Noether's theorem for discrete symmetry

In summary, the conversation discusses the possibility of a discrete version of Noether's symmetry for potentials with discrete symmetries. The purpose is to find a way to describe the evolution of the phase space without solving equations numerically. However, it is mentioned that Noether's theorem is based on continuous groups and may not apply in the discrete case. The conversation also touches on the concept of conservation laws in Hamiltonian mechanics and the potential for finding invariant quantities along trajectories. The possibility of using lattice theory or lattice gauge theory to gain insights is also mentioned.
  • #1
Dalor
9
0
I am wondering if it existes some discret version of the Noether symmetry for potential with discrete symmetry (like $C_n$ ).

The purpose is to describe the possible evolution of the phase space over the time without having to solve equations numerically (since even if the potential may have some symmetry solving the equation may be long and exhibe some chaotic behavior).

Thank's
 
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  • #2
So no one has any insight for that ? Even a small one ?
 
  • #3
Dalor said:
So no one has any insight for that ? Even a small one ?
No, I am not sure about it, but I cannot imagine it. Noether's theorem is a result of a variation process, continuously changing differential coordinates. It is a statement about Lie groups, i.e. analytical groups - originally called continuous groups. So major ingredients simply break away in any discrete case. Maybe one can consider different connection components or coverings to map discrete behavior, but that's more a guess than an insight.
 
  • #4
Thank you for the answer, yes I came to the same conclusion it does not seems easy to go from continuous to discret case. And may be Noether Theorem is not the correct approach.

But I said Noether Theorem at first because I am interesting in concervation law in my Hamiltonian.

Maybe there is other "easy" way to find invariant quantity along the trajectory in hamiltonian mechanic ? (This is for classical case not quantum one).
And by considering symmetry we could maybe conclude some interesting things on the solution even if the full solution is not found.

I am interesting in this because even if I know that the solution exhib chaotic solution it does not mean than the full phase space is reachable from any initial condition.
 
  • #6
To me this discussion is the clearest in Hamiltonian mechanics. A constant of motion is some function ##S(q,p)## of your generalised coordinates that satisfies ##\dot S = 0##. From the equations of motion, this derivative is given by ##\dot S = \{S,H\}## and therefore ##\{S,H\} = 0## for a constant of motion. Due to the anti-symmetry of the Poisson bracket, this also means that ##\{H,S\} = 0##, which means that ##H## does not change under the canonical transformations generated by ##S##, i.e., those canonical transformations are a one-parameter set of transformations that are a symmetry of the Hamiltonian. In other words, you can go both ways, continuous symmetry ⇔ constant of motion.
 

FAQ: Noether's theorem for discrete symmetry

What is Noether's theorem for discrete symmetry?

Noether's theorem for discrete symmetry is a mathematical theorem that states that for every discrete symmetry of a physical system, there exists a corresponding conserved quantity. This means that if a system remains unchanged under a certain transformation, then there is a physical quantity that remains constant throughout the system's evolution.

What are discrete symmetries?

Discrete symmetries are transformations that leave a physical system unchanged, but only in discrete steps. This means that the system will remain unchanged if it is rotated, reflected, or translated by a certain amount, but not by any arbitrary amount.

What is the significance of Noether's theorem for discrete symmetry?

Noether's theorem for discrete symmetry is significant because it provides a powerful tool for understanding the underlying symmetries and conservation laws of physical systems. It allows scientists to make predictions about the behavior of a system based on its symmetries, and has been used to uncover new fundamental laws of nature.

How is Noether's theorem for discrete symmetry applied in physics?

Noether's theorem for discrete symmetry is applied in physics by identifying the discrete symmetries of a system and using them to determine the corresponding conserved quantities. These conserved quantities can then be used to make predictions about the behavior of the system and to uncover new physical laws.

Can Noether's theorem for discrete symmetry be applied to all physical systems?

Yes, Noether's theorem for discrete symmetry can be applied to all physical systems, as long as they possess discrete symmetries. This includes classical systems, quantum systems, and relativistic systems. However, the specific conserved quantities and their corresponding symmetries may differ depending on the type of system being studied.

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