A Tale of Two Theorems - School of MathematicsEmmy Noether 1918 and after \In 1918, E. Noether proved two remarkable theorems relating symmetry groups of a variational integral to

$Page 1: A Tale of Two Theorems - School of MathematicsEmmy Noether 1918 and after \In 1918, E. Noether proved two remarkable theorems relating symmetry groups of a variational integral to$
A Tale of Two Theorems

Yvette Kosmann-SchwarzbachCentre de Mathematiques Laurent Schwartz, Ecole Polytechnique, France

Symmetries of Differential Equations:Frames, Invariants and Applications

A Conference in Honor of the 60th Birthday of Peter OlverSchool of Mathematics, University of Minnesota, Minneapolis, 17-20 May 2012

Conference in Honor of Peter Olver A Tale of Two Theorems

Emmy Noether 1918 and after

“In 1918, E. Noether proved two remarkable theorems relatingsymmetry groups of a variational integral to properties of itsassociated Euler–Lagrange equations.”

Peter Olver,Introduction to Applications of Lie Groups to Differential

Equations, GTM 107, 1986, 1993.

I Emmy Noether’s “Invariante Variationsprobleme”Nachrichten von der Koniglichen Gesellschaft derWissenschaften zu Gottingen, Mathematisch-physikalischeKlasse, 1918, pp. 235-257.

I What motivated Noether’s article?

I What did Noether prove?

I Have these theorems been generalized?

I The generalization of Noether’s two theoems.


A reference and a review

I have studied Noether’s article. See The Noether Theorems.Invariance and Conservation Laws in the Twentieth Century,Springer, 2011.

This book was reviewed by Peter Olver for the Bulletin of the AMS.See Bull. Amer. Math. Soc. (N. S.), posted on November 4, 2011.

Review to appear in print. Until it is printed, visible athttp://www.ams.org/journals/bull/0000-000-00/


What prompted Noether’s article in 1918?

In 1915 Felix Klein and David Hilbert invited Noether to Gottingento help them understand the full implications of the general theoryof relativity.

• Noether’s expertise in invariant theory.

Her thesis (1907) under Paul Gordan in Erlangen was titled“On the Construction of the System of Forms of a TernaryBiquadratic Form”.It was the search for the invariants of a ternary biquadratic form,i.e., of a homogeneous polynomial of degree 4 in 3 variables.

• Elucidation of the further implications of Einstein’s generaltheory of relativity.

She proved a conjecture made by Hilbert concerning the nature ofthe law of conservation of energy.


Noether’s postcard to Felix Klein, 15 February 1918 (verso)


What variational problems was Noether considering?

“We consider variational problems which are invariant under acontinuous group (in the sense of Lie). [...] What follows thusdepends upon a combination of the methods of the formal calculusof variations and of Lie’s theory of groups.”

Noether considers a general n-dimensional variational problem oforder κ for an Rµ-valued function (n, µ and κ arbitrary integers),

I =

∫· · ·∫

f

(x , u,

∂u

∂x,∂2u

∂x2, · · ·

)dx ,

where x = (x1, . . . , xn) = (xα) are the independent variables, andu = (u1, . . . , uµ) = (ui ) are the dependent variables.Noether states her conventions: “I omit the indices here, and in

the summations as well whenever it is possible, and I write∂2u

∂x2for

∂2uα∂xβ∂xγ

, etc.” and “I write dx for dx1 . . . dxn for short.”


The theorems of Noether in English translation

“In what follows we shall examine the following two theorems:

I. If the integral I is invariant under a [group] Gρ, then there are ρlinearly independent combinations among the Lagrangianexpressions which become divergences – and conversely, thatimplies the invariance of I under a [group] Gρ. The theoremremains valid in the limiting case of an infinite number ofparameters.

II. If the integral I is invariant under a [group] G∞ρ dependingupon arbitrary functions and their derivatives up to order σ, thenthere are ρ identities among the Lagrangian expressions and theirderivatives up to order σ. Here as well the converse is valid1.”

1For some trivial exceptions, see §2, note 13.Conference in Honor of Peter Olver A Tale of Two Theorems

Noether’s proof of Theorem I

Noether assumes that the action integral I =∫

fdx is invariant.Actually, she assumes a more restrictive hypothesis, the invarianceof the integrand, fdx , which is to say δ(fdx) = 0. This hypothesisis expressed by the relation,

δf + Div(f ·∆x) = 0,

Here δf is the variation of f for the variation

δui = ∆ui −∑ ∂ui

∂xλ∆xλ.

• Noether introduces the components of the evolutionaryrepresentative δ of the vector field δ.

Remark. δ is also called the “vertical representative” of δ.

• δf is the Lie derivative of f in the direction of δ.


δ is a generalized vector field

• δ is a generalized vector field, not a vector field on the trivialvector bundle Rn × Rµ → Rn.In fact, ifδ =

∑nα=1 Xα(x) ∂

∂xα +∑µ

i=1 Y i (x , u) ∂∂ui ,

thenδ =

∑µi=1

(Y i (x , u)− Xα(x)ui

α

)∂∂ui , where ui

α = ∂ui

∂xα .

Generalized vector fields would be re-discovered much later byHarold Johnson (1964), Robert Hermann (1965), then in, a jointpaper, by Robert L. Anderson, Sukeyuki Kumei and Carl Wulfman(1972). In 1979, Robert Anderson and Nail Ibragimov called themLie-Backlund transformations (a misnomer).Locally, they are written,

X =n∑

α=1

Xα(x)∂

∂xα+

µ∑i=1

Y i

(x , u,

∂u

∂x,∂2u

∂x2, · · ·

)∂

∂ui.


End of Noether’s proof of Theorem I

By integrating by parts, Noether obtains the identity∑ψi δui = δf + Div A,

where the ψi ’s are the “Lagrangian expressions”, i.e., thecomponents of the Euler–Lagrange derivative of L,and A is linear in δu and its derivatives.In view of the invariance hypothesis which is expressed byδf + Div(f ·∆x) = 0, this identity can be written∑

ψi δui = Div B, with B = A− f ·∆X .

Therefore B is a conserved current for the Euler–Lagrangeequations of L.

QEDThe equations Div B = 0 are the conservation laws that aresatisfied when the Euler–Lagrange equations ψi = 0 are satisfied.


The converse of Theorem I

Noether proves that the existence of ρ “linearly independentdivergence relations” implies the invariance under a (Lie) group ofsymmetries of dimension ρ, by passing from the infinitesimalsymmetries to invariance under their flows, provided the vectorfields ∆u and ∆x are ordinary vector fields.

In the general case, the existence of ρ linearly independentconservation laws yields infinitesimal invariance under a Lie algebraof infinitesimal symmetries of dimension ρ.


Theorem II

Theorem II deals with a symmetry group depending on arbitraryfunctions—such as the group of diffeomorphisms of the space-timemanifold and, more generally, the groups of all gauge theories thatwould be developped later. Noether shows that to such symmetriesthere correspond identities satisfied by the variational derivatives,and conversely.The assumption is that “the integral I is invariant under a [group]G∞ρ depending upon arbitrary functions and their derivatives upto order σ”, i.e., Noether assumes the existence of ρ infinitesimalsymmetries of the Lagrangian, each of which depends linearly onan arbitrary function p (depending on λ = 1, 2, . . . , ρ) of thevariables x1, x2, . . . , xn, and its derivatives up to order σ.Each such symmetry is defined by a vector-valued linear differentialoperator of order σ, D, with components Di , i = 1, 2, . . . , µ.


The adjoint operator

Noether then introduces, without giving it a name or a particularnotation, the adjoint operator, (Di )

∗, of each of the Di ’s. Byconstruction, (Di )

∗ satisfies

ψi Di (p) = (Di )∗(ψi )p + Div Γi ,

where Γi is linear in p and its derivatives. The symmetryassumption and an integration by parts imply

µ∑i=1

ψi Di (p) = Div B.

This relation implies

µ∑i=1

(Di )∗(ψi ) p = Div(B −

µ∑i=1

Γi ).


Identities

Since p is arbitrary, by Stokes’s theorem and the Du Bois-Reymondlemma,

µ∑i=1

(Di )∗(ψi ) = 0,

Thus, for each λ = 1, 2, . . . , ρ, there is a differential relationamong the components ψi of the Euler–Lagrange derivative of theLagrangian f that is identically satisfied.

Noether explains the precautions that must be taken—theintroduction of an equivalence relation on the symmetries—for theconverse to be valid.


Improper conservation laws

Noether observes that each identity may be written∑µi=1 aiψi = Divχ, where χ is defined by a linear differential

operator acting on the ψi ’s. She then shows that each divergenceDiv B is equal to the divergence of a quantity C , where C vanishesonce the Euler–Lagrange equations, ψi = 0, are satisfied.Furthermore, from the equality of the divergences of B and C , itfollows that

B = C + D

for some D whose divergence vanishes identically, which is to say,independently of the satisfaction of the Euler–Lagrange equations.These are the conservation laws that Noether called improperdivergence relations.


Trivial conservation laws

The quantity C and not only its divergence vanish on ψi = 0.

• C is a trivial conservation law of the first kind, in the sense ofOlver.

The divergence of D vanishes identically, i.e., whether ψi = 0 ornot.

• D is a trivial conservation law of the second kind (Div D is a nulldivergence), in the sense of Olver.


What became of Noether’s improper conservation laws?

In general relativity the improper conservation laws which aretrivial of the second kind are called strong, while the conservationlaws obtained from the first theorem are called weak laws.The strong laws play an important role in the basic papers ofPeter G. Bergmann (1958), Andrzej Trautman (1962) andJoshua N. Goldberg (1980).

In gauge theory, the identities in Noether’s second theorem are atthe basis of Stasheff’s “cohomological physics”. They are used asthe antighosts in the BV construction for Lagrangians withsymmetries. See Ron Fulp, Tom Lada and Jim Stasheff,“Noether’s variational theorem II and the BV formalism” (2003).


Noether’s proof of Hilbert’s conjecture

Hilbert asserted (without proof) in early 1918 that, in the case ofgeneral relativity and in that case only, there are no properconservation laws.Noether shows that the situation is better understood “in the moregeneral setting of group theory.”She explains the apparent paradox that arises from theconsideration of the finite-dimensional subgroups of groups thatdepend upon arbitrary functions.“Given I invariant under the group of translations, then the energyrelations are improper if and only if I is invariant under an infinitegroup which contains the group of translations as a subgroup.”Noether concludes with the striking formula:“The term relativity that is used in physics should be replaced byinvariance with respect to a group.”


After Noether’s article “Invariant Variationsprobleme”

Noether submitted the “Invariant Variationsprobleme” for herHabilitation, finally obtained in 1919,but she never referred to her article in her subsequent publications.

Hermann Weyl, in Raum, Zeit, Materie, performed computationsvery similar to hers, but referred to Noether only in a footnote inthe third (1919) and subsequent editions.

Richard Courant must have been aware of her work because a briefsummary of a limited form of both theorems appears in allGerman, and later English editions of “Courant-Hilbert”, firstpublished in 1924.


Who “invented” divergence symmetries?

In Gottingen, Noether had only one immediate follower, ErichBessel-Hagen (1898-1946), who was Klein’s student.

In 1921, Bessel-Hagen published an article on conservation laws inelectrodynamics in the Mathematische Annalen in which hedetermined in particular those which were the result of theconformal invariance of Maxwell’s equations.

He explained that Klein had posed the problem of “the applicationto Maxwell’s equations of the theorems stated by Miss EmmyNoether regarding the invariant variational problems.”

He formulated the two Noether theorems “slightly more generally”than they had been formulated in her article, and added“I owe these to an oral communication by Miss Emmy Noetherherself.”


Divergence symmetries

Bessel-Hagen considers the divergence symmetries that correspond,not to the invariance of the Lagrangian fdx , but to the invarianceof the action integral

∫fdx , i.e., infinitesimal transformations

satisfying, instead of δ(fdx) = 0, the weaker conditionδ(fdx) = DivC , where C is a vectorial expression.

Noether’s fundamental relation remains valid under this weakerassumption provided that

B = A− f ·∆x

be replaced byB = A + C − f ·∆x .


Have the Noether theorems been generalized?

Except for Bessel-Hagen, not until the 1970’s.

Until then, the so-called “generalizations” were all due to physicistsand mathematicians who had no direct knowledge of her article andthought that they were generalizing it, while they were generalizingthe truncated and restricted version of her first theorem that theyhad read in Edward L. Hill’s article, “Hamilton’s principle and theconservation theorems of mathematical physics” (1951).


A one-to-one map between equivalence classes ofdivergence symmetries and of conservation laws

In the late70’s and early 80’s, using different languages, bothmathematically and linguistically, Peter Olver and Cheri Shakiban,in Minneapolis, and Alexander M. Vinogradov, in Moscow, madegreat advances in the Noether theory.

Define a symmetry of a differential equation to be trivial if itsevolutionary representative vanishes on the solutions of theequation. Then one can formulate the Noether-Olver-Vinogradovtheorem (ca. 1985):

For Lagrangians such that the Euler–Lagrange equations are anormal system, Noether’s correspondence induces a one-to-onemap between equivalence classes of divergence symmetries andequivalence classes of conservation laws.


Non-variational equations

What Franco Magri showed in an article in Italian in 1978 is that,if D is a differential operator and VD is its linearization, searchingfor the restriction of the kernel of the adjoint (VD)∗ of VD to thesolutions of D(u) = 0 is an algorithmic method for thedetermination of the conservation laws for a possibly nonvariationalequation D(u) = 0.For an Euler–Lagrange operator, the linearized operator isself-adjoint. Therefore this result generalizes the first Noethertheorem.This idea is to be found and much developped in the work ofseveral mathematicians, most notably Alexander Vinogradov, ToruTsujishita, Ian Anderson, and ... Peter Olver.


In the modern language of differential geometry

• Trautman in his “Noether equations and conservation laws”(1967, 1972) was the first to present even a part of Noether’sarticle in the language of manifolds, fiber bundles and, inparticular, the jet bundles that had been defined and studied byCharles Ehresmann and his student Jacques Feldbau, and byNorman Steenrod about 1940.

• Stephen Smale published the first part of his article on “Topologyand mechanics” in 1970, in which he proposed a geometricframework for mechanics on the tangent bundle of a manifold.

• Hubert Goldschmidt and Shlomo Sternberg wrote a landmarkpaper in 1973 in which they formulated the Noether theory forfirst-order Lagrangians in an intrisic, geometric fashion.

• Jerrold Marsden published extensively on the theory andapplications of Noether’s correspondence from 1974 until his death.


Moment maps

In the 1970’s, many authors contributed to the “geometrization”of Noether’s first theorem, notably Jedrzej Sniatycki, DemeterKrupka, and Pedro Garcıa.

Also in the 1970’s the theory of the moment map was developped.Moment maps were first defined independently in 1970 byJean-Marie Souriau and Bertram Kostant.The conservation of the moment of a Hamilotnian action is theHamiltonian version of Noether’s first theorem. Souriau called thisresult “le theoreme de Noether symplectique”, although there isnothing Hamiltonian or symplectic in Noether!


Cohomological interpretation

It was Alexander Vinogradov who showed in 1977 that generalizedvector fields are nothing other than ordinary vector fields on thebundle of jets of infinite order of sections of a bundle. BothLagrangians and conservation laws then appear as special types offorms.The divergence operator may be interpreted as a horizontaldifferential, one that acts on the independent variables only.Whence one obtains a cohomological interpretation of Noether’sfirst theorem.The study of the exact sequence of the calculus of variations, andof the variational bicomplex, which constitutes a vastgeneralization of Noether’s theory, was developed in 1975 and laterby W lodzimierz Tulczyjew in Warsaw, by Paul Dedecker inBelgium, by Vinogradov in Moscow, by Tsujishita in Japan, and inthe United States by Anderson and by ... Peter Olver.


Discrete versions of the Noether theorems

A pioneer,

John David Logan, First integrals in the discrete variationalcalculus, Aequationes Mathematicae, 9 (1973), pp. 210–220.

The differentiation operation is replaced by the shift operator.The independent variables are now integers, and the integral isreplaced by a sum, L[u] =

∑n L(n, [u]), where [u] denotes u(n)

and finitely many of its shifts.The variational derivative is expressed in terms of the inverse shift.


Recent advances

For up-to-date results, see

Peter Hydon and Elizabeth Mansfield,A variational complex for difference equations, Found. Comput.Math. 4 (2004), no. 2, 187–217.

Elizabeth Mansfield, Noether’s theorem for smooth, difference andfinite element systems, in Foundations of ComputationalMathematics (Santander, 2005), London Math. Society LectureNote Series 331, 2006.

Peter Hydon and Elizabeth Mansfield, Extensions of Noether’ssecond theorem: from continuous to discrete systems, Proc. R.Soc. Lond. Ser. A Math. Phys. Eng. Sci. 467 (2011), no. 2135.(contains a modern re-formulation of Noether’s second theorem aswell as a discrete version)


Towards a Noether theorem on Lie algebroids

The geometric framework for the variational calculus for first-orderLagrangians in one independent variable, and the symmetries andconservation laws of mechanics, is the tangent bundle of theconfiguration manifold.

There have been several generalizations of this approach, wherethe tangent bundle is replaced by an arbitrary Lie algebroid:Alan Weinstein (Lagrangian mechanics and groupoids, 1992),Paulette Libermann (Lie algebroids and mechanics, 1996).

I shall briefly describe another approach, that of Eduardo Martınez,Janusz Grabowski, Manuel de Leon and collaborators (2001 to2010).See in particular, Eduardo Martınez, Lagrangian mechanics on Liealgebroids, Acta Appl. Math. (2001).


Mechanics on Lie algebroids

A Lie algebroid is a vector bundle π : A→ M equipped with• a Lie bracket on sections, and• an anchor, a vector bundle morphism ρ to the tangent bundle ofthe base,satisfying the Leibniz rule,

∀X ,Y ∈ Γ(A), ∀f ∈ C∞(M), [X , fY ] = f [X ,Y ] + (ρ(X ) · f )Y .

Lie algebroids generalize tangent bundles as well as Lie algebras.

The geometry of first-order Lagrangians in one independentvariable can be extended from TM → M to an arbitrary Liealgebroid A→ M as follows.


Second-order geometry on Lie algebroids

The role of the double tangent bundle T (TM) is played by a Liealgebroid LA, the pull-back Lie algebroid of A over the projectionπ : A→ M, a special case of a construction due to Higgins andMackenzie (1990), for which see Kirill Mackenzie’s book, GeneralTheory of Lie Groupoids and Lie Algebroids, 2005.LA = TA⊕TM A is the subbundle of TA⊕ A that completes thecommutative diagram:

TA⊕TM A → A↓ ↓ ρ

TATπ→ TM

The elements of LA are pairs (X , a) ∈ TA× A such that

(Tπ)(X ) = ρ(a).

• If A = TM, then LA = LTM reduces to TA = T (TM).


The Lie algebroid LA

LA can be made into a Lie algebroid over A, withanchor the projection onto TA, andbracket such that, for X ,Y ∈ Γ(TA) and for section u, vof A which satisfy Tπ ◦ X = ρ ◦ u ◦ π and Tπ ◦ Y = ρ ◦ v ◦ π,

[(X , π∗u), (Y , π∗v)]LA = ([X ,Y ]TA, π∗[u, v ]A) .

An admissible element of LA is a pair (X , a) ∈ TA⊕TM A suchthat X is tangent to A at a ∈ A.Admissible sections play the role of second-order differentialequations on A.


The geometry of LA

There is a vertical endomorphism V of LA that maps a pair(X , a) ∈ LA to the vertical lift of X .The dual map V ∗ is an endomorphism of (LA)∗.There is a Liouville vector field, Z , on LA whose value at (X , a) isthe vertical lift of a.If L is a Lagrangian function on A, pulled back to LA,

θL = V ∗(dL)

is a section of (LA)∗, the Cartan section.If ωL = dθL is nondegenerate, define a section XL of LA by

iXLωL = dEL,

where EL = iZ dL− L is the energy.Then XL is admissible and satisfies the Euler–Lagrange equation,

LXLθL = dL.


Particular cases

• Lie algebra: Euler–Poincare equations,

• Tangent bundle: Euler–Lagrange equations,

• Foliation: holonomic mechanics,

• Action Lie algebroid: Euler–Poisson–Poincare equations(Holm–Marsden–Ratiu, 1998).


The (elementary) Noether theorem on Lie algebroids

An (infinitesimal) symmetry of L is defined as a section a of Asuch that Lac L = 0, where ac is the complete lift of a. In otherwords, a is a symmetry of L if < dL, ac >= 0.The Noether current associated with a section a of A is < θL, a

c >.

Noether’s theorem on Lie algebroids (Eduardo Martınez, 2001)If a section of A is a symmetry of the regular Lagrangian L, thenthe associated Noether current is invariant under the flow of XL.


An appreciation

I quote from Gregg Zuckerman’s “Action principles and globalgeometry”, in Mathematical Aspects of String Theory, S. T. Yau,ed. (World Scientific, 1987):

“E. Noether’s famous 1918 paper, “Invariant variational problems”crystallized essential mathematical relationships amongsymmetries, conservation laws, and identities for the variational or‘action’ principles of physics. [. . . ] Thus, Noether’s abstractanalysis continues to be relevant to contemporary physics, as wellas to applied mathematics (see Olver, 1986).”


Conclusion

• Peter Olver’s book of 1986 (second edition 1993) is still the bestavailable text for a modern, yet legible exposition of the Noethertheory.



Extract form Peter’s letter of June 26, 1984(page 1)

UNIVERSITY OF MINNESOTA School of MathematicsTWIN CITIES 127 Vincent Hall

206 Church Street S.E.Minneapolis, Minnesota 55455

June 26, 1984

Dear Yvette,...Most of my effort now is being spent on the book, which I stillhope to have completed by the end of the summer!...


Another extract from Peter’s letter of June 26, 1984(page 2)

...Sheehan is now 1 year old, and an unbelievable amount of trouble –he gets into everything!...

Time flies!

Happy birthday Peter!


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A Tale of Two Theorems - School of MathematicsEmmy Noether 1918 and after \In 1918, E. Noether proved two remarkable theorems relating symmetry groups of a variational integral to