) Noether’s Theorem is*super rad . The theorem is, colloquially,*

>>Continuous symmetries imply conserved quantities.(Let’s dig into the origin of this powerful theorem and list a couple of examples. I’ll restrict my attention to a subclass of symmetries for the sake of space, buuuut if there’s interest, I could do a more general post in the future. (*Heads up:*because this post is written using LaTeX, it’s probably easiest to read directly from my blog.) • Defining the Lingo:ALagrangianis a function that (after some manipulation) yields the physical evolution of a system. A generic Lagrangian $ L $ can be a real continuous function of …******************** ($ m $ realparameters$ t_j $, where $ j=1, cdots, m $, and $ n $ pairs of

paths, which we label $ q_i (t_1, cdots, t_m) $ and $ tilde {q} _i (t_1, cdots, t_m) $, where $ i=1, cdots, n $ and a path means “a real continuous function of the aforementioned parameters”

In this exceedingly general case, we might write

$$ L=L (q_1, cdots, q_n, tilde {q} _1, cdots, tilde {q} _n, t_1, cdots, t_m) $$

However

, for the sake of clarity, let’s restrict ourselves to one parameter $ t $ and one function pair $ (q, tilde {q}) $. Let’s further suppose $ L $ depends on its parameter only through the function pair, aka let us write $ L=L (q, tilde {q}) $ (as opposed to $ L=L (q, tilde {q} , t) $). This is the case relevant to a classical particle confined to a friction-free line, wherein $ q (t) $ is the particle’s**positionalong the line and $ t $ records the**time (*****************.

While $ L $ is constructed to be. a function of position $ q (t) $ and some other path $ tilde {q} (t) $, we always intend to eventually set $ tilde {q} (t) $ equal to the velocity $ dot {q} equiv dq / dt $. Furthermore, of the many combinations $ (q (t), tilde {q} (t)) $ equaling $ (q (t), dot {q} (t)) $, nature chooses the pair satisfying the (Euler-Lagrange equation:)

$$ left. frac { partial L} { partial q} right | _ {{ tilde {q}= dot {q}} = frac {dp} {dt} $$

where

$$ p equiv left. frac { partial L} { partial tilde {q}} right | _ {{tilde {q}= dot {q}} $$

The differential equation we get from plugging a specific $ L $ into the Euler-Lagrange equation is called the

equation of motionof the system, and $ p $ is the**momentum conjugate to $ q $.**

• The Main Idea:No matter the specific form of $ L $, we can plot it over $ (q, tilde {q}) $ space. (To be clear: this is the space of values to which the paths $ q (t) $ and $ tilde {q} (t) $ are mapping, so essentially the real plane.) The resulting plot will look something like the left plot in the following image:

*********************** I’ve made a point to highlight contours of constant $ L $ in orange. The right plot projects those contours onto the $ (q, tilde {q}) $ plane. These contours lie at the heart of Noether’s Theorem: imagine we slide the coordinates $ (q, tilde {q}) $ around in the plane so that each point moves continuously along a contour of constant $ L $. Usually the shape of $ L $ is changed when we distort the $ (q, tilde {q}) $ plane below it, but by forcing each $ (q, tilde {q}) $ to flow along a contour, $ L $ is unaffected! A continuous transformation that leaves $ L $ unchanged is called a**continuous symmetry**of the Lagrangian.*We can make this whole business quantitative by parameterizing the sliding operation with a real variable $ alpha $. Let’s choose $ alpha $ so that $ alpha=0 $ corresponds our initial unchanged $ (q, tilde {q}) $ plane. Then the statement “$ alpha $ parameterizes contours of constant $ L $” is mathematically expressible as*

$$ left. frac {dL} {d alpha} right | _ { alpha=0}=0 $$

which equals, by the chain rule of differentiation,

$$ frac { partial L} { partial q} hspace {3 pt} left. Frac {dq} {d alpha} right | _ { alpha=0} frac { partial L} { partial tilde {q}} hspace {3 pt} left. frac {d tilde {q}} {d alpha} right | _ { alpha=0}=0 $$

Now, this equation holds true across*all*of $ (q, tilde {q} ) $ space, including along the physical path solving the equations of motion. For that path, we may use the Euler-Lagrange equation, the definition of conjugate momentum $ p equiv partial L / partial tilde {q} $, and the fact that $ partial dot {q} / partial alpha=d / dt ( partial q / partial alpha) $ to write,

$$ frac { dp} {dt} hspace {3 pt} frac { partial q} { partial alpha} p hspace {3 pt} frac {d} {dt} left [frac{partial q}{partial alpha}right]=0 $$

aka, according to the product rule of differentiation,

$$ frac {d} {dt} left [phspace{3 pt}frac{partial q}{partial alpha}right]=0 $$

In other words, the value of $ p hspace {3 pt} ( partial q / partial alpha) $ doesn’t change in time – it’s a**conserved quantity!**This is a special case of Noether’s Theorem.

If we repeat the above calculation with $ n $ paths $ q_i (t) $, we instead derive

$$ text {*} hspace { pt} frac {d} {dt} left [vec{p}cdot frac{dvec{q}}{dalpha}right]=0 hspace { pt} text {*} $$

where $ vec {q} equiv (q_1, cdots, q_n) $ and $ vec {p} equiv (p_1, cdots, p_n) $, and we find that

>>Momentum along a continuous symmetry direction is conserved.

This powerful statement allows us to identify conserved quantities from symmetries alone. For example …

• 1. Coordinate Translation Symmetry:Suppose $ L $ doesn’t depend on the coordinate $ q_i $ for some value of $ i $. Then the translation $ q_i mapsto q_i alpha $ is a continuous symmetry transformation. Because $ dq_k / d alpha $ is nonzero only for the coordinate we’re transforming ($ k=i $), the associated conserved quantity is…

$$ vec {p} cdot frac {d vec {q}} {d alpha}=p_i $$

which is the $ i $ th conjugate momentum. And so,**coordinate translation invariance necessarily implies conservation of the corresponding momentum.**

*(****************** • 2. Rotational Symmetry:Suppose $ L $ depends on the coordinates $ (x, y, z) $ and is invariant under the rotation of $ x $ and $ y $ into one-another:*

$$ x mapsto ( cos alpha) x ( sin alpha) y hspace { pt} y mapsto ( cos alpha) y – ( sin alpha) x $$

*Then we may calculate, $$ left. frac {dx} {d alpha} right | _ { alpha=0}=y hspace {50 pt} left. frac {dy} {d alpha} right | _ { alpha=0}=-x $$*

from which Noether’s Theorem implies conservation of…

$$ vec {p} cdot frac {d vec {q}} {dt}=p_x y – p_y x=- L_z $$

where $ L_z $ is the angular momentum in the $ z $ -direction.Rotational invariance necessarily implies angular momentum conservation.

* • Closing Comment:Note how we didn’t need an explicit form for $ L $ to make these arguments. Noether’s Theorem tells us that conserved quantities naturally emerge not as coincidences, but from the presence of symmetries. Consequently, if I experimentally observe a conserved quantity (like electric charge, or color charge, or even things like lepton number) then I can make my model consistent with that observation by encoding certain symmetries into my Lagrangian. In this way, Noether’s Theorem is impressive in both its elegance and practical power.*

Thanks for the ask, anonymous! I hope this helped. Although I did not have space to discuss it here, Noether’s Theorem also extends to Lagrangians with explicit parameter dependence, systems with multiple evolution parameters (like QFT), instances where $ (q, tilde {q}) $ transformations change $ L $ by a total derivative, and more. One extension allows us to demonstrate how time translation invariance yields energy conservation.

*) Have a physics question you think I might be able to help answer? Send me an ask. Until then, have a great day! Best wishes, my friend!*

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