# What is the difference between dynamical symmetry and geometrical symmetry?

i really can not understand it

tiny-tim
Homework Helper
Hi wdlang! What's "dynamical symmetry"?

I googled and wiki'd it, but couldn't find anything. Hi,

are you refering to symmetry breaking ? A symmetry can be dynamically or spontaneously broken.

Hi,

are you refering to symmetry breaking ? A symmetry can be dynamically or spontaneously broken.

Thanks a lot

dynamically broken?

i seldom hear it. However, spiontaneously symmetry broken is well know.

could you explain it in detail? thanks！

Here is my humble understanding. A dynamical symmetry is a _hidden_ symmetry. The classic example would be the Hydrogen atom. Naively, we would only expect an SO(3) symmetry associated with rotational symmetry. This would be the geometrical symmetry, which leads to the conserved angular momentum vector. In fact, the full symmetry of the system is SO(4); this is exhibited by there being another conserved vector, the Laplace-Runge-Lenz (LRL) vector.

Since the LRL vector is peculiar to the particular potential of the hydrogen atom and does not emerge as the result of some general geometrical feature shared by a whole class of systems (like rotational symmetry), it is termed a _dynamical_ symmetry. If one were naively observing the Hydrogen atom, then one would only notice the extra symmetry in studying its dynamics.

Disclaimer: This is only what I have gleaned from reading some papers on dynamical symmetry; I have never read an actual definition.

Side notes:
* If I recall correctly, the SO(4) symmetry of the Hydrogen atom can be realized by starting in a four dimensional space and dimensionally reducing. In which case the dynamical symmetry starts out as a geometrical symmetry.

* Dynamical symmetry breaking is a type of spontaneous symmetry breaking and is an unrelated topic.

I think symmetry is due to dynamic interactions within creation. for example, hydrogen atoms are correct, symmetry in the isospin of nuclear related to the group SU (2)