How Did Emmy Noether Derive Her Equation for Variational Symmetries in 1915?

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SUMMARY

Emmy Noether derived her equation for variational symmetries in 1915, demonstrating that if infinitesimals X and Y exist for an ordinary differential equation L(x,y,y'), the Euler-Lagrange Equation can be reformulated using these variables. This approach allows for a more efficient method of solving differential equations by substituting directly into her derived equation. The significance of symmetry in solving differential equations is emphasized, making it a crucial concept for mathematics and physics students.

PREREQUISITES
  • Understanding of the Euler-Lagrange Equation
  • Familiarity with ordinary differential equations (ODEs)
  • Basic knowledge of variational principles
  • Concept of infinitesimals in calculus
NEXT STEPS
  • Research Emmy Noether's original paper on variational symmetries
  • Study the implications of symmetry in differential equations
  • Explore Leon M. Lederman's book "Symmetry and the Beautiful Universe"
  • Learn about advanced topics in symmetry classes in mathematics
USEFUL FOR

Mathematics and physics students, researchers in theoretical physics, and anyone interested in the application of symmetry in solving differential equations.

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I am in need of Noether's paper she wrote in around 1915 about variational symmetries.

I need to know how she found that [for an ode L(x,y,y')]if an X and Y (infinitesimals) exists then you can rewrite the Euler-Lagrange Equation in terms of L(x,y,y'), X(x,y) and Y(x,y).

This is really amazing how she didn't this. So instead of solving for y(x) in the E-L equ you can just plug it into the equation she came up with.

If anyone could find the paper she wrote that would be awesome!

Thanks
 
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Ok, I found it. thanks anyways.(http://www.physics.ucla.edu/~cwp/lists/accDB_su.html)

Symmetry is THE most powerful tool in solving differential equations. If you are a young math major you should difinitely think about going to a school where symmetry classes are avaliable.
 
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Speaking of Emmy Noether...

Leon Lederman has written an excellent, non-technical book dedicated to symmetry in physics. It is a long-overdue, popular homage to Emmy Noether!

Symmetry and the Beautiful Universe
by Leon M. Lederman & Christopher T. Hill​
 

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