Do All Automorphisms in Physics Imply Symmetries in Equations?

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All physical symmetries correspond to automorphisms in axioms, but not all automorphisms in physical equations imply symmetries. Many physical phenomena are described by differential equations, where an automorphism can be viewed as an isomorphism of the solution set. This isomorphism typically represents a coordinate transformation that maps solutions into one another without altering the governing equation's form. Such transformations are classified as symmetries of the differential equation. Understanding this relationship is crucial for comprehending the underlying principles of physical laws.
Garrulo
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All the physical simmetries implicate an automorphism in axioms. But ¿all the automorphisms from a physical equation do implicate a simmetry?
 
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I am not sure I understand, but I think I may know what you have in mind. Many phenomena in physics are governed by differential equations, and we can consider an isomorphism of the solution set of a given equation into itself (an automorphism). Usually such an isomorphism expresses a coordinate transformation of some sort. This transformation maps solutions into each other, and the form of the governing differential equation does not change. We define such a map as a symmetry of the differential equation.
 

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