One variation of R. Miron's spray theory in OsckM

2003 ◽  
Vol 40 (4) ◽  
pp. 443-462
Author(s):  
Irena Čomić
Keyword(s):  
Vector Fields ◽  
Transformation Group ◽  
Higher Order

Lately a big attention has been payed on the higher order geometry. Some relevant papers are mentioned in the references. R. Miron and Gh. Atanasiu in [16], [17] studied the geometry of OsckM. R. Miron in [19] gave the comprehend theory of higher order geometry and its application. The whole theory of sprays in OsckM M is established. Here, using R. Miron's method, a variation of this theory is given. The transformation group is slightly different from that used in [19] and it will change the geometry. The adapted basis, the Liouville vector fields, the equation of sprays, will have different form. We give the relations between coefficients of S and the Liouville vector fields.

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Vector Fields ◽  
Higher Order ◽  
Beam Equation ◽  
Series Solutions ◽  
Local Conservation ◽  
Power Series Method ◽  
Potential System ◽  
Power Series Solutions ◽  
The Difference

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields

2017 ◽  
Vol 27 (14) ◽  
pp. 1750224
Author(s):  
Jing Li ◽  
Liying Kou ◽  
Duo Wang ◽  
Wei Zhang
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
Three Dimensional ◽  
Recursive Formula ◽  
Higher Order ◽  
Dimensional Vector ◽  
Symbolic Calculations

In this paper, we mainly focus on the unique normal form for a class of three-dimensional vector fields via the method of transformation with parameters. A general explicit recursive formula is derived to compute the higher order normal form and the associated coefficients, which can be achieved easily by symbolic calculations. To illustrate the efficiency of the approach, a comparison of our result with others is also presented.

scholarly journals Modeling n -Symmetry Vector Fields using Higher-Order Energies

2018 ◽  
Vol 37 (2) ◽  
pp. 1-18 ◽  
Author(s):  
Christopher Brandt ◽  
Leonardo Scandolo ◽  
Elmar Eisemann ◽  
Klaus Hildebrandt
Keyword(s):  
Vector Fields ◽  
Higher Order

scholarly journals Representing Higher-Order Singularities in Vector Fields on Piecewise Linear Surfaces

2006 ◽  
Vol 12 (5) ◽  
pp. 1315-1322 ◽  
Author(s):  
Wan-chiu Li ◽  
Bruno Vallet ◽  
Nicolas Ray ◽  
Bruno Levy
Keyword(s):  
Vector Fields ◽  
Piecewise Linear ◽  
Higher Order

scholarly journals Liftings of Some Types of Tensor Fields and Connections to Tangent Bundles of pr-Velocities

1970 ◽  
Vol 40 ◽  
pp. 13-31 ◽  
Author(s):  
Akihiko Morimoto
Keyword(s):  
Vector Fields ◽  
Higher Order ◽  
Tensor Fields ◽  
Multi Index ◽  
Differentiable Functions ◽  
Tangent Bundles

In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to generalize the results of [6] to the tangent bundles of pr-velocities in a manifold M— notions due to C. Ehresmann [1] (see also [2]). In §1, we explain the pr-velocities in a manifold and define the (Λ)-lifting of differentiable functions for any multi-index λ -(λ1, λ2,…,λp) of non-negative integers λi satisfying ΣΛi≤r. In § 2, we construct ‹Λ›-lifts of any vector fields and ‹Λ›-lifts of 1-forms. The ‹Λ›-lift is a little bit different from the ‹Λ›-lift of vector fields in [6].

scholarly journals HIGHER-ORDER NOETHER SYMMETRIES IN k-SYMPLECTIC HAMILTONIAN FIELD THEORY

2013 ◽  
Vol 10 (08) ◽  
pp. 1360013
Author(s):  
NARCISO ROMÁN-ROY ◽  
MODESTO SALGADO ◽  
SILVIA VILARIÑO
Keyword(s):  
Field Theory ◽  
Conservation Laws ◽  
Vector Fields ◽  
Higher Order ◽  
Field Theories ◽  
Noether Symmetries ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Infinitesimal Transformations

For k-symplectic Hamiltonian field theories, we study infinitesimal transformations generated by some kinds of vector fields which are not Noether symmetries, but which allow us to obtain conservation laws by means of suitable generalizations of Noether's theorem.

Higher order weak linearizations of stochastically driven nonlinear oscillators

2007 ◽  
Vol 463 (2083) ◽  
pp. 1827-1856 ◽  
Author(s):  
Nilanjan Saha ◽  
D Roy
Keyword(s):  
Vector Fields ◽  
Probability Distributions ◽  
Strong Solutions ◽  
Nonlinear Oscillators ◽  
Higher Order ◽  
Computational Effort ◽  
Local Error ◽  
True Solution ◽  
Multiplicative Noises ◽  
Backward Euler

We present derivative-free weak and strong solutions of stochastically driven nonlinear oscillators of engineering interest using higher order forms of the locally transversal linearization (LTL) method. Unlike strong solutions, weak stochastic solutions attempt to predict only mathematical expectations of functions of the true solution and are obtainable with much less computational effort. The linearized equations corresponding to the higher order implicit LTL schemes are arrived at using backward Euler–Maruyama and Newmark expansions in order to conditionally replace nonlinear drift and multiplicative diffusion vector fields. We also briefly describe alterations through which explicit forms of such higher order linearizations are obtained. In the weak forms, the Gaussian stochastic integrals, appearing in the linearized solutions, are replaced by random variables with simpler and discrete probability distributions. The resulting local approximations to the true solution are of significantly higher formal order of accuracy, as reflected through local error estimates. Numerical illustrations are presently provided for the Duffing and Van der Pol oscillators driven by additive and multiplicative noises, which are indicative of the numerical accuracy, computational speed and algorithmic simplicity.

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