arXiv research

The paper simplifies symmetries in complex geometric structures.

Abstract reviews symmetry and reduction in dynamical systems.

This work extends reduction processes for nonholonomic discrete mechanical systems.

A reduction method of ODEs not possessing Lie point symmetries makes use of the so called $λ$-symmetries (C. Muriel and J. L. Romero, \emph{IMA J. Appl. Math.} \textbf{66}, 111-125, 2001). The notion of covering for an ODE $\mathcal{Y}$ is used here to recover $λ$-symmetries of $\mathcal{Y}$ as nonlocal symmetries. In …

Abstract: Generalized reduction methods for symmetries in graded geometry.

We survey the role of symmetry in diffeomorphic registration of landmarks, curves, surfaces, images and higher-order data. The infinite dimensional problem of finding correspondences between objects can for a range of concrete data types be reduced resulting in compact representations of shape and spatial structure. Th…

Study of symmetry distributions in Lorentzian naturally reductive nilmanifolds.

Study of symplectic trivialization and reduction of bundles with symmetry and connection.

We show that the distribution of symmetry of a naturally reductive nilpotent Lie group coincides with the invariant distribution induced by the set of fixed vectors of the isotropy. This extends a known result on compact naturally reductive spaces. We also address the study of the quotient by the foliation of symmetry.

We discuss the use of Dirac structures to obtain a better understanding of the geometry of a class of optimal control problems and their reduction by symmetries. In particular we will show how to extend the reduction of Dirac structures recently proposed by Yoshimura and Marsden [Yo09] to describe the reduction of a cl…

In this work we introduce a category of discrete Lagrange--Poincare systems LP_d and study some of its properties. In particular, we show that the discrete mechanical systems and the discrete mechanical systems obtained by the Lagrangian reduction of symmetric discrete mechanical systems are objects in LP_d. We introdu…

The paper discusses reducing Hamiltonian systems by scaling and standard symmetries, leading to Kirillov Hamiltonian systems.

Develops a reduction theory for covariant field theories with gauge symmetries.

Paper reduces nonholonomic systems with symmetries.

New method extends invariant reduction to rescaled geometric structures.

Transformers reduce redundancy by focusing on invariant relational quantities.

Two reduction schemes for symplectic manifolds are shown equivalent.

We study control systems invariant under a Lie group with application to the problem of nonlinear trajectory planning. A theory of symmetry reduction of exterior differential systems is employed to demonstrate how symmetry reduction and reconstruction is effective in the explicit, exact construction of planned system t…

We approach the construction of Backlund transformations for Darboux integrable hyperbolic partial differential equations in the plane through the reduction of exterior differential systems. For example it is shown that all the Backlund transformations in arXiv:0707.4408v2 can be constructed using symmetry reduction.

Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.

We provide some insights in the study of branching problems of reductive groups, and a method of investigations into symmetry breaking operators. First, we give geometric criteria for finiteness property of linearly independent continuous (respectively, differential) operators that intertwine two induced representation…

Study on geodesics in a specific sub-Riemannian structure with two types of behavior.

We discuss smooth nonlinear control systems with symmetry. For a free and proper action of the symmetry group, the reduction of symmetry gives rise to a reduced smooth nonlinear control system. If the action of the symmetry group is only proper, the reduced nonlinear control system need not be smooth. Using the smooth …

We give a new mechanism for constructing Backlund transformations by using symmetry reduction of differential systems. We then characterize a family of Backlund transformations between Darboux integrable systems where the Backlund transformation can be constructed by the proposed symmetry reduction method.

Study 2D viscoelastic equations using Lie group theory.

Paper develops reduction theory for controlled Lagrangian systems with symmetry and momentum map.

In this paper, from the viewpoint of completeness of Marsden-Weinstein reduction, we illustrate how to give the definitions of a controlled Hamiltonian (CH) system and a reducible controlled Hamiltonian system with symmetry; and how to describe the dynamics of a CH system and the controlled Hamiltonian equivalence; as …

We propose a reduction procedure for symplectic connections with symmetry. This is applied to coadjoint orbits whose isotropy is reductive.

The paper studies symmetry reduction of control systems and its implications for feedback linearization.

A controlled magnetic Hamiltonian (CMH) system is a regular controlled Hamiltonian (RCH) system with magnetic symplectic form, it is an important special case of RCH system. Note that there is a magnetic term on the cotangent bundle of the Heisenberg group, such that we can define a CMH system with symmetry of the Heis…

Survey of recent developments in symmetric reductions and controls for Hamiltonian systems.

Variational reduction simplifies Lagrangian systems with scaling symmetries.

The paper explores polysymplectic structures and their reductions in field theories.

We describe a new algebraic multisymplectic formulation of the classical BRST symmetry. The analogue of Marsden-Weinstein reduction for multisymplectic manifolds is described. We then give a homological description of Multisymplectic Marsden-Weinstein reduction.

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type…

We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.

Symmetry reduction of Painlevé IV to Flaschka-Newell Painlevé II

The paper simplifies complex mechanical systems with external forces.

Reduces multisymplectic Lie systems through symmetry analysis.

We realise the first and second Grushin distributions as symmetry reductions of the 3-dimensional Heisenberg distribution and 4-dimensional Engel distribution respectively. Similarly, we realise the Martinet distribution as an alternative symmetry reduction of the Engel distribution. These reductions allow us to derive…

Study new symmetries in non-symmetric spaces and discontinuous groups.

Improves Marsden-Weinstein reduction theory for k-polysymplectic manifolds.

Invariant reduction preserves Poisson structures in PDEs.

The paper studies symmetry reduction and optimal control on Riemannian manifolds.

A general study of symmetries in optimal control theory is given, starting from the presymplectic description of this kind of system. Then, Noether's theorem, as well as the corresponding reduction procedure (based on the application of the Marsden-Weinstein theorem adapted to the presymplectic case) are stated both in…

An analogue of geometric quantization of Poisson algebras obtained by algebraic reduction of symmetries is developed. Interpretation of the obtained results and their application to the problem of commutativity of quantization and reduction are given

Develops neural networks for reductive Lie groups, enhancing symmetry respect.

Reduces symplectic Hamiltonian systems to contact systems, realizing Poincaré's dream.