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I modified the question because it was confused.

On my book there is this mathematical definition of symmetry transformation:

"The equations of motion have a symmetry, if the solutions of the equations transformed by the symmetry are still solutions of the equations of motion, namely, there is a symmetry if the transformed equations of motions have the same form of the original".

I don't understand the meaning of this sentence, do you think is it a good (and easy) mathematical definition of symmetry transformation? Anyway what "equal in form" means? Then, i know rotation of an isolated system is a symmetry, can you make an easy example of an isolated system in which if we apply a rotation the mathematical definition of symmetry apply? Can you make an example of non symmetric transformation and show me why the mathematical definition doesn't apply?

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  • $\begingroup$ Which textbook? $\endgroup$ – Qmechanic Jun 13 '19 at 15:59
  • $\begingroup$ Title “Lezioni di fisica teorica” author: Nino Zanghì page 79. It’s in italian anyway: ge.infn.it/~zanghi/FT/ZUM.pdf $\endgroup$ – SimoBartz Jun 13 '19 at 16:32
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Suppose you keep track of the total energy in the system at hand and it remains unchanged with time. This is Conservation of energy. However you transform the time coordinate, the energy remains constant. Symmetry is in the system if some transformation leaves certain elements of the system unchanged. This includes translations in time.

Now suppose you consider the energy as it changes with position. It's important to consider the energy measurements in the SAME coordinate system. Coordinates refer to a real, singular point. You need to compare the change in energy at different points, not the same point at different coordinates. Keeping this in mind, we know that momentum remains constant in time if the space derivative of the Hamiltonian is 0 everywhere in the system.

This applies to every conjugate momentum. So If the energy is constant under rotations, then angular momentum is conserved.

Look up Hamiltonian Mechanics and conjugate coordinates for more information.

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