# When Lie Teams Grew to become Physics

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### Summary

We clarify by easy examples (one-parameter Lie teams), partly within the unique 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 approach that’s far nearer to the language of physics, and people papers usually are not as old school as you may count on.

### Introduction

$$

operatorname{SU(3)}occasions operatorname{SU(2)} occasions operatorname{U(1)}

$$

is probably essentially the most outstanding instance of a Lie group in trendy physics. To emphasize the truth that SM isn’t the ultimate phrase leads generally 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 degree that doesn’t must cite quantum physics. Lie spoke of *the idea of invariants of tangent transformations*. Properly, he truly 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 guide 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 *constantly group in Lie’s sense* in her groundbreaking paper [6] that led to the connection between conservation legal guidelines and the idea of tangent transformations, Lie idea.

### Peter John Olver – Minneapolis 1986

The instinct is straightforward. 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)##. As a way to develop into a bodily legislation, now we have to exhibit by experiments that the options of the differential equation system are in accordance with the outcomes of the experiments. This implies now we have 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, nevertheless, mustn’t rely upon the coordinate system we used since nature can not know what we used. Therefore any change of coordinates might not alter the experiment. This implies mathematically {that a} transformation of coordinates leads to the identical options of the differential equation system, i.e. ##mathcal{L}(x,dot x,t)## is strong 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 area determines a variational symmetry group of the practical ##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.## Particularly, 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 practical [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 fluctuate constantly, the impact of the corresponding transformations on ##x,y## will probably be to remodel them constantly, too; i.e. for a small enough change of ##a,## the adjustments in ##x,y## are as small as we wish. 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 relentless level whereas ##(x_1,y_1)## is variable, then ##T_a## is the parameterized orbit via ##(x,y).## The orbit similar 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 similar 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*}

adjustments ##x,y## by an infinitesimal quantity. The Taylor sequence 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 area ##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 basic 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 sequence 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*}

Particularly ##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 (stream) 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 via 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## kind a differential equation whose resolution 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 similar to ##(x,y).## Say we remedy 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{circumstances}

u(x_1,y_1)=u(x,y)

v(x_1,y_1)=v(x,y)+t

finish{circumstances}

$$

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{circumstances}

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{circumstances}

$$

There may be one other resolution 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 Methodology of Discovering the Group from its Infinitesimal Transformation

We get from the MacLaurin sequence 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 alleged 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 required and ample 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 resolution ##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 comprise ##f(x,y)## as an element (assuming ##f## comprises no repeated components)

$$

U.f=omega (x,y)cdot f(x,y)

$$

and ##U^2## comprises ##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## each time ##f(x,y)## does is each the required and ample situation that ##f(x,y)=0## be an invariant equation.

**Theorem:** The required and ample 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 components. 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## each time ##f(x,y)=0## this curve consists of invariant factors. Curves of this sort usually are not included among the many orbits of the group. In all different circumstances, ##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 alleged 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## might equally properly 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 will probably 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 now we have discovered that ##U.phi=F(phi).## Furthermore, if the curves of the household usually are not orbits of the group, the equation of the household could be chosen in such kind that ##F(phi)## shall develop into any desired operate of ##phi##. Particularly, there isn’t any loss

in assuming the equation so chosen that that is ##1.## For if a given alternative ##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 identical 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 leads to

$$

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 now we have 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 trendy 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 capable of show it for arbitrary differential expressions by a special methodology [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 important theorems say of their unique wording [6]

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

**2.** If the integral ##I## is invariant with respect to a [Lie group] ##G_{inftyrho }##, through which the arbitrary features seem as much as the ##sigma##-th by-product, then there are ##rho## similar 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 contemporary 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 of. Now we take into account easy 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^*

$$

For the reason that practical ##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 named 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 option to proceed is:

(a) Decide the features ##psi,varphi##, i.e. the transformations, that are thought of.

(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 relentless ##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 motive why the vitality of the particle is conserved, and what’s its vitality?

(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 rely 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 now we have 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 reveals that invariance below totally different transformations leads to totally different dialog portions.

### Sources

Sources

[1] P.J. Olver, Functions of Lie Teams to Differential Equations, New York 1986, Springer, GTM 107

[2] M.S. Lie, Begründung einer Invarianten-Theorie der Berührungs-Transformationen, Mathematische Annalen 1874, Vol. 8, pages 215-303

[3] M.S. Lie, Classification und Integration von gewöhnlichen Differentialgleichungen zwischen xy, die eine Gruppe von Transformationen gestatten, Leipzig 1883

[4] A. Cohen, An Introduction to Lie Principle of One-Parameter Teams, Baltimore 1911

[5] A.E. Noether, Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1918, Invarianten beliebiger Differentialausdrücke, pages 37-44

[6] A.E. Noether, Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 1918, Invariante Variationsprobleme, pages 235-257

[7] Instance for Noether’s theorem, Full Resolution Guide, July 2018, I-3, pages 507ff.

https://www.physicsforums.com/threads/solution-manuals-for-the-math-challenges.977057/

[8] A.Cohen, Elementary Treatise on Differential Equations, Baltimore 1906

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