Diffeomorphism

Diffeomorphism

Diffeomorphism refers to a smooth and invertible function that can map one differential manifold to another. It is considered a diffeomorphism if it is a homeomorphism from one smooth manifold to another, and both the function and its inverse are smooth. In essence, diffeomorphism is a way to describe the equivalence of two manifolds in terms of their smoothness and continuity.From: Autonomous Underwater Vehicles [2017], Manifold Learning Theory and Applications [2019], Introduction to Topological Quantum Matter & Quantum Computation [2019]

Introduction

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Published in Sabiha Wadoo, Pushkin Kachroo, Autonomous Underwater Vehicles, 2017

Sabiha Wadoo, Pushkin Kachroo

Let G be a Lie group with identity I, and let XI be a tangent vector to G at I. For any g ∈ G, define the left translation by g to be a map Lg: G → G such that Lg(x) = gx, where x ∈ G. Since G is a Lie group, Lg is a diffeomorphism of G for each g. Diffeomorphism is a smooth inveritable function that maps one differential manifold to another. Taking the differential of Lg at e results in a map from the tangent space of G at e to the tangent space of G at g: dLg:TgG→TgG such that Xg=dLg(Xg)

Noether-type theorem for fractional variational problems depending on fractional derivatives of functions

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Journal Information

Published in Applicable Analysis, 2021

M. J. Lazo, G. S. F. Frederico, P. M. Carvalho-Neto

In a seminal work, Emmy Noether obtained a correspondence linking symmetry groups of a Lagrangian system and its corresponding constants of motion (also known as its first integrals). Actually, when a Lagrangian is invariant under the action of a symmetry group, then the system exhibits a corresponding conservation law. A symmetry is defined by a one-parameter group of diffeomorphisms under which the action functional is invariant:

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Introduction

Noether-type theorem for fractional variational problems depending on fractional derivatives of functions