Jet Single‐Time Lagrange Geometry and Its Applications

Balan, Vladimir; Neagu, Mircea

doi:10.1002/9781118143759

Cited by 18 publications

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“…where we have the following physical meanings: • m is the mass of the particule; • V is the LB-monolayer compressing rate; • p is a constant monolayer parameter given by the physical formula p = π 2 q 2 εε 0 The differential geometry (in the sense of nonlinear connections, Cartan linear connections, d-torsions, d-curvatures etc.) produced by an arbitrary jet Lagrangian function L : J 1 (R, M n ) → R is now completely done by Balan and Neagu in the monograph [3]. The geometrical ideas from [3] are similar (but however distinct ones) to those exposed by Miron and Anastasiei in the classical Lagrangian geometry on tangent bundles (see [8]).…”

Section: Introductionmentioning

confidence: 88%

“…produced by an arbitrary jet Lagrangian function L : J 1 (R, M n ) → R is now completely done by Balan and Neagu in the monograph [3]. The geometrical ideas from [3] are similar (but however distinct ones) to those exposed by Miron and Anastasiei in the classical Lagrangian geometry on tangent bundles (see [8]). More accurately, these jet geometrical Lagrangian ideas were initiated by Asanov in [2] and developed further in multi-parameter way in the book [10], and in single-time way in [3].…”

Section: Introductionmentioning

confidence: 88%

“…More accurately, these jet geometrical Lagrangian ideas were initiated by Asanov in [2] and developed further in multi-parameter way in the book [10], and in single-time way in [3]. In such a context, this paper is devoted to the development on the 1-jet vector bundle J 1 (T, R 2 ) of the single-time Lagrange geometry (see [3]) produced by the 2D-monolayer Lagrangian (1.2).…”

Section: Introductionmentioning

confidence: 99%

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Jet theoretical Yang-Mills energy in the geometric dynamics of two-dimensional monolayer

Neagu

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Grushevskaya

2013

Journal of Mathematical Physics

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Langmuir-Blodgett films (LB-films) consist from few LB-monolayers which are high structured nanomaterials that are very promising materials for applications. We use a geometrical approach to describe structurization into LB-monolayers. Consequently, we develop on the 1-jet space J 1 ([0, ∞), R 2 ) the single-time Lagrange geometry (in the sense of distinguished (d-) connection, d-torsions and an abstract anisotropic electromagnetic-like d-field) for the Lagrangian governing the 2D-motion of a particle of monolayer. One assumed that an expansion near singular points for the constructed geometrical Lagrangian theory describe phase transitions to LB-monolayer. Trajectories of particles in a field of the electrocapillarity forces of monolayer have been calculated in a resonant approximation utilizing some Jacobi equations. A jet geometrical Yang-Mills energy is introduced and some computer graphic simulations are exposed.Mathematics Subject Classification (2010): 53C60, 53C80, 81T13.

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Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

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Jet theoretical Yang-Mills energy in the geometric dynamics of two-dimensional monolayer

Neagu

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Grushevskaya

2013

Journal of Mathematical Physics

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“…Proof. For the energy action functional E, the associated Euler-Lagrange equations can be written in the equivalent form (see [5,2])…”

Section: )mentioning

confidence: 99%

An anisotropic geometrical approach for non-relativistic extended dynamics

2013

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In this paper we present the distinguished (d-) Riemannian geometry (in the sense of nonlinear connection, Cartan canonical linear connection, together with its d-torsions and d-curvatures) for a possible Lagrangian inspired by optics in non-uniform media. The corresponding equations of motion are also exposed, and some particular solutions are given. For instance, we obtain as geodesic trajectories some circular helices (depending on an angular velocity ω), certain circles situated in some planes (ones are parallel with xOy, and other ones are orthogonal on xOy), or some straight lines which are parallel with the axis Oz. All these geometrical geodesics are very specific because they are completely determined by the non-constant index of refraction n(x). (2010): 53C60, 53C80, 83C10. Mathematics Subject Classification

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“…A well-known example is Berwald-Moor space (M, f ), where f (x i ;ẋ i ) = (ẋ 1 · · ·ẋ n ) 1 n , n = m; see, e.g., [6,14,23]. The paper starts with the discussion of the notion of geodesics in Finsler and pseudoFinsler spaces with n ≥ 3 (Section 1).…”

Section: Introductionmentioning

confidence: 99%