Foundations of higher-order variational theory on Grassmann fibrations

2014 ◽  
Vol 11 (07) ◽  
pp. 1460023 ◽  
Author(s):  
Zbyněk Urban ◽  
Demeter Krupka
Keyword(s):  
Vector Fields ◽  
Variational Principles ◽  
Variational Calculus ◽  
Higher Order ◽  
Noether Theorem ◽  
Variational Theory ◽  
Lagrange Equations ◽  
Variation Formula ◽  
One Dimensional ◽  
The First Variation

A setting for higher-order global variational analysis on Grassmann fibrations is presented. The integral variational principles for one-dimensional immersed submanifolds are introduced by means of differential 1-forms with specific properties, similar to the Lepage forms from the variational calculus on fibred manifolds. Prolongations of immersions and vector fields to the Grassmann fibrations are defined as a geometric tool for the variations of immersions, and the first variation formula in the infinitesimal form is derived. Its consequences, the Euler–Lagrange equations for submanifolds and the Noether theorem on invariant variational functionals are proved. Examples clarifying the meaning of the Noether theorem in the context of variational principles for submanifolds are given.

A FRACTAL VARIATIONAL THEORY FOR ONE-DIMENSIONAL COMPRESSIBLE FLOW IN A MICROGRAVITY SPACE

Fractals ◽  
2020 ◽  
Vol 28 (02) ◽  
pp. 2050024 ◽  
Author(s):  
JI-HUAN HE
Keyword(s):  
Compressible Flow ◽  
Lagrange Multiplier ◽  
Variational Formulation ◽  
Inverse Method ◽  
Variational Principles ◽  
Microgravity Condition ◽  
Multiplier Method ◽  
Variational Theory ◽  
One Dimensional ◽  
The One

The semi-inverse method is adopted to establish a family of fractal variational principles of the one-dimensional compressible flow under the microgravity condition, and Cauchy–Lagrange integral is successfully derived from the obtained variational formulation. A suitable application of the Lagrange multiplier method is also elucidated.

scholarly journals VARIATIONAL PRINCIPLES FOR INVOLUTIVE SYSTEMS OF VECTOR FIELDS

2004 ◽  
Vol 01 (03) ◽  
pp. 201-232
Author(s):  
GIUSEPPE GAETA ◽  
PAOLA MORANDO
Keyword(s):  
Vector Field ◽  
Variational Principle ◽  
Vector Fields ◽  
Variational Principles ◽  
Hamiltonian Dynamics ◽  
Higher Order

In many relevant cases — e.g., in Hamiltonian dynamics — a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar way by means of a higher-order variational principle, and how this extends to involutive systems of vector fields.

GEOMETRIC STUDY OF HAMILTON'S VARIATIONAL PRINCIPLE

1991 ◽  
Vol 03 (04) ◽  
pp. 379-401 ◽  
Author(s):  
JOSÉ F. CARIÑENA ◽  
CARLOS LÓPEZ
Keyword(s):  
Variational Calculus ◽  
Higher Order ◽  
Lagrangian Mechanics ◽  
Lagrange Equations ◽  
Order Case ◽  
Helmholtz Conditions ◽  
Higher Order Tangent Bundle ◽  
Order Tangent ◽  
Geometric Study

Higher order tangent bundle geometry and sections along maps are used in Geometrical Mechanics in order to develop an intrinsic variational calculus. The role of variational derivative as the bundle operator associated to exterior differential on the set of trajectories is remarked. Euler-Lagrange equations and Poincaré-Cartan form are rederived in this way. Helmholtz conditions for the inverse problem of Lagrangian Mechanics are geometrically obtained for the general higher order case.

scholarly journals Willmore-Like Tori in Killing Submersions

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Manuel Barros ◽  
Óscar J. Garay ◽  
Álvaro Pámpano
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Total Space ◽  
Lagrange Equations ◽  
Base Surface ◽  
Variation Formula ◽  
First Variation ◽  
Invariant Potential ◽  
The First Variation

The first variation formula and Euler-Lagrange equations for Willmore-like surfaces in Riemannian 3-spaces with potential are computed and, then, applied to the study of invariant Willmore-like tori with invariant potential in the total space of a Killing submersion. A connection with generalized elastica in the base surface of the Killing submersion is found, which is exploited to analyze Willmore tori in Killing submersions and to construct foliations of Killing submersions made up of Willmore tori with constant mean curvature.

scholarly journals Cosmic Analogues of Classic Variational Problems

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 71 ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Relativistic Particle ◽  
Friedmann Equation ◽  
Variational Calculus ◽  
Variational Problems ◽  
Particle Mechanics ◽  
One Dimensional ◽  
Spatially Homogeneous

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

scholarly journals Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1016
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Applied Sciences ◽  
Degree Of Approximation ◽  
Higher Order ◽  
Second Order ◽  
Dimensional Case ◽  
Moduli Of Continuity ◽  
One Dimensional ◽  
Multidimensional Generalization ◽  
The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

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.

Symmetries from the solution manifold

2015 ◽  
Vol 12 (08) ◽  
pp. 1560016 ◽  
Author(s):  
Víctor Aldaya ◽  
Julio Guerrero ◽  
Francisco F. Lopez-Ruiz ◽  
Francisco Cossío
Keyword(s):  
Symplectic Structure ◽  
Vector Fields ◽  
Sigma Model ◽  
Canonical Quantization ◽  
General Concept ◽  
Noether Theorem ◽  
Preliminary Step ◽  
Hamiltonian Vector Fields ◽  
Solution Manifold

We face a revision of the role of symmetries of a physical system aiming at characterizing the corresponding Solution Manifold (SM) by means of Noether invariants as a preliminary step towards a proper, non-canonical, quantization. To this end, "point symmetries" of the Lagrangian are generally not enough, and we must resort to the more general concept of contact symmetries. They are defined in terms of the Poincaré–Cartan form, which allows us, in turn, to find the symplectic structure on the SM, through some sort of Hamilton–Jacobi (HJ) transformation. These basic symmetries are realized as Hamiltonian vector fields, associated with (coordinate) functions on the SM, lifted back to the Evolution Manifold through the inverse of this HJ mapping, that constitutes an inverse of the Noether Theorem. The specific examples of a particle moving on S3, at the mechanical level, and nonlinear SU(2)-sigma model in field theory are sketched.

Stress–energy–momentum tensors in higher order variational calculus

2000 ◽  
Vol 34 (1) ◽  
pp. 41-72 ◽  
Author(s):  
A. Fernández ◽  
P.L. Garcı́a ◽  
C. Rodrigo
Keyword(s):  
Variational Calculus ◽  
Higher Order ◽  
Stress Energy

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.

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