Mechanical systems in the generalized Lie algebroids framework

Arcuş, Constantin M.

doi:10.1142/s0219887814500236

Cited by 12 publications

(12 citation statements)

References 29 publications

(37 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The (ρ, η)-semispray S will be called the canonical (ρ, η)-semispray associated to the mechanical (ρ, η)-system ((E, π, M ) , F e , (ρ, η) Γ) and locally invertible vector bundles morphism (g, h) (see [4]). …”

Section: Remark 42 Using the Above Definition It Is Easy To See Thatmentioning

confidence: 99%

Weyl’s theory in the generalized Lie algebroids framework

Arcuş¹,

Peyghan

Sharahi

2014

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical (ρ, η)-system with respect to a (ρ, η)-spray, the Berwald (ρ, η)-derivative operator and its mixed curvature, we obtain main results to conceptualize the Weyl's method in this general framework. Finally, we obtain two new results of Weyl type for the geometry of mechanical (ρ, η)-systems.

show abstract

Section: Remark 42 Using the Above Definition It Is Easy To See Thatmentioning

confidence: 99%

Weyl’s theory in the generalized Lie algebroids framework

Arcuş¹,

Peyghan

Sharahi

2014

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Extending the notion of Lie algebroid from one base manifold to a pair of diffeomorphic base manifolds, the second author introduced the generalized Lie algebroid [1,2]. Using the lift of a differentiable curve defined on the base of a generalized Lie algebroid, he developed a new theory of mechanical systems with many applications in physics [4]. The space used for developing this theory of mechanical systems is the Lie algebroid generalized tangent bundle (((ρ, η)T F, (ρ, η)τ F , F ), [, ] (ρ,η)T F , (ρ, Id F )), of a generalized Lie algebroid ((F, ν, N ), [, ] F,h , (ρ, η)).…”

Section: Introductionmentioning

confidence: 99%

Vertical and complete lifts of sections of a (dual) vector bundle and Legendre duality

Peyghan¹,

Nourmohammadifar²,

Arcuş³

2018

Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics

View full text Add to dashboard Cite

Supplementary comments about generalized Lie algebroids are presented and a new point of view over the construction of the Lie algebroid generalized tangent bundle of a (dual) vector bundle is introduced. Using the general theory of exterior differential calculus for generalized Lie algebroids, a covariant derivative for exterior forms of a (dual) vector bundle is introduced. Using this covariant derivative, the complete lift of an arbitrary section of a (dual) vector bundle is discovered. A theory of Legendre type and Legendre duality between vertical and complete lifts is presented. Finally, a duality between Lie algebroids structures is developed.

show abstract

“…The notion of Lie algebroid is a natural generalization of the tangent bundle and Lie algebra. In the last decades the Lie algebroids [27,28] are the objects of intensive studies with applications to mechanical systems or optimal control [2,10,13,23,26,31,32,33,35,36,37,38,39,45] and are the natural framework in which one can develop the theory of differential equations, where the notion of symmetry plays a very important role.…”

Section: Introductionmentioning

confidence: 99%