Lie groups | Symmetry

Lie point symmetry

Lie point symmetry is a concept in advanced mathematics. Towards the end of the nineteenth century, Sophus Lie introduced the notion of Lie group in order to study the solutions of ordinary differential equations (ODEs). He showed the following main property: the order of an ordinary differential equation can be reduced by one if it is invariant under one-parameter Lie group of point transformations. This observation unified and extended the available integration techniques. Lie devoted the remainder of his mathematical career to developing these continuous groups that have now an impact on many areas of mathematically based sciences. The applications of Lie groups to differential systems were mainly established by Lie and Emmy Noether, and then advocated by Élie Cartan. Roughly speaking, a Lie point symmetry of a system is a local group of transformations that maps every solution of the system to another solution of the same system. In other words, it maps the solution set of the system to itself. Elementary examples of Lie groups are translations, rotations and scalings. The Lie symmetry theory is a well-known subject. In it are discussed continuous symmetries opposed to, for example, discrete symmetries. The literature for this theory can be found, among other places, in these notes. (Wikipedia).

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Teach Astronomy - Symmetry

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Symmetry in Physics | Noether's theorem

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From playlist Symmetry

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Introduction to Symmetry about the x-axis, y-axis, and the origin Using Points

This video introduces symmetry about the x-axis, y-axis, and the origin using points on the coordinate plane. Site: http://mathispower4u.com

From playlist Determining Odd and Even Functions

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Symmetries show up everywhere in physics. But what is a symmetry? While the symmetries of shapes can be interesting, a lot of times, we are more interested in symmetries of space or symmetries of spacetime. To describe these, we need to build "invariants" which give a mathematical represen

From playlist Relativity

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Axioms of Lie algebra theory

In this video I write down the axioms of Lie algebras and then discuss the defining anti-symmetric bilinear map (the Lie bracket) which is zero on the diagonal and fulfills the Jacobi identity. I'm following the compact book "Introduction to Lie Algebras" by Erdmann and Wildon. https://gi

From playlist Algebra

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Lie groups: Lie algebras

This lecture is part of an online graduate course on Lie groups. We define the Lie algebra of a Lie group in two ways, and show that it satisfied the Jacobi identity. The we calculate the Lie algebras of a few Lie groups. For the other lectures in the course see https://www.youtube.co

From playlist Lie groups

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Lie derivatives of differential forms

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From playlist Symplectic geometry and mechanics

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Lie derivative pt. 2: Properties and general tensors

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From playlist Lie derivative

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Simple groups, Lie groups, and the search for symmetry II | Math History | NJ Wildberger

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From playlist MathHistory: A course in the History of Mathematics

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Lie Groups for Deep Learning w/ Graph Neural Networks

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From playlist PH1121

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Lie Groups and Lie Algebras: Lesson 42 Group Theory Review #1

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From playlist Lie Groups and Lie Algebras

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Is the variety of singular tuples of matrices a null cone? - Viswambhara Makam

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From playlist Mathematics

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Nick Sheridan: Counting curves using the Fukaya category

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From playlist Fall 2017

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Mod-01 Lec-2 Symmetry in Perfect Solids

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On the symmetries of and equivalence test for design polynomials by Nikhil Gupta

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From playlist Workshop on Algebraic Complexity Theory 2019

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Lie Groups and Lie Algebras: Lesson 43 Group Theory Review #2 (improved video quality)

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From playlist Lie Groups and Lie Algebras

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Lie Groups and Lie Algebras: Lesson 20 - Finite transformation example

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From playlist Lie Groups and Lie Algebras

Related pages

Continuous symmetry | Lie group | Translation (geometry) | Differential form | Product rule | Dynamical system | Invariant (mathematics) | Lie bracket of vector fields | Discrete symmetry | Algebraic variety | Élie Cartan | State variable | Algebra | Algebraic system | Scaling (geometry) | Maple (software) | Mathematics | Contact geometry | Moving frame | Lie algebra | Manifold | Flow (mathematics) | Partial differential equation | Vector field