scholarly journals MOVING FRAMES AND NOETHER’S CONSERVATION LAWS—THE GENERAL CASE

2016 ◽  
Vol 4 ◽  
Author(s):  
TÂNIA M. N. GONÇALVES ◽  
ELIZABETH L. MANSFIELD
Keyword(s):  
Conservation Laws ◽  
Group Action ◽  
Lie Group ◽  
Moving Frame ◽  
Moving Frames ◽  
Lagrange Equations ◽  
Linear Action ◽  
Independent Variables ◽  
Lie Group Action ◽  
Standard Linear

In recent works [Gonçalves and Mansfield, Stud. Appl. Math., 128 (2012), 1–29; Mansfield, A Practical Guide to the Invariant Calculus (Cambridge University Press, Cambridge, 2010)], the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler–Lagrange equations and the space of conservation laws in terms of vectors of invariants and the Adjoint representation of a moving frame. In this paper, we show how these calculations extend to the general case where the independent variables may participate in the action. We take for our main expository example the standard linear action of SL(2) on the two independent variables. This choice is motivated by applications to variational fluid problems which conserve potential vorticity. We also give the results for Lagrangians invariant under the standard linear action of SL(3) on the three independent variables.

scholarly journals Moving frames and Noether’s finite difference conservation laws I

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
E L Mansfield ◽  
A Rojo-Echeburúa ◽  
P E Hydon ◽  
L Peng
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Group Action ◽  
Lie Group ◽  
Difference Equations ◽  
Continuum Limit ◽  
Moving Frame ◽  
Moving Frames ◽  
Lagrange Equations ◽  
The Difference

Abstract We consider the calculation of Euler–Lagrange systems of ordinary difference equations, including the difference Noether’s theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We introduce the difference moving frame, a natural discrete moving frame that is adapted to difference equations by prolongation conditions. For any Lagrangian that is invariant under a Lie group action on the space of dependent variables, we show that the Euler–Lagrange equations can be calculated directly in terms of the invariants of the group action. Furthermore, Noether’s conservation laws can be written in terms of a difference moving frame and the invariants. We show that this form of the laws can significantly ease the problem of solving the Euler–Lagrange equations, and we also show how to use a difference frame to integrate Lie group invariant difference equations. In this Part I, we illustrate the theory by applications to Lagrangians invariant under various solvable Lie groups. The theory is also generalized to deal with variational symmetries that do not leave the Lagrangian invariant. Apart from the study of systems that are inherently discrete, one significant application is to obtain geometric (variational) integrators that have finite difference approximations of the continuous conservation laws embedded a priori. This is achieved by taking an invariant finite difference Lagrangian in which the discrete invariants have the correct continuum limit to their smooth counterparts. We show the calculations for a discretization of the Lagrangian for Euler’s elastica, and compare our discrete solution to that of its smooth continuum limit.

scholarly journals Moving frames and Noether’s finite difference conservation laws II

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
E L Mansfield ◽  
A Rojo-Echeburúa
Keyword(s):  
Finite Difference ◽  
Conservation Laws ◽  
Lie Group ◽  
Difference Equations ◽  
Adjoint Action ◽  
Moving Frame ◽  
Integration Step ◽  
Moving Frames ◽  
Lagrange Equations ◽  
Common Structure

Abstract In this second part of the paper, we consider finite difference Lagrangians that are invariant under linear and projective actions of $SL(2)$, and the linear equi-affine action that preserves area in the plane. We first find the generating invariants, and then use the results of the first part of the paper to write the Euler–Lagrange difference equations and Noether’s difference conservation laws for any invariant Lagrangian, in terms of the invariants and a difference moving frame. We then give the details of the final integration step, assuming the Euler Lagrange equations have been solved for the invariants. This last step relies on understanding the adjoint action of the Lie group on its Lie algebra. We also use methods to integrate Lie group invariant difference equations developed in Part I. Effectively, for all three actions, we show that solutions to the Euler–Lagrange equations, in terms of the original dependent variables, share a common structure for the whole set of Lagrangians invariant under each given group action, once the invariants are known as functions on the lattice.

scholarly journals The stratified spaces of a symplectic lie group action

2006 ◽  
Vol 58 (1) ◽  
pp. 51-75 ◽  
Author(s):  
Juan-Pablo Ortega ◽  
Tudor S. Ratiu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Stratified Spaces ◽  
Lie Group Action

SINGULAR SWITCHED LINEAR SYSTEMS: SOME GEOMETRIC ASPECTS

2012 ◽  
Vol 26 (25) ◽  
pp. 1246006
Author(s):  
H. DIEZ-MACHÍO ◽  
J. CLOTET ◽  
M. I. GARCÍA-PLANAS ◽  
M. D. MAGRET ◽  
M. E. MONTORO
Keyword(s):  
Linear Systems ◽  
Group Action ◽  
Lie Group ◽  
Equivalence Relation ◽  
Geometric Approach ◽  
Differentiable Manifold ◽  
Equivalence Classes ◽  
Switched Linear Systems ◽  
Lie Group Action

We present a geometric approach to the study of singular switched linear systems, defining a Lie group action on the differentiable manifold consisting of the matrices defining their subsystems with orbits coinciding with equivalence classes under an equivalence relation which preserves reachability and derive miniversal (orthogonal) deformations of the system. We relate this with some new results on reachability of such systems.

scholarly journals Certain Types of Complex Lie Group Action

2018 ◽  
Vol 10 (3) ◽  
Author(s):  
Ahmed Khalaf Radhi ◽  
Taghreed Hur Majeed
Keyword(s):  
Tensor Product ◽  
Group Action ◽  
Lie Group ◽  
The Other ◽  
Smooth Representation ◽  
Equivalent Relation ◽  
Complex Group ◽  
Lie Group Action ◽  
Group 10 ◽  
Definition Of

     The main aim in this paper is to look for a novel action with new properties on       from the  , the Literature are concerned with studying the action of  of two representations , one is usual and the other is the dual, while our  interest in this work  is focused on some actions on complex Lie group[10] . Let G be a matrix complex  group , and  is representation of   In this study we will present and analytic  the  concepts of action of complex  group on    We recall the definition of  tensor  product of two representations of  group and construct  the definition of action of   group on , then by using the equivalent  relation   between  and  , we get a new action : The two actions are forming smooth  representation of    This  we have new action which called     denoted by    which acting on      This  is smooth representation of   The theoretical Justifications are developed and prove supported by some concluding  remarks and illustrations.

On Certain Polynomials of Gaussian Type

1979 ◽  
Vol 31 (2) ◽  
pp. 274-281 ◽  
Author(s):  
Daniel Reich
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Orbit Space ◽  
Partition Functions ◽  
Poincaré Polynomial ◽  
Complete Answer ◽  
Lie Group Action ◽  
Image Position ◽  
Positive Coefficients ◽  
Positive Integers

Introduction. We shall consider functions of the formwhere {ri} and {si} are sets of positive integers. Such functions were studied by E. Grosswald in [2], who took {si} to be pairwise relatively prime, and asked the following two questions:(a) When is ƒ(t) a polynomial?(b) When does ƒ(t) have positive coefficients?These questions arise naturally from the work of Allday and Halperin, who show in [1] that under suitable circumstance ƒ(t) will be the Poincare polynomial of the orbit space of a certain Lie group action. Grosswald gives a complete answer to (a), but (b) is a much harder question, and a complete answer is provided only for the case m = 2. His treatment involves the representation of the coefficients of ƒ(t) by partition functions, and uses a classical description by Sylvester of the semigroup generated by {si}.

Integration of Equivariant Forms

2020 ◽  
pp. 228-232
Author(s):  
Loring W. Tu
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Differential Form ◽  
Circle Action ◽  
Manifolds With Boundary ◽  
Boundary Points ◽  
Manifold With Boundary ◽  
Lie Group Action ◽  
Cartan Model ◽  
Interior Points

This chapter illustrates integration of equivariant forms. An equivariant differential form is an element of the Cartan model. For a circle action on a manifold M, it is a polynomial in u with coefficients that are invariant forms on M. Such a form can be integrated by integrating the coefficients. This can be called equivariant integration. The chapter shows that under equivariant integration, Stokes's theorem still holds. Everything done so far in this book concerning a Lie group action on a manifold can be generalized to a manifold with boundary. An important fact concerning manifolds with boundary is that a diffeomorphism of a manifold with boundary takes interior points to interior points and boundary points to boundary points.

scholarly journals Inexact trajectory planning and inverse problems in the Hamilton–Pontryagin framework

2013 ◽  
Vol 469 (2160) ◽  
pp. 20130249 ◽  
Author(s):  
Christopher L. Burnett ◽  
Darryl D. Holm ◽  
David M. Meier
Keyword(s):  
Lie Group ◽  
Trajectory Planning ◽  
Quantum Control ◽  
Higher Order ◽  
Integration Scheme ◽  
Planning Problem ◽  
Lagrange Equations ◽  
Lie Group Action ◽  
Potential Applications ◽  
New Interpretation

We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler–Lagrange equations is obtained from a higher-order Hamilton–Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre–Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton–Pontryagin principle and preserves the geometric properties of the continuous-time solution.

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