The aim of this paper is to develop on the 1-jet space J^1(R,M^4) the jet Generalized Lagrange Geometry for the rheonomic Chernov metric. The associated gravitational and electromagnetic field models based on the rheonomic Finsler Chernov metric tensor are developed and discussed.
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48 results for “rheonomic Lagrange manifolds”
Geodesic sprays and frozen metrics defined for time-dependent Lagrange manifolds.
problem Modeling wildfire spread under varying conditions.
method Define systems of pre-extremals for energy functional, induce Hamilton orthogonal nets, and derive frozen metrics.
result Energy pre-extremals become Finsler geodesics of the frozen metric.
In this paper we construct the differential equations of the stream lines that characterize plasma regarded as a non-isotropic medium geometrized by a jet rheonomic time-invariant Berwald-Moor metric. Section 1 contains historical notes regarding the Plasma Physics and its geometrical description. Section 2 analyzes th…
The paper contains a geometrization of a time dependent Lagrangian function defined on the 1-jet space J^1(R,M) which identifies with R\times TM. The reader is invited to compare this geometrization with that developped by Miron and Anastasiei.
The aim of this paper is to develop on the 1-jet space J^1(R,M^4) the Finsler-like geometry (in the sense of d-connection, d-torsions and d-curvatures) of the rheonomic Berwald-Moor metric. A natural geometrical gravitational field theory produced by the rheonomic Berwald-Moor metric is also constructed.
The aim of this paper is to develop on the 1-jet space J^1(R,M^3) the Finsler-like geometry (in the sense of distinguished (d-) connection, d-torsions and d-curvatures) of the rheonomic Berwald-Moor metric of order three. Some natural geometrical field theories (gravitational and electromagnetic) produced by the preced…
A new model uses Lorentz-Finsler geometry to predict wave propagation.
problem Modeling wave propagation in anisotropic and rheonomic media.
method Identifying wave trajectories as lightlike pregeodesics of a specific Lorentz-Finsler metric, solving ODE systems.
result Wave trajectories can be easily computed in real time.
Proves Euler-Lagrange equations for complex functionals on Fréchet manifolds.
problem Variational calculus on Fréchet manifolds.
method Proves Euler-Lagrange equations for functionals on Fréchet manifolds.
result Validates Euler-Lagrange equations for specific functionals.
Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper we study a generalization, which consists of replacing th…
We relate certain universal curvature identities for Kaehler manifolds to the Euler-Lagrange equations of the scalar invariants which are defined by pairing characteristic forms with powers of the Kaehler form.
The paper introduces discrete Dirac structures for mechanics, simplifying dynamics.
problem Formulating discrete mechanics with constraints.
method Developed -discrete Dirac structures and induced Dirac structures.
result Discrete Lagrange--Dirac systems are equivalent to -discrete Lagrange--d'Alembert equations.
Researchers study surface area functionals in CR manifolds, deducing equations for various cases.
problem Investigating surface area functionals in 3D CR manifolds.
method Deduced Euler-Lagrange equations for energy functionals in various 3D CR manifolds.
result New equations deduced for surface area functionals on disk bundles, Rossi spheres, and 3D tori.
Generalizes Einstein-Gauss-Bonnet theory to pseudo-Kaehler manifolds.
problem Extending Lovelock theory to pseudo-Kaehler manifolds.
method Extending Euler-Lagrange equations from Kaehler to pseudo-Kaehler manifolds.
result Generalized Einstein-Gauss-Bonnet theory for pseudo-Kaehler manifolds.
New statistical biharmonic maps derived from a variation problem.
problem Variation problem for mappings between statistical manifolds.
method Statistical biharmonic maps derived from the Euler-Lagrange equation.
result Improper affine hyperspheres induce examples of statistical biharmonic maps.
Paper studies critical points of curvature energies in 4D.
problem Critical points of conformally invariant extrinsic energies on 4-manifolds.
method Converted Euler-Lagrange equations to a system with favourable structures using invariances and Noether's theorem.
result Generalized Tristan Rivière's work on Willmore energy to 4D.
The paper studies -harmonic maps between almost Hermitian manifolds and derives their Euler-Lagrange equation.
problem Energy minimization of maps between almost Hermitian manifolds.
method Derives the -harmonic map equation and proves analogous results to harmonic maps.
result The -harmonic map equation coincides with the harmonic map equation up to first order terms.
It is shown that the Euler-Lagrange equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of reduction and the relation with Lagrange multiplier method are also studied.
Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associate with a geometric structure.…
Dirac structures on tangent bundles provide a unified framework for Lagrange--Dirac dynamical systems.
problem Unified geometric framework for Lagrange--Dirac dynamical systems
method Introducing a Lagrange--Dirac structure on the tangent bundle
result Unified framework for nonholonomic, degenerate Lagrangian, and symmetric systems
In this paper, we will see that the symplectic creed by Weinstein "everything is a Lagrangian submanifold" also holds for Hamilton-Poincaré and Lagrange-Poincaré reduction. In fact, we show that solutions of the Hamilton-Poincaré equations and of the Lagrange-Poincaré equations are in one-to-one correspondence with dis…
Study nonholonomic systems with collisions using variational principles.
problem Variational problems on nonholonomic systems with collisions.
method Extended variational principle, introduced connection on principal bundles, applied Lagrange–Poincaré–Pontryagin reduction.
result Implicit Lagrange–d'Alembert–Pontryagin equations for nonholonomic systems with collisions.
The paper studies Einstein-Hilbert action on complex manifolds.
problem Deriving equations for the Einstein-Hilbert action on almost -product manifolds.
method Adapted variations of metric, deriving Euler-Lagrange equations.
result Presented a nice form of Einstein equation.
In this paper we introduce a natural definition for the affine maps between two Finsler manifolds and and we give some geometrical properties of these affine maps. Starting from the equations of the affine maps, we construct a natural Berwald-Riemann-Lagrange geometry on the 1-jet space $J^1(TM;…
In this paper, it is elaborated the theory the Ricci flows for manifolds enabled with nonintegrable (nonholonomic) distributions defining nonlinear connection structures. Such manifolds provide a unified geometric arena for nonholonomic Riemannian spaces, Lagrange mechanics, Finsler geometry, and various models of grav…
Defines a new conformally invariant Yang-Mills type energy for 6-manifolds.
problem No specific problem stated; focuses on defining a new energy.
method Defines a conformally invariant action S on gauge connections on a 6-manifold M, leading to higher-order conformally invariant Yang-Mills equations.
result The Euler-Lagrange equations of S provide a conformally invariant analogue of Yang-Mills equations, with special cases recovering known invariants.
The paper studies variations of metrics on foliated manifolds and finds solutions to specific actions.
problem Variations of metrics on foliated pseudo-Riemannian manifolds.
method Developed variation formulas and applied to Einstein-Hilbert type actions.
result Found solutions like twisted products, conformal submersions, and isoparametric foliations.
Optimization with inequality constraints using embedded gradient vector field method
problem Optimization with inequality constraints
method Geometric framework using quadratic slack variables
result Derives Lagrange multiplier functions and second-order optimality conditions
Study variational problems for integral invariants of maps between pseudo-Riemannian manifolds.
problem Understanding variational properties of integral invariants defined from the second fundamental form.
method Derive first variational formulae for integral invariants of degree two, show Euler-Lagrange equation for Chern-Federer energy, and provide examples of submanifolds.
result The Euler-Lagrange equation of the Chern-Federer energy functional reduces to a second order PDE.
Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.
problem Understanding the relationship between the Schwarzian derivative and variational equations.
method Analyzing the Schwarzian derivative as a first integral and Euler-Lagrange operator for specific variations.
result The Schwarzian derivative is both a first integral and the Euler-Lagrange operator for a certain class of variations.
We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and desc…
The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilt…
Study on generalized ξ-parallel maps in Riemannian geometry.
problem Characterizing and understanding generalized ξ-parallel maps.
method Defined energy functional, derived first variation formula, and Euler-Lagrange equation.
result Established fundamental properties and relationships with harmonic and biharmonic maps.
The Lagrangian formalism on a arbitrary non-fibrating manifold is considered. The kinematical description of this generic situation is based on the concept of (higher-order) Grassmann manifolds which is the factorization of the regular velocity manifold to the action of the differential group. Here we introduce in this…
Extends Gauduchon's result to higher dimensions, showing balanced metrics.
problem Understanding critical metrics in higher-dimensional Hermitian manifolds.
method Analyzes the functional of -norm of torsion -form and full Chern torsion.
result Critical metrics are balanced in all dimensions.
Global minimizers exist for Tonelli Lagrangians on half-Lie groups.
problem Existence and properties of minimizers for Lagrangians on infinite-dimensional spaces.
method Introduced Tonelli Lagrangians on half-Lie groups, proved existence of minimizers and flow lines.
result Global minimizers exist above certain energy thresholds.
Metric nonlinear connectionsmath.DG
For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange struc…
The paper studies almost complex structures on ACH Einstein manifolds and finds deformation results.
problem The study of canonical almost complex structures on ACH Einstein manifolds.
method Variational problem with the Dolbeault Laplacian acting on -forms.
result Deformation results of Einstein ACH metrics associated with critical almost complex structures.
Paper extends variational problem to spheres using control methods.
problem Extending classical variational problem to spheres.
method Variational and optimal control approaches.
result Established relation between Hamiltonian and Euler-Lagrange equations.
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…
Researchers extend Godbillon-Vey functional to almost contact manifolds, finding critical structures.
problem Finding optimal almost contact manifolds using the Godbillon-Vey functional.
method Introduced a Godbillon-Vey type functional for 3D almost contact manifolds and found its Euler-Lagrange equations.
result Constructed critical 3D almost contact manifolds with double-twisted product structure.
Discrete Lagrange problems solved with Lie group constraints.
problem Solving discrete Lagrange problems with Lie group constraints.
method Proving critical sections are solutions of unconstrained variational problems, applying Noether theory and multisymplectic forms.
result Critical sections of discrete Lagrange problems are solutions of unconstrained variational problems.
In this paper are studied the harmonic maps between two generalized Lagrange spaces. At the same time, it is proved that the solutions of class of certain ODEs or PDEs are harmonic maps between certain convenient generalized Lagrange spaces.
In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the…
Develops Lagrange-Hamilton geometry for COVID-19 disease dynamics.
problem Modeling the spread of COVID-19 disease.
method Least squares variational method, nonlinear connections, d-torsions, Lagrangian Yang-Mills.
result Jacobi stability of the dynamical system.
The paper explores Lagrangians with simplified Euler-Lagrange equations.
problem Variational problems with Euler-Lagrange equations of reduced order.
method Geometrical construction to derive a family of Lagrangians.
result Lagrangians with reduced-order Euler-Lagrange equations are polynomials in the highest-order derivatives.
Lagrange's map construction ideas influenced later mathematicians.
problem Improving geographical map construction methods.
method Analyzing the impact of Lagrange's memoir on subsequent mathematical works.
result Lagrange's ideas were influential in later mathematical developments, particularly in geography.
The sum of Lagrange numbersmath.GT
Sum of Lagrange numbers equals a specific formula.
problem Proving the Markov Uniqueness Conjecture (MUC).
method Combining McShane's identity and Schmutz's work.
result MUC is equivalent to the given sum formula.
We extend the correspondence between Hessian and Kähler metrics and curvatures to Lagrange spaces.