scholarly journals Lagrange Spaces with (γ,β)-Metric

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Suresh K. Shukla ◽  
P. N. Pandey
Keyword(s):  
Lagrange Equations ◽  
Metric Tensor ◽  
Nonlinear Connection

We study Lagrange spaces with (γ,β)-metric, where γ is a cubic metric and β is a 1-form. We obtain fundamental metric tensor, its inverse, Euler-Lagrange equations, semispray coefficients, and canonical nonlinear connection for a Lagrange space endowed with a (γ,β)-metric. Several other properties of such space are also discussed.

scholarly journals On Almost φ-Lagrange Spaces

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-16
Author(s):  
P. N. Pandey ◽  
Suresh K. Shukla
Keyword(s):  
Differential Equation ◽  
Lagrange Equations ◽  
Metric Tensor ◽  
Nonlinear Connection

We initiate a study on the geometry of an almost φ-Lagrange space (APL-space in short). We obtain the expressions for the symmetric metric tensor, its inverse, semispray coefficients, solution curves of Euler-Lagrange equations, nonlinear connection, differential equation of autoparallel curves, coefficients of canonical metrical d-connection, and h- and v-deflection tensors in an APL-space. Corresponding expressions in a φ-Lagrange space and an almost Finsler Lagrange space (AFL-space in short) have also been deduced.

Vector-tensor field theories and the Einstein-Maxwell field equations

1974 ◽  
Vol 341 (1626) ◽  
pp. 285-297 ◽  
Keyword(s):  
Dimensional Space ◽  
Field Theories ◽  
Maxwell Field ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Lagrange Equations ◽  
Third Order ◽  
Metric Tensor ◽  
First Order ◽  
First Derivative

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of the metric tensor g ij and its first two derivatives together with the first derivative of a vector field ψ i are investigated. In general, the Euler-Lagrange equations obtained by variation of g ij are of fourth order in g ij and third order in ψ i . It is shown that in a four dimensional space the only Euler-Lagrange equations which are of second order in g ij and first order in ψ i are the Einstein field equations (with an energy-momentum term). Various conditions are obtained under which the Einstein-Maxwell field equations are then an inevitable consequence.

scholarly journals On an R-Randers mth-Root Space

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
P. N. Pandey ◽  
Shivalika Saxena
Keyword(s):  
Finsler Space ◽  
Riemannian Metric ◽  
Root Space ◽  
Metric Tensor ◽  
Nonlinear Connection

We consider an n-dimensional Finsler space Fn(n>2) with the metric L(x,y)=F(x,y)+α(x,y), where F is an mth-root metric and α is a Riemannian metric. We call such space as an R-Randers mth-root space. We obtain the expressions for the fundamental metric tensor, Cartan tensor, geodesic spray coefficients, and the coefficients of nonlinear connection in an R-Randers mth-root space. Some other properties of such space have also been discussed.

scholarly journals AN AP-STRUCTURE WITH FINSLERIAN FLAVOR I: THE PRINCIPAL IDEA

2009 ◽  
Vol 24 (22) ◽  
pp. 1749-1762 ◽  
Author(s):  
M. I. WANAS
Keyword(s):  
Geometric Structure ◽  
Building Blocks ◽  
The Other ◽  
Metric Tensor ◽  
Nonlinear Connection ◽  
Absolute Parallelism ◽  
Cartan Type ◽  
Linear Connections

A geometric structure (FAP-structure), having both absolute parallelism and Finsler properties, is constructed. The building blocks of this structure are assumed to be functions of position and direction. A nonlinear connection emerges naturally and is defined in terms of the building blocks of the structure. Two linear connections, one of Berwald type and the other of the Cartan type, are defined using the nonlinear connection of the FAP. Both linear connections are nonsymmetric and consequently admit torsion. A metric tensor is defined in terms of the building blocks of the structure. The condition for this metric to be a Finslerian one is obtained. Also, the condition for an FAP-space to be an AP-one is given.

Non-Newtonian effects in axially symmetric oscillatory flows of some elastico-viscous liquids

1970 ◽  
Vol 68 (3) ◽  
pp. 731-750 ◽  
Author(s):  
J. R. Jones
Keyword(s):  
Elastic Properties ◽  
Equations Of State ◽  
Time Lag ◽  
Physical Constant ◽  
Oscillatory Motion ◽  
Axially Symmetric ◽  
Metric Tensor ◽  
Oscillatory Flows ◽  
Viscous Liquids ◽  
Deformation Stress

In (general) elastico-viscous liquids the response to stress at any instant will depend on previous rheological history, the equations of state needed to describe the rheological properties of a typical material element at any instant t being expressible in the form of a (properly invariant†) set of integro-differential equations relating its deformation, stress and temperature histories (as defined by a metric tensor (of a convected frame of reference), a stress tensor and the temperature measured in the element as functions of previous time t'( < t)) together with the time lag (t – t') and physical constant tensors associated with the element (1). Thus in any type of oscillatory motion a rheological history will necessarily be a function of the frequency of the forcing agent, the rheological history of a number of different types of elastico-viscous liquids in some simple shearing oscillatory flows being a rather simple oscillatory history (see, for example, (2–4)). It is, therefore, to be expected that a liquid with elastic properties will behave somewhat differently from any inelastic viscous liquid when subjected to any kind of oscillatory motion, and it is for this reason that oscillatory motions have been used extensively to detect and measure the elastic properties of liquids (see, for example, (2–5)).

scholarly journals Second-order integrable Lagrangians and WDVV equations

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
E. V. Ferapontov ◽  
M. V. Pavlov ◽  
Lingling Xue
Keyword(s):  
Lagrangian Density ◽  
Theta Functions ◽  
Second Order ◽  
Elementary Functions ◽  
Lagrange Equations ◽  
Wdvv Equations ◽  
Integrability Conditions ◽  
Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

scholarly journals Introduction to Macroscopic Optimal Design in the Mechanics of Composite Materials and Structures

2021 ◽  
Vol 5 (2) ◽  
pp. 36
Author(s):  
Aleksander Muc
Keyword(s):  
Composite Materials ◽  
Composite Structures ◽  
Constitutive Relations ◽  
A Priori ◽  
Chemical Properties ◽  
Composite Plates ◽  
Failure Criteria ◽  
Lagrange Equations ◽  
Homogenization Methods ◽  
Mechanics Of Composite Materials

The main goal of building composite materials and structures is to provide appropriate a priori controlled physico-chemical properties. For this purpose, a strengthening is introduced that can bear loads higher than those borne by isotropic materials, improve creep resistance, etc. Composite materials can be designed in a different fashion to meet specific properties requirements.Nevertheless, it is necessary to be careful about the orientation, placement and sizes of different types of reinforcement. These issues should be solved by optimization, which, however, requires the construction of appropriate models. In the present paper we intend to discuss formulations of kinematic and constitutive relations and the possible application of homogenization methods. Then, 2D relations for multilayered composite plates and cylindrical shells are derived with the use of the Euler–Lagrange equations, through the application of the symbolic package Mathematica. The introduced form of the First-Ply-Failure criteria demonstrates the non-uniqueness in solutions and complications in searching for the global macroscopic optimal solutions. The information presented to readers is enriched by adding selected review papers, surveys and monographs in the area of composite structures.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 170-184
Author(s):  
Richard Atkins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shortest Paths ◽  
Equivalence Problem ◽  
Second Order ◽  
System Of Differential Equations ◽  
Lagrange Equations ◽  
Invariant Functions ◽  
Underlying Geometry ◽  
The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

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