Abstract
In this paper, we will trace the development of the use of symmetry in discussing the theory of motion initiated by Emmy Noether in 1918. Though it started with its use in classical mechanics, and has been heavily used in engineering applications of mechanics, it came into its own in relativity, and quantum theory and their applications in particle physics and field theory. It will be beyond the scope of this article to explain the quantum field theory applications in any detail, but the base for understanding it will be provided here. We will also go on to discuss an insight from some more mathematical developments related to Noether symmetry.
Highlights
Physics started as a study of motion in Greek times and was formalized by Aristotle (c. 350 BC) by a set of “laws”, which he declared as “self-evident truths”, based on his view of the universe as it was visualized
The law for motion on the Earth was based on the nature of the object moving, and for motion in the heavens on the “truth” that heavenly objects are made of the perfect element, aether, and move along perfect circles, unless they are contaminated by proximity to the Earth, in which case epicycles develop
Writing the invariant as an arbitrary constant, it can be used to write, say, the highest derivative in the combination, in terms of the other variables. It is needed for partial differential equations (PDEs), as “solving them” without the boundary conditions is not very meaningful, and the invariants can incorporate the boundary conditions
Summary
Physics started as a study of motion in Greek times and was formalized by Aristotle (c. 350 BC) by a set of “laws”, which he declared as “self-evident truths”, based on his view of the universe as it was visualized. “Tyger, tyger burning bright; In the forests of the night; What immortal hand or eye Could frame thy fearful symmetry?” This is almost as common is its use in the geometric sense of leaving a figure or shape invariant under some transformation, such as reflection or rotation. While considering solutions of polynomial equations of degree 5 or more in 1771, Lagrange extended the concept to invariance of polynomials under permutations of its elements [8] While this led to many other developments in Algebra, I am here concerned with its use by Abel [9] and Galois [10] to invent Group Theory, so as to prove that there could be no solution of quintic or higher degree polynomial equations by means of radicals.
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