we provide examples when this is not the case. We also study the relations between dynamical symmetries, Newtonoid vector fields, Cartan symmetries and conservation laws, showing when one of them will imply the others. We use several examples of partial differential equations to illustrate when these concepts are related and when they are not.

math-ph↗