Q: What is the physical meaning of “symmetries”? Why is there one-to-one correspondence between laws of conservation and symmetries? Why is it important that there is such correspondence?

Physicist: This is the shortest answer yet: “Noether“.

When a physicist talks about symmetry, they don’t usually mean symmetry the way everyone else in the world does.  The backbone of mechanics (both classical and quantum) is the “Lagrangian”, \mathscr{L}.  Basically, the Lagrangian is \mathscr{L}=T-V where T is kinetic energy and V is potential energy (some systems are easier to describe than others).  You can use the Lagrangian as a shortcut for describing all kinds of physical phenomena and dynamics, using the principle of least action.  This is a course on its own (several courses), so I won’t go into it.  If you can change some variable without changing the dynamics the Lagrangian describes, then you’ve found a “symmetry”.

For example, near the Earth’s surface, gravitational potential energy is given by V=mgz (mass times gravity times height), and as always kinetic energy is given by T=\frac{1}{2}mv^2 (one half mass times velocity squared).  So \mathscr{L}=\frac{1}{2}mv^2-mgz.

Right off the bat you’ll notice that \mathscr{L} has no x or y (just z), so if you change the x or y it doesn’t change \mathscr{L} at all.  Symmetry!  It turns out that this gives you conservation of momentum in the x and y directions (that’s not obvious, btw).

You’ll also notice that there is a z in \mathscr{L}.  As a result, changing z changes \mathscr{L}, and you don’t have conservation of momentum in the z direction (up-down direction).  Try holding something out and letting it go, it will suddenly start moving toward the ground (as if by magic!), which is a definitely not conserved momentum.

Notice also there are no “t’s” involved (no time dependence) in \mathscr{L}.  This one gives you conservation of energy!

This was an example of a very straightforward, simple Lagrangian.  But, with a bit of slickness, and some well written (really nasty complex) Lagrangians, you can find dozens of symmetries that lead to conservation laws for energy, momentum, angular momentum, electric charge, particle number, Baryon number, Lepton flavor, all kinds of stuff!

The theorem that describes the correspondence between symmetries and conservation laws is “Noether’s theorem“.  It’s arguably one of the most important theorems evers.

Emmy Noether: crazy smart

The dynamics of a system are completely governed by the Lagrangian of that system which, frankly, you can often guess (“pulled it out of my keister” is a standard technique in physics circles).  This makes things easy.  When you hear about the “aesthetics of equations” physicists are often talking about Lagrangians.

However, there’s a big difference between having an equation, and having a solved equation.  So, being unable to find a solution, if you can find a symmetry (and thus a conserved quantity) you’re a lot closer to being able to model your system.

The Lagrangian that describes the gravitational interaction of all the stars in our galaxy includes a different term for every pair of stars.  So since there are 500,000,000,000 stars in our galaxy (give or take), there are approximately 12,500,000,000,000,000,000,000,000 terms in \mathscr{L}.  Most of these terms are really small, but still…  Luckily, this Lagrangian still has the symmetries that lead to conservation of momentum, angular momentum, and energy, which is enough to build pretty solid computer models.  Huzzah for Noether!

If you’d really like to learn how to use Lagrangians, then 1) learn some basic calculus of variations and Euler-Lagrange, and 2) get ready to use the principle of least action without knowing why it works (spookiest damn thing in physics).

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