Summary

We clarify by easy examples (one-parameter Lie teams), partly within the authentic language, and alongside the historic papers of Sophus Lie, Abraham Cohen, and Emmy Noether how Lie teams grew to become a central matter in physics. Physics, in distinction to arithmetic, didn’t expertise the Bourbakian transition so the language of for instance differential geometry didn’t change fairly as a lot over the last hundred years because it did in arithmetic. This additionally signifies that arithmetic at the moment has been written in a method that’s far nearer to the language of physics, and people papers aren’t as old style as you would possibly anticipate.

Introduction

$$
operatorname{SU(3)}occasions operatorname{SU(2)} occasions operatorname{U(1)}
$$
is probably essentially the most outstanding instance of a Lie group in fashionable physics. To emphasize the truth that SM isn’t the ultimate phrase leads typically to
$$
operatorname{SU(3)}occasions operatorname{SU(2)} occasions operatorname{U(1)} subset operatorname{SU(5)}
$$
relying on somebody’s favourite candidate for a GUT, right here ##operatorname{SU(5)}##. Nevertheless, Lie teams entered physics on a a lot much less subtle, extra rudimentary stage that doesn’t have to cite quantum physics. Lie spoke of the speculation of invariants of tangent transformations. Properly, he really stated touching transformations [2]. These days, we frequently learn mills and language doesn’t all the time correctly distinguish between the teams of analytical coordinate transformations and their linear representations. Lie developed his idea between 1870 and 1880 [3], referencing [2] Jacobi’s work on partial differentials. Cohen already spoke of Lie teams in his e book about one-parameter teams 1911 [4], whereas Noether in her well-known papers 1918 simply known as them the group of all analytical transformations of the variables (Lie group), which corresponds to the group of all linear transformations of the differentials (linear illustration of the Lie group on its Lie algebra) [5], or repeatedly group in Lie’s sense in her groundbreaking paper [6] that led to the connection between conservation legal guidelines and the speculation of tangent transformations, Lie idea.

Peter John Olver – Minneapolis 1986

The instinct is easy. A bodily phenomenon in nature is a process described by location, time, and the change of location in time, shortly: by a differential equation system ##mathcal{L}(x,dot x,t)##. So as to develop into a bodily legislation, we have now to display by experiments that the options of the differential equation system are in accordance with the outcomes of the experiments. This implies we have now to measure sure portions. A measurement is a comparability with a reference body. A reference body is a set of coordinates within the laboratory which we used to explain the differential equation system. The end result of the experiments, nonetheless, shouldn’t depend upon the coordinate system we used since nature can not know what we used. Therefore any change of coordinates could not alter the experiment. This implies mathematically {that a} transformation of coordinates ends in the identical options of the differential equation system, i.e. ##mathcal{L}(x,dot x,t)## is powerful aka invariant below coordinate transformations inside our reference body. It turned out that these transformations are even linked to bodily conservation legal guidelines, i.e. invariant portions. How does this learn in a contemporary textbook?

Theorem: A generalized vector discipline determines a variational symmetry group of the useful ##mathcal{L}[u]=int L,dx## if and provided that its attribute is the attribute of a conservation legislation ##operatorname{Div}P=0## for the corresponding Euler-Lagrange equations ##E(L)=0.## Specifically, if ##mathcal{L}## is a nondegenerate variational downside, there’s a one-to-one correspondence between equivalence courses of nontrivial conservation legal guidelines of the Euler-Lagrange equations and equivalence courses of variational symmetries of the useful [1].

That’s fairly a discrepancy between instinct and technical particulars. The purpose is, if we now begin to enter the jungle of these technical particulars and show the concept, chances are high excessive that we’ll lose instinct. As an alternative, let the time warp start and see how the topic has been launched to physics a century in the past.

Abraham Cohen – Baltimore 1911

Group of Transformations

The set of parameterized coordinate transformations [and our example in brackets of a rotation with the angle as the parameter]
start{align*}
T_a, : ,x_1&=phi(x,y,a), , ,y_1=psi(x,y,a)
[x_1&= xcos a – y sin a, , , y_1=x sin a +ycos a]
T_b, : ,x_2&=phi(x_1,y_1,b), , ,y_2=psi(x_1,y_1,b)
[x_2&= x_1cos b – y_1 sin b, , , y_2=x_1sin b +y_1cos b]
finish{align*}
carries a gaggle construction if the results of performing one after which the opposite transformation is once more of the shape
start{align*}
T_aT_b=T_c, : ,x_2&=phi(phi(x,y,a),psi(x,y,a),b)=phi(x,y,c)
y_2&=psi(phi(x,y,a),psi(x,y,a),b)=psi(x,y,c)
[x_2&= xcos (a+b) – y sin (a+b), , , y_2=xsin (a+b) +ycos (a+b)]
finish{align*}
the place the parameter ##c## relies upon solely on the parameters ##a,b.## Since ##phi,psi## are steady features of the parameter ##a,## if we begin with the worth ##a_0,## and permit ##a## to range repeatedly, the impact of the corresponding transformations on ##x,y## can be to rework them repeatedly, too; i.e. for a small enough change of ##a,## the modifications in ##x,y## are as small as we would like. A variation of the parameter ##a## generates a metamorphosis of the purpose ##(x,y)## to numerous factors on some curve, which we name the orbit of the group. If ##(x,y)## is taken into account as a continuing level whereas ##(x_1,y_1)## is variable, then ##T_a## is the parameterized orbit by means of ##(x,y).## The orbit comparable to any level ##(x,y)## could also be obtained by eliminating ##a## from the 2 equations of ##T_a.## Cohen known as the orbit path-curve of the group.

Infinitesimal Transformation

Let ##a_0## be the worth of ##a## that corresponds to the an identical transformation, and ##delta a## an infinitesimal. [##a_0=0## in our example.] As ##phi,psi## are analytical, the transformation
start{align*}
x_1=phi(x,y,a_0+delta a), &, ,y_1=psi(x,y,a_0+delta a)
[x_1=xcos(delta a)-ysin(delta a), &, ,y_1=xsin(delta a)+ycos(delta a)]
finish{align*}
modifications ##x,y## by an infinitesimal quantity. The Taylor collection turns into
start{align*}
underbrace{x_1-underbrace{phi(x,y,a_0)}_{=x}}_{=:delta x}&=underbrace{left(left. dfrac{partial phi}{partial a}proper|_{a_0}proper)}_{=:xi(x,y)} delta a +O(delta^2 a)=delta x=xi(x,y)delta a+ldots
underbrace{y_1-underbrace{psi(x,y,a_0)}_{=y}}_{=:delta y}&=underbrace{left(left. dfrac{partial psi}{partial a}proper|_{a_0}proper)}_{=:eta(x,y)} delta a +O(delta^2 a)=delta y=eta(x,y)delta a+ldots
finish{align*}
start{align*}
[delta x&=(-xsin(0)-ycos(0))delta a+O(delta^2 a)=-y,delta a+O(delta^2 a)]
[delta y&=(xcos(0)-ysin(0))delta a+O(delta^2 a)=x,delta a+O(delta^2 a)]
finish{align*}
[Note that ##(x,y)perp (delta x,delta y)## in our example as expected for a rotation.]

Neglecting the upper powers of ##delta a,## we get the infinitesimal transformation (generator, vector discipline ##U##)
start{align*}
U, : ,delta x=xi(x,y)delta a, &, ,delta y=eta(x,y)delta a
U, : ,delta x=xi delta a=left(left. dfrac{partial x_1}{partial a}proper|_{a=a_0}proper)delta a, &, ,delta y=eta delta a=left(left. dfrac{partial y_1}{partial a}proper|_{a=a_0}proper)delta a
finish{align*}

$$[U, : ,delta x=xidelta a=-ydelta a, , ,delta y=etadelta a=xdelta a]$$

Image of Infinitesimal Transformation U.f

##delta ## is the image of differentiation with respect to the parameter ##a## within the restricted sense that it designates the worth which the differential of the brand new variable ##x_1## or ##y_1## assumes when ##a=a_0.## If ##f(x,y)## is a common analytical operate, the impact of the infinitesimal transformation on it, ##U.f,## is to exchange it by ##f(x+xidelta a,y+eta delta a).## The Taylor collection at ##a=a_0## is thus
start{align*}
underbrace{f(x+xidelta a,y+eta delta a)-f(x,y)}_{=:delta f}=underbrace{left(xidfrac{partial f}{partial x}+etadfrac{partial f}{partial y}proper)}_{=:U.f}delta a+O(delta^2 a)
finish{align*}
and with ##f_1=f(x_1,y_1)##
start{align*}
left.dfrac{partial f_1}{partial a}proper|_{a_0}&=dfrac{partial f(x_1,y_1)}{partial x_1}cdot left. dfrac{partial x_1}{partial a}proper|_{a_0}+dfrac{partial f(x_1,y_1)}{partial y_1}cdot left. dfrac{partial y_1}{partial a}proper|_{a_0}&=xidfrac{partial f(x_1,y_1)}{partial x_1}+etadfrac{partial f(x_1,y_1)}{partial y_1}=xidfrac{partial f(x,y)}{partial x}+etadfrac{partial f(x,y)}{partial y}=U.f
finish{align*}
Specifically ##U.x = xi, , ,U.y=eta.##

##U.f## could be written if the infinitesimal transformation ##delta x=xidelta a, delta y=etadelta a ## is thought, and conversely, the infinitesimal transformation is thought if ##U.f## is given. We are saying that ##U.f## represents the infinitesimal transformation.

[Say ##f(x,y)=x^2+y^2## for our example. Then
begin{align*}
partial f&=(x+xidelta a)^2+(y+eta delta a)^2-(x^2+y^2)=underbrace{2left(xxi+yetaright)}_{=U.f}delta a+O(delta^2 a )
U.f&=2left(xxi+yetaright)=2(-xy+yx)delta aequiv 0 quad
end{align*}
The effect of an infinitesimal rotation on a circle is zero.]

Group Generated by an Infinitesimal Transformation

The infinitesimal transformation
$$
U.f=xidfrac{partial f}{partial x}+eta dfrac{partial f}{partial y} ;textual content{ or }; delta x=xi(x,y)delta t, , ,delta y=eta(x,y) delta t
$$
carries the purpose ##(x,y)## to the neighboring place ##(x+xidelta t,y+etadelta t).## The repetition of this transformation an indefinite variety of occasions has the impact of carrying the purpose alongside an orbit which is exactly that integral curve (circulation) of the system of differential equations
$$
dfrac{d x_1}{d t}=xi(x_1,y_1), , ,dfrac{d y_1}{d t}=eta(x_1,y_1)
$$
which passes by means of the purpose ##(x,y).## Now ##dfrac{dx_1}{xi(x_1,y_1)}=dfrac{dy_1}{eta(x_1,y_1)}## being free from ##t## type a differential equation whose answer could also be written $$u(x_1,y_1)=fixed=u(x,y)$$ since ##x_1=x,y_1=y## when ##t=0.## That is the equation of the orbit comparable to ##(x,y).## Say we clear up the equation for ##x_1=omega(y_1,c)## then
$$
dt=dfrac{d y_1}{eta(omega (y_1,c),y_1)}Longrightarrow t=int dt= int dfrac{1}{eta(omega (y_1,c),y_1)}dy_1 +c’
$$
and an answer takes the shape (##c## changed by its expression in ##x_1,y_1## once more)
$$
v(x_1,y_1)-t=fixed =v(x,y)
$$
Contemplating
$$
start{instances}
u(x_1,y_1)=u(x,y)
v(x_1,y_1)=v(x,y)+t
finish{instances}
$$
as a metamorphosis from ##(x,y)## at ##t=0## to ##(x_1,y_1)##, we see that these outline a one-parameter Lie group, translation by ##(0,t).##

[The integral curve of the differential equations in our example is given by
begin{align*}
dfrac{dx_1}{dt}=-y_1, &, ,dfrac{dy_1}{dt}=x_1
-dfrac{x_1}{y_1}=int dfrac{dx_1}{-y_1}&=int dfrac{dy_1}{x_1}=dfrac{y_1}{x_1}+c[6pt]
u(x_1,y_1)=x^2_1+y^2_1&=c
finish{align*}
Let ##x_1=sqrt{c-y_1^2}=omega (y_1,c)## so
start{align*}
dt&=dfrac{dy_1}{x_1(omega (y_1,c),y_1)}=dfrac{dy_1}{x_1(sqrt{c-y_1^2},y_1)}=dfrac{dy_1}{sqrt{c-y_1^2}}
t&=int dfrac{dy_1}{sqrt{c-y_1^2}} =arcsinleft(dfrac{y_1}{sqrt{c}}proper)+c’=arcsinleft(dfrac{y_1}{sqrt{x_1^2+y_1^2}}proper)+c’
finish{align*}
$$
start{instances}
x_1^2+y_1^2=x^2+y^2
arcsinleft(dfrac{y_1}{sqrt{x_1^2+y_1^2}}proper)=arcsinleft(dfrac{y}{sqrt{x^2+y^2}}proper)+t
finish{instances}
$$
There’s one other answer to the differential equation. We get from
start{align*}
dfrac{dx}{P}&=dfrac{dy}{Q}=dfrac{dt}{R}=dfrac{lambda dx+mu dy+nu dt}{lambda P+mu Q+nu R}[6pt]
dfrac{dx_1}{-y_1}&=dfrac{dy_1}{x_1}=dfrac{dt}{1}=dfrac{-y_1lambda dx_1+x_1mu dy_1}{lambda y_1^2+mu x_1^2}[6pt]
t&=-dfrac{sqrt{lambda}}{sqrt{mu}}arctandfrac{sqrt{mu}x_1}{sqrt{lambda}y_1}+dfrac{sqrt{mu}}{sqrt{lambda}} arctandfrac{sqrt{lambda}y_1}{sqrt{mu}x_1}[6pt]
v(x_1,y_1)&=arctan dfrac{y_1}{x_1}-arctandfrac{x_1}{y_1}=arctandfrac{y}{x}-arctandfrac{x}{y}+t=v(x,y)+t quad ]
finish{align*}

One other Technique of Discovering the Group from its Infinitesimal Transformation

We get from the MacLaurin collection for ##f_1##

start{align*}
f_1&=f+left. dfrac{partial f_1}{partial t}proper|_{t=0}t+left. dfrac{partial^2 f_1}{partial t^2}proper|_{t=0}dfrac{t^2}{2!}+left. dfrac{partial^3 f_1}{partial t^3}proper|_{t=0}dfrac{t^3}{3!}+ldots
f_1&=f+U.f,t+U^2.f,dfrac{t^2}{2!}+U^3.f,dfrac{t^3}{3!}+ldots=exp(tU).f
finish{align*}
start{align*}
[,text{Example: }U.f&=-ydfrac{partial f}{partial x} +xdfrac{partial f}{partial y}
U.x&=-y,,U^2.y=U.(-y)=-x,,U^3.x=U.(-x)=ytext{ etc.}[6pt]
x_1&=xleft(1-dfrac{t^2}{2!}+dfrac{t^4}{4!}+ldotsright)-yleft(t-dfrac{t^3}{3!}+dfrac{t^5}{5!}-ldotsright)
&=xcos t-ysin t
y_1&=xleft(t-dfrac{t^3}{3!}+dfrac{t^5}{5!}-ldotsright)+yleft(1-dfrac{t^2}{2!}+dfrac{t^4}{4!}+ldotsright)
&=xsin t+ycos t quad]
finish{align*}

Invariants

A operate of the variables is claimed to be an invariant of a gaggle (or invariant below the group) whether it is left unaltered by each transformation of the group, i.e. ##f(x_1,y_1)=f(x,y).##

Theorem: The mandatory and adequate situation that ##f(x,y)## be invariant below the group ##U.f## is
$$
U.f =xi dfrac{partial f}{partial x}+etadfrac{partial f}{partial y}equiv 0.
$$
[We have already seen that ##u(x,y)=x^2+y^2=c## is an integral curve of the differential equations in our example. A general solution of ##U.f=0## is then given by ##f=F(u)=F(x^2+y^2),## see [8,  §79].]

Orbits. Invariant Factors and Curves

The differential equations for orbits have been obtained by options to
$$
dfrac{dx}{xi}=dt =dfrac{dy}{eta} Longleftrightarrow dfrac{dy}{dx}=dfrac{eta}{xi}.
$$
The overall answer ##u(x,y)=c## represents a household of orbits. [##x^2+y^2=c## in our example; circles are the invariants of rotations.] If ##f(x,y)=0## is an invariant equation, then ##f(x_1,y_1)=0## for all factors ##(x_1,y_1)## and ##U.f=0 .## This implies ##U.f## should include ##f(x,y)## as an element (assuming ##f## accommodates no repeated elements)
$$
U.f=omega (x,y)cdot f(x,y)
$$
and ##U^2## accommodates ##f## as an element, too, by
$$
U^2.f=U.(U.f)=(U.omega ) f + omega (U.f)=(U.omega +omega^2).f
$$
This course of could be inductively repeated
$$
U^n.f=theta(x,y)cdot f(x,y), , ,U^{n+1}.f=(U.theta+theta omega )f,
$$
therefore the vanishing of ##U.f## at any time when ##f(x,y)## does is each the required and adequate situation that ##f(x,y)=0## be an invariant equation.

Theorem: The mandatory and adequate situation that ##f(x,y)=0## be invariant below the group ##U.f## is that ##U.f=0## for all values ##x,y## for which ##f(x,y)=0##, it being presupposed that ##f(x,y)## has no repeated elements. Factors whose coordinates fulfill the 2 equations ##xi(x,y)=0,,eta(x,y)=0## are invariant below the group. If ##xi(x,y)=0,,eta(x,y)=0## at any time when ##f(x,y)=0## this curve consists of invariant factors. Curves of this sort aren’t included among the many orbits of the group. In all different instances, ##f(x,y)=0## is an orbit.

If ##U.f=0## for all values ##x,y##, ##f(x,y)## is an invariant, and ##f(x,y)=c## fixed (together with zero) is an orbit.

[##xi=-y,,eta=x,,u(x,y)=x^2+y^2=c## are the equations of all orbits of rotations. There are no other invariant curves. The point ##(x,y)=(0,0)## is invariant.]

Invariant Household of Curves

A household of curves is claimed to be invariant below a gaggle, if each transformation of the group transforms every curve ##f(x,y)=c.## into some curve of the household
start{align*}
f(x_1,y_1)=f(phi(x,y,t),psi(x,y,t))=omega(x,y,t)=c’
finish{align*}
These equations have to be options of the identical differential equation
$$
dfrac{partial f}{partial x}dx+dfrac{partial f}{partial y}dy=0 textual content{ and }dfrac{partial omega }{partial x}dx+dfrac{partial omega }{partial y}dy=0.
$$
which is the case if
$$
detbegin{pmatrix}f_x&f_y omega_x&omega_yend{pmatrix}=0 Longleftrightarrow omega =U.f=F(f).
$$
The household of curves ##f(x,y)=c## could equally effectively be written ##Phi[f(x,y)]=c,## the place ##Phi(f)## is any holomorphic operate of ##f.## From ##U.f=F(f)## and the chain rule we get
$$
U.Phi(f) = dfrac{dPhi}{df}U.f=dfrac{dPhi}{df}F(f).
$$
This can be any desired operate of ##f,## say ##Omega(f), ## if the household of orbits is excluded, i.e. if ##F(f)neq 0## then
$$
dfrac{dPhi}{df}F(f)=Omega(f) Longleftrightarrow Phi(f)=intdfrac{Omega(f)}{F(f)}df.
$$

##left[-ydfrac{partial f}{partial x}+xdfrac{partial f}{partial y}=F(f)right.## leads to ##dfrac{dx}{-y}=dfrac{dy}{x}=dfrac{df}{F(f)},## so the general solution is of the form
$$
arctanleft(dfrac{y}{x}right)-phi(f)=psi(x^2+y^2) text{ or }
f=Phileft(arctanleft(dfrac{y}{x}right)-psi(x^2+y^2)right).
$$
The equation ##dfrac{y}{x}=c## representing the family of straight lines through the origin is a simple example.]

Alternant (Commutator)

Let ##U_1,U_2## be any two homogeneous linear partial differential operators
$$
U_1=xi_1(x,y)dfrac{partial }{partial x}+eta_1(x,y)dfrac{partial }{partial y}; , ;U_2=xi_2(x,y)dfrac{partial }{partial x}+eta_2(x,y)dfrac{partial }{partial y}
$$
Then
start{align*}
U_1U_2.f&=(U_1.xi_2)left(dfrac{partial f}{partial x}proper)+(U_1.eta_2)left(dfrac{partial f}{partial y}proper)+
xi_1xi_2dfrac{partial^2 f}{partial x^2}+ (xi_1eta_2+xi_2eta_1)dfrac{partial^2f }{partial x partial y}+eta_1eta_2dfrac{partial^2 f}{partial y^2}
U_2U_1.f&=(U_2.xi_1)left(dfrac{partial f}{partial x}proper)+(U_2.eta_1)left(dfrac{partial f}{partial y}proper)+
xi_2xi_1dfrac{partial^2 f}{partial x^2}+ (xi_2eta_1+xi_1eta_2)dfrac{partial^2f }{partial x partial y}+eta_2eta_1dfrac{partial^2 f}{partial y^2}
finish{align*}
and
$$
[U_1,U_2]=U_1U_2-U_2U_1=(U_1.xi_2-U_2.xi_1)dfrac{partial }{partial x}+(U_1.eta_2-U_2.eta_1)dfrac{partial }{partial y}
$$
is once more a homogeneous linear partial differential operator.

Integrating Issue

If ##phi(x,y)=c## is a household of curves invariant below the group
$$
U.f=xidfrac{partial f}{partial x}+etadfrac{partial f}{partial y}
$$
then we have now realized that ##U.phi=F(phi).## Furthermore, if the curves of the household aren’t orbits of the group, the equation of the household could be chosen in such type that ##F(phi)## shall develop into any desired operate of ##phi##. Specifically, there isn’t a loss
in assuming the equation so chosen that that is ##1.## For if a given selection ##phi(x,y)=c## results in ##F(phi),## the choice ##Phi(phi)=c= displaystyle{intdfrac{1}{F(phi)}dphi}## will give ##U.Phi(phi)=1.## Suppose now that
$$
M,dx+N, dy=0 quad (*)
$$
is a differential equation whose household of integral curves ##phi(x,y)=c## is invariant below the group ##U.f,## the integral curves not being orbits of the latter. Let additional ##phi## be chosen such that
$$
U.phi=xidfrac{partial phi}{partial x}+etadfrac{partial phi}{partial y}=1
$$
Since ##phi## is an answer of ##(*),##
$$
dphi = xi dfrac{partial phi}{partial x},dx+etadfrac{partial phi}{partial y},dy =0
$$
is similar equation as ##(*)## and thus
$$
dfrac{partial phi /partial x}{M}=dfrac{partial phi /partial y}{N} textual content{ or } Ndfrac{partial phi}{partial x}-Mdfrac{partial phi}{partial y}=0.
$$
Fixing the equation system ends in
$$
dfrac{partial phi}{partial x}=dfrac{M}{xi M+eta N}, , ,dfrac{partial phi}{partial y}=dfrac{N}{xi M+eta N}, , ,
dphi =dfrac{M,dx+N,dy}{xi M+eta N}
$$
Therefore we have now confirmed

Marius Sophus Lie – Christiania 1874

Theorem: If the household of integral curves of the differential equation ##M,dx + N,dy = 0## is left unaltered by the group ##Uf equiv xi dfrac{df}{dx}+eta dfrac{df}{dy},## ##dfrac{1}{xi M+eta N}## is an integrating issue of the differential equation.

This was the place, when, and by whom all of it obtained began.

Amalie Emmy Noether – Göttingen 1918

Noether spoke in [5] about differential expressions and meant features
$$
f(x,dx)=f(x_1,ldots,x_n; dx_1,ldots,dx_n)
$$
which are analytical in all arguments and investigated the analytical transformations of the variables and the corresponding linear transformations of their differentials concurrently
start{align*}
f(x,dx) &longrightarrow g(y,dy)
x_i=x_i(y_1,ldots,y_n), &, ,dx_i=sum_{okay=1}^n dfrac{partial x_k}{partial y_k}dy_k
finish{align*}
and an invariant of ##f## as an analytical operate
start{align*}
Jleft(f,dfrac{partial f}{partial dx}cdots dfrac{partial^{rho+sigma}f }{partial x^rho partial dx^sigma}cdots dx,delta x,d^2x,ldotsright)=Jleft(g,dfrac{partial g}{partial dy}cdots dfrac{partial^{rho+sigma}g }{partial y^rho partial dy^sigma}cdots dy,delta y,d^2y,ldotsright)
finish{align*}
which already seems like our fashionable expression ##mathcal{L}(x,dot x,t).## The questions in regards to the group of all invariants and their equivalence courses have been decreased to questions of the linear idea of invariants by Christoffel and Ricci within the case of particular differential equations. Noether known as it a discount theorem and was in a position to show it for arbitrary differential expressions by a distinct technique [5]. The essence of Lie’s idea can finest be described by the next diagram
start{equation*} start{aligned} G &longrightarrow GL(mathfrak{g}) dfrac{d}{dx}downarrow & quad quad quad uparrowexp mathfrak{g} &longrightarrow mathfrak{gl(g)} finish{aligned} finish{equation*}
Noether’s primary theorems say of their authentic wording [6]

1. If the integral ##I## is invariant with respect to a [Lie group] ##G_rho,## then ##rho## linearly unbiased connections of the Lagrangian expressions develop into divergences. Conversely, it follows that ##I## is invariant with respect to a [Lie group] ##G_rho.## The theory additionally holds within the restrict of infinitely many parameters.

2. If the integral ##I## is invariant with respect to a [Lie group] ##G_{inftyrho }##, during which the arbitrary features seem as much as the ##sigma##-th by-product, then there are ##rho## an identical relations between the Lagrangian expressions and their derivatives as much as the ##sigma##-th order; the converse additionally applies right here.

Epilogue – Noether Cost

Allow us to end with an instance of recent language.

The motion on a classical particle is the integral of an orbit ##gamma, : ,t rightarrow gamma(t)##
$$
S(gamma)=S(x(t))= int mathcal{L}(t,x,dot{x}),dt
$$
over the Lagrange operate ##mathcal{L}##, which describes the system thought-about. Now we contemplate clean coordinate transformations
start{align*}
x &longmapsto x^* := x +varepsilon psi(t,x,dot{x})+O(varepsilon^2)
t &longmapsto t^* := t +varepsilon varphi(t,x,dot{x})+O(varepsilon^2)
finish{align*}
and we evaluate
$$
S=S(x(t))=int mathcal{L}(t,x,dot{x}),dttext{ and }S^*=S(x^*(t^*))=int mathcal{L}(t^*,x^*,dot{x}^*),dt^*
$$
Because the useful ##S## determines the legislation of movement of the particle, $$S=S^*$$ means, that the motion on this particle is unchanged, i.e. invariant below these transformations, and particularly
start{equation*}
dfrac{partial S}{partial varepsilon}=0 quadtext{ resp. }quad left. dfrac{d}{dvarepsilon}proper|_{varepsilon =0}left(mathcal{L}left(t^*,x^*,dot{x}^*proper)cdot dfrac{dt^*}{dt} proper) = 0
finish{equation*}
Emmy Noether confirmed precisely 100 years in the past, that below these circumstances (invariance), there’s a conserved amount ##Q##. ##Q## is known as the Noether cost.
$$
S=S^* Longrightarrow left. dfrac{d}{dvarepsilon}proper|_{varepsilon =0}left(mathcal{L}left(t^*,x^*,dot{x}^*proper)cdot dfrac{dt^*}{dt} proper) = 0 Longrightarrow dfrac{d}{dt}Q(t,x,dot{x})=0
$$
with
$$
Q=Q(t,x,dot{x}):= sum_{i=1}^N dfrac{partial mathcal{L}}{partial dot{x}_i},psi_i + left(mathcal{L}-sum_{i=1}^N dfrac{partial mathcal{L}}{partial dot{x}_i},dot{x}_iright)varphi = textual content{ fixed}
$$
The overall method to proceed is:
(a) Decide the features ##psi,varphi##, i.e. the transformations, that are thought-about.
(b) Test the symmetry by equation.
(c) If the symmetry situation holds, then compute the conservation amount ##Q## with ##mathcal{L},psi,varphi,.##

Instance: Given a particle of mass ##m## within the potential ##U(vec{r})=dfrac{U_0}{vec{r,}^{2}}## with a continuing ##U_0##. At time ##t=0## the particle is at ##vec{r}_0## with velocity ##dot{vec{r}}_0,.##

The Lagrange operate with ##vec{r}=(x,y,z,t)=(x_1,x_2,x_3,t)## of this downside is
$$
mathcal{L}=T-U=dfrac{m}{2},dot{vec{r}},^2-dfrac{U_0}{vec{r,}^{2}},.
$$
1. Give a cause why the power of the particle is conserved, and what’s its power?

(a) Power is homogeneous in time, so we selected ##psi_i=0 , varphi=1##
(b) and examine
start{equation*}
left. dfrac{d}{dvarepsilon}proper|_{varepsilon = 0} left(mathcal{L}^*,cdot,dfrac{d}{dt},(t+varepsilon )proper)=left. dfrac{d}{dvarepsilon}proper|_{varepsilon = 0} left(mathcal{L}^*,cdot,1right) = 0
finish{equation*}
since ##mathcal{L}^*## doesn’t depend upon ##t^*## and thus not on ##varepsilon##, and calculate
(c) the Noether cost as
start{align*}
Q(t,x,dot{x})&=mathcal{L}- sum_{i=1}^Ndfrac{partial mathcal{L}}{partial dot{x}_i} ,dot{x}_i=T-U-dfrac{m}{2}left( dfrac{partial}{partial dot{x}_i}left( sum_{i=1}^3 dot{x}^2_i proper),dot{x}_i proper)
&=dfrac{m}{2}, dot{vec{r,}}^2 – U -m,dot{vec{r,}}^2=-T-U=-E&=-dfrac{m}{2}, dot{vec{r,}}^2- dfrac{U}{vec{r,}^2}=-dfrac{m}{2}, dot{vec{r,}}_0^2- dfrac{U}{vec{r,}_0^2}
finish{align*}
by time invariance.

2. Think about the next transformations with infinitesimal ##varepsilon##
$$vec{r} longmapsto vec{r},^*=(1+varepsilon),vec{r},, , ,,tlongmapsto t^*=(1+varepsilon)^2,t$$
and confirm the situation of E. Noether’s theorem.

##dot{vec{r}},^*=dfrac{dvec{r},^*}{dt^*}=dfrac{(1+varepsilon),dvec{r}}{(1+varepsilon)^2, dt }=dfrac{1}{1+varepsilon},dot{vec{r}},## and thus ##,mathcal{L}^*=dfrac{1}{(1+varepsilon)^2},mathcal{L}, ##, i.e.
start{align*}
left. dfrac{d}{dvarepsilon}proper|_{varepsilon =0}&left(mathcal{L}left(t^*,x^*,dot{x}^*proper)cdot dfrac{dt^*}{dt} proper) = left. dfrac{d}{dvarepsilon}proper|_{varepsilon =0} mathcal{L}^*,dfrac{dt^*}{dt} &=left. dfrac{d}{dvarepsilon}proper|_{varepsilon =0} dfrac{mathcal{L}}{(1+varepsilon)^2}cdot (1+varepsilon)^2=left. dfrac{d}{dvarepsilon} proper|_{varepsilon =0}mathcal{L} = 0
finish{align*}
and the situation of Noether’s theorem holds.

3. Compute the corresponding Noether cost ##Q## and consider ##Q## for ##t=0##.

The transformations we have now are
start{align*}
x &longmapsto x^* = (1+varepsilon)x & Longrightarrow quad& psi_x=x
y &longmapsto y^* = (1+varepsilon)y & Longrightarrow quad& psi_y=y
z &longmapsto z^* = (1+varepsilon)z & Longrightarrow quad& psi_z=z
t &longmapsto t^* = (1+2varepsilon)t & Longrightarrow quad& varphi=2t
finish{align*}
and the Noether cost is thus given by
start{align*}
Q(t,x,dot{x})&= sum_{i=1}^N dfrac{partial mathcal{L}}{partial dot{x}_i},psi_i + left(mathcal{L}-sum_{i=1}^N dfrac{partial mathcal{L}}{partial dot{x}_i},dot{x}_iright)varphi
&=sum_{i=1}^3 dfrac{partial}{partial dot{x}_i}left(dfrac{m}{2},dot{vec{r},}^2-dfrac{U_0}{vec{r,}^{2}}proper),psi_i ,+
&+ left(dfrac{m}{2},dot{vec{r}},^2-dfrac{U_0}{vec{r,}^{2}}-sum_{i=1}^3 dfrac{partial }{partial dot{x}_i},left(dfrac{m}{2},dot{vec{r}},^2-dfrac{U_0}{vec{r,}^{2}}proper)dot{x}_iright)varphi
&=m(dot{x}x+dot{y}y+dot{z}z) ,+
&+left( dfrac{m}{2}dot{vec{r,}}^2-dfrac{U_0}{vec{r,}^{2}}-m(dot{x}^2+dot{y}^2+dot{z}^2)proper)2t&=m, dot{vec{r}},vec{r},+left( -dfrac{m}{2}dot{vec{r,}}^2-dfrac{U_0}{vec{r,}^{2}} proper)2t=m, dot{vec{r}},vec{r}, -(T+U)2t
&=m, dot{vec{r}},vec{r}, -2Et;stackrel{t=0}{=}; m, dot{vec{r}}_0,vec{r}_0
finish{align*}
which exhibits that invariance below totally different transformations ends in totally different dialog portions.

Sources