Radu Miron

The geometry of Lagrange spaces : theory and applications

I. Fibre Bundles, General Theory. II. Connections in Fibre Bundles. III. Geometry of the Total Space of a Vector Bundle. IV. Geometrical Theory of Embeddings of Vector Bundles. V. Einstein Equations. VI. Generalized Einstein--Yang--Mills Equations. VII. Geometry of the Total Space of a Tangent Bundle. VIII. Finsler Spaces. IX. Lagrange Spaces. X. Generalized Lagrange Space. XI. Applications of the GLn Spaces with the Metric Tensor e2sigma(x,y)gammaij(x,y). XII. Relativistic Geometrical Optics. XIII. Geometry of Time Dependent Lagrangians. Bibliography. Index.

The Geometry of hamilton and lagrange spaces

Preface. 1. The geometry of tangent bundle. 2. Finsler spaces. 3. Lagrange spaces. 4. The geometry of cotangent bundle. 5. Hamilton spaces. 6. Cartan spaces. 7. The duality between Lagrange and Hamilton spaces. 8. Symplectic transformations of the differential geometry of T* M. 9. The dual bundle of a k-osculator bundle. 10. Linear connections on the manifold T*2M. 11. Generalized Hamilton spaces of order 2. 12. Hamilton spaces of order 2. 13. Cartan spaces of order 2. Bibliography. Index.

Higher Order Lagrange Spaces

The notion of Lagrange space of order k, (k ∈ N *), is an immediate extension of that given in Chapter 6 for the Lagrange space of order 2.

Lagrange geometry of second order

Motivated by concrete problems in variational calculus, the differential geometry of second order Lagrange spaces is introduced and studied. First, the osculator bundle of order 2, its two distinguished Liouville vector fields, the 2-almost tangent structure, and the notion of a 2-spray are studied. Second, appropriate nonlinear and linear connections (N-connections) preserving the horizontal and vertical distributions, and their curvatures and Ricci identities are given. Finally, the Lagrange space of order 2 is defined and several important examples involving second order variational problems are provided, along with an almost product space H^3^n regarded as a geometrical model of such a space.

Relativistic geometrical optics

We investigate the gravitational and electromagnetic fields on the generalized Lagrange space endowed with the metricgij(x, y) = γij(x) + {1 + 1/n2(x, y)}yiyj. The generalized Lagrange spacesMm do not reduce to Lagrange spaces. Consequently, they cannot be studied by methods of symplectic geometry. The restriction of the spacesMm to a sectionSν(M) leads to the Maxwell equations and Einstein equations for the electromagnetic and gravitational fields in dispersive media with the refractive indexn(x, V) endowed with the Synge metric. Whenn(x, V) = 1 we have the classical Einstein equations. If 1/n2=1−1/c2 (c being the light velocity), we get results given previously by the authors. The present paper is a detailed version of a work in preparation.

The geometry of Ingarden spaces

Abstract The Randers spaces RF n were introduced by R. S. Ingarden. They are considered as Finsler spaces F n = ( M , α + β ) equipped with the Cartan nonlinear connection. In the present paper we define and study what we call the Ingarden spaces, I F n , as Finsler spaces I F n = ( M , α + β ) equipped with the Lorentz nonlinear connection. The spaces R F n and I F n are completely different. For I F n we discuss: the variational problem, Lorentz nonlinear connection, canonical N -metrical connection and its structure equations, the Cartan 1-form ω , the electromagnetic 2-form tF and the almost symplectic 2-form 0. The formula dω = F + θ is established. It has as a consequence the generalized Maxwell equations. Finally, the almost Hermitian model of I F n is constructed.

General Randers Spaces

The Finsler spaces with the fundamental function where F(x, y) \(\sqrt {{a_{ij}}\left( x \right){y^i}{y^i}} + {b_i}\left( x \right){y^i},\left( {x,y} \right) \in \widetilde {TM} = TM\backslash \left\{ O \right\},\) Where a ij (x) is a Riemannian metric tensor, were introduced by R.G. Ingarden, [4], [1]. These were suggested by Randers’ studies [8] on the geometrical model of the gravitational and electromagnetic fields, a reason to call them “Randers spaces”.

Noether theorem in higher-order Lagrangian mechanics

We study higher-order Lagrangian mechanics on thek-velocity manifold. The variational problem gives rise to new concepts, such as main invariants, Zermelo conditions, higher-order energies, and new conservation laws. A theorem of Noether type is proved for higher-order Lagrangians. The invariants to the infinitesimal symmetries are explicitly written. All this construction is a natural extension of classical Lagrangian mechanics.

New aspects of lagrangian relativity

The geometry of Lagrange spaces is applied to the description of classical general relativity and electrodynamics. First the Einstein equations are given in a new form, where the geometrical objects related to the internal variables are separated from those related to the external variables. After this, several special Lagrange spaces are analyzed. The almost Riemannian Lagrange spaces are rather simple for explicit calculations, and they recover all classical results of general relativity and electrodynamics. Further, the almost locally Minkowski Lagrange spaces and the almost Finsler Lagrage spaces are discussed. Although the latter are complicated from the geometrical point of view, they are interesting candidates for physical interpretations.

Nonlinear connections for nonconservative mechanical systems

The geometry of a Lagrangian mechanical system is determined by its associated evolution semispray. We uniquely determine this semispray using the symplectic structure and the energy of the Lagrange space and the external force field. We study the variation of the energy and Lagrangian functions along the evolution and the horizontal curves and give conditions by which these variations vanish. We provide examples of mechanical systems which are dissipative and for which the evolution nonlinear connection is either metric or symplectic.The geometry of a nonconservative mechanical system is determined by its associated semispray and the corresponding nonlinear connection. The semispray is uniquely determined by the symplectic structure and the energy of the corresponding Lagrange space and the external force field. We prove that the corresponding nonlinear connection is uniquely determined by its compatibility with the metric tensor and the symplectic structure of the Lagrange space. We study the variation of the energy and Lagrangian functions along the evolution curves and the horizontal curves and give conditions by which these variations vanish. We provide examples of mechanical systems which are dissipative and for which the evolution nonlinear connection is either metric or symplectic.