Lecture 5: Noether-type theorems - University of Kent...Lecture 5: Noether-type theorems Constrained variational symmetries Example The interaction of a scalar particle of mass m and

Lecture 5: Noether-type theorems

Lecture 5: Noether-type theoremsSymmetry Methods for Differential and Difference Equations

Peter Hydon

University of Kent


Outline

1 Noether’s Theorems for P∆Es

2 Constrained variational symmetries

3 A discretization that preserves gauge symmetries

4 Why does everything transfer to P∆Es?

5 Summary: the main results in Lecture 5


Noether’s Theorems for P∆Es

A brief reminder The difference Euler operator with respect toprincipal variables v = (v 1, . . . , vK ) ∈ RK is

Ev = (Ev 1 , . . . ,EvK ) , where Ev α = S−J∂

∂v αJ.

All other variables are considered subsidiary; they are not affectedby variations in v.

Lemma: If A = 0 is in Kovalevskaya form then (under a mildtechnical condition) a CLaw is trivial if and only if

Q (I) ≡ Q(I, [0]) ≡{

EA(C (I, [A])

)}∣∣∣A=0

= 0.

The function Q (I) is called the root of the CLaw.

Corollary: If A = 0 is in Kovalevskaya form then

equal roots←→ equivalent characteristics←→ equivalent CLaws.




For difference variational problems, the action is

L[u] =∑

n

L(n, [u]

),

and the Euler–Lagrange (E–L) equations are

Euα(L) ≡ S−J

{∂L

∂uαJ

}= 0, α = 1, . . . , q. (1)

Generalized symmetries of the E–L equations have generators

X = SJ

(Qα(n, [u])

) ∂

∂uαJ

that satisfy the linearized symmetry condition,

X(Euα(L)

)= 0 when (1) holds.



Symmetries are variational if the E–L equations are unchanged:

X (L) ≡ SJ

(Qα(n, [u])

) ∂L

∂uαJ=

N∑i=1

(Si − I)P i0

(n, [u]

).

Sum by parts, to obtain the Noether equation:

QαEuα(L) =N∑i=1

(Si − I)P i(n, [u]

). (2)

Noether’s Theorem (for P∆Es)A prolonged vector field X is a variational symmetry generator ifand only if its characteristic Q is the characteristic of aconservation law for the Euler–Lagrange equations.



Example Noether’s Theorem applies equally to variational O∆Es(and ODEs), the only difference being that the ‘conservation law’is a first integral, φ.The McMillan map, is an E–L equation:

L = uu1 − α ln(u2 + 1

)−→ Eu(L) ≡ u1 + u−1 −

2αu

u2 + 1= 0

This has a dynamical (that is, generalized – for O∆Es) variationalsymmetry,

Q = (u2+1)(u1−u−1) −→ XL = (Sn−I){

u2 +(u2 − 2α + 1

)u1u−1

}.

The Noether equation is QEu(L) = (Sn − I)φ, where

φ = u2 +(u2 − 2α + 1

)u1u−1 − Q S−1

n

∂L

∂u1

= u2 − 2αu1u−1 +(u2 + 1

)(u−1)2 .



Noether’s Theorem establishes the link between variationalsymmetries and CLaws. It leads to a stronger result for E–Lequations that can be written in Kovalevskaya form.

Theorem If the Euler–Lagrange equations can be written inKovalevskaya form, there is a one-to-one correspondence betweenthe equivalence classes of variational symmetries and theequivalence classes of CLaws of the Euler–Lagrange equations.

Noether’s Second Theorem (for overdetermined systems) also hasa difference analogue.

Definition A linear difference operator is an operator of the form

D = aJ(n) SJ

that has at least two terms.



Noether’s Second Theorem: The Lie algebra of variationalsymmetry generators depends on R independent arbitrary functionsgr (n) if and only if there exist linear difference operators Dαr thatgive R independent difference relations between the E–L equations:

Dαr Eα(L) ≡ 0, r = 1, . . . ,R, (3)

Given the relations (3), the corresponding characteristics are

Qα(n, [u; g]) =(Dαr)†

(gr ), (4)

where † denotes the adjoint. In the formal theory of P∆Es (resp.PDEs), the adjoint is taken using the `2 (resp. L2) inner product.

Conversely, given the characteristics, the corresponding differencerelations (3) have

Dαr =∑

J

{S−J

(∂Qα(n, [u; g])

∂SJgr

)}S−J .



A key part of the proof of Noether’s Second Theorem uses the factthat Euler–Lagrange operators annihilate divergences.

For each r , apply Egr to the Noether equation (2), to get

0 = Egr{

Qα(n, [u; g]) Euα(L)}≡ S−J

(∂Qα(n, [u; g])

∂grJEuα(L)

).

(5)This gives the difference operators Dαr explicitly.

It is convenient to regard (5) as ‘new’ Euler–Lagrange equationsthat arise when g is varied in the functional

L [u; g] =∑

L(n, [u; g]),

which has the Lagrangian

L(n, [u; g]) = Qα(n, [u; g]) Euα(L(n, [u])).



Example Consider the variational problem whose Lagrangian is

L = (v − u1,0)(w − u0,1)− µ(v0,1 − w1,0)2,

where µ is a non-zero constant. Each variational point symmetrygenerator that depends on one or more completely arbitraryfunctions of (m, n) is of the form

X = g(m, n) ∂u + g(m + 1, n) ∂v + g(m, n + 1) ∂w ;

here XL ≡ 0. The Euler–Lagrange equations are

Eu(L) ≡− w−1,0 + u−1,1 − v0,−1 + u1,−1 = 0,

Ev (L) ≡ w − u0,1 − 2µ(v − w1,−1) = 0,

Ew (L) ≡ v − u1,0 + 2µ(v−1,1 − w) = 0.



As X depends on a single arbitrary function, Noether’s SecondTheorem gives one difference relation between the Euler–Lagrangeequations. So (5) is

Eg

{gEu(L) + g1,0Ev (L) + g0,1Ew (L)

}≡ 0,

which amounts to the difference relation

Eu(L) + S−1m

(Ev (L)

)+ S−1

n

(Ew (L)

)≡ 0.



The three Euler–Lagrange equations are related because theLagrangian involves only two dependent variables (and theirshifts), namely v = v − u1,0 and w = w − u0,1. If we write

L = v w − µ(v0,1 − w1,0)2,

the Euler–Lagrange equations corresponding to variations in v andw are

Ev (L) ≡ w−2µ(v−w1,−1

)= 0, Ew (L) ≡ v +2µ

(v−1,1−w

)= 0.

These two equations determine v and w completely, but v and ware determined only up to an arbitrary choice of u. This is anexample of gauge freedom, which is expressed by the gaugesymmetry generator X .


Constrained variational symmetries

A gap: Variational symmetry generators may depend on functionsof the independent variables that are constrained in some way.

Suppose that R functions, gr (n), are subject to a set of S lineardifference constraints,

Ksrgr = 0, s = 1, . . . ,S , (6)

and that this set is complete (that is, no additional constraints areneeded). The constraints may be applied by taking variations withrespect to g of the augmented Lagrangian

L(n, [u; g;λ]) = Qα(n, [u; g]) Euα(L(n, [u]))− λsKsr (gr ); (7)

here λ1, . . . , λS are Lagrange multipliers.

This leads to a result that bridges the two Noether theorems.



A bridging theorem (H. & Mansfield)

Suppose that the Lie algebra of variational symmetry generatorsdepends on R independent functions gr (n) that are subject to acomplete set of linear difference constraints,

Ksrgr = 0, s = 1, . . . ,S .

Then there are R − S independent difference relations between theE–L equations, which are obtained by eliminating all λs from

S−J

(∂Qα(n, [u; g])

∂SJgrEα(L)

)≡ (Ksr )†(λs), r = 1, . . . ,R.

The above relations also give the conservation laws correspondingto Noether’s (First) Theorem.



Example The Euler–Lagrange equations for the Lagrangian

L =(u1,0 + v0,1

)2+ 2(u1,0 + v0,1

)(w1,0 + z0,1

)− (u − w)(v − z)

areEu(L) ≡ 2

(u + v−1,1 + w + z−1,1

)− v + z = 0,

Ev (L) ≡ 2(u1,−1 + v + w1,−1 + z

)− u + w = 0,

Ew (L) ≡ 2(u + v−1,1

)+ v − z = 0,

Ez(L) ≡ 2(u1,−1 + v

)+ u − w = 0.

There is a variational symmetry generator,

X = g1(m, n)(∂u + ∂w

)+ g2(m, n)

(∂v + ∂z

),

for every pair of functions g1, g2 that satisfy g11,0 + g2

0,1 = 0, so theaugmented Lagrangian (7) is

L = g1{

Eu(L) + Ew (L)}

+ g2{

Ev (L) + Ez(L)}− λ

(g1

1,0 + g20,1

).



Taking variations with respect to g1 and g2 yields the identities

Eu(L) + Ew (L)− λ−1,0 ≡ 0,

Ev (L) + Ez(L)− λ0,−1 ≡ 0.(8)

Eliminate λ from (8) to obtain the difference relation

Sm{

Eu(L) + Ew (L)}− Sn

{Ev (L) + Ez(L)

}≡ 0.

The remainder of (8) amounts to

λ = 4(u1,0 + v0,1

)+ 2(w1,0 + z0,1

). (9)

This gives a conservation law for every g1, g2 that satisfy theconstraint, namely

C =(Sm − I

) (g2−1,0 λ

)+(Sn − I

) (g1

0,−1λ),

where λ is given by (9).



What is different for PDEs?

The Noether theorems and the bridging theorem all have analoguesfor PDEs, which are derived in the same manner.

Integration by parts replaces summation by parts, and differentialoperators replace difference operators. In particular, SJ and itsadjoint S−J are replaced (respectively) by DJ and its adjoint,

(−D)J = (−D1) j1(−D2) j2 · · · (−DN) jN = (−1)|J|DJ.

For instance, the PDE formula for variational symmetries that aresubject to S differential constraints, Ksr (gr ) = 0, is

(−D)J

(∂Qα(x, [u; g])

∂gr,JEα(L)

)≡ (Ksr )†(λs). (10)

If the arbitrary functions are unconstrained, set λs = 0 to get thedifferential relations in Noether’s Second Theorem.



Example The interaction of a scalar particle of mass m andcharge e with an electromagnetic field has a variationalformulation with well-known gauge symmetries.

The independent variables are the standard flat space-timecoordinates x = (x0, x1, x2, x3), where x0 denotes time. Themetric is η = diag{−1, 1, 1, 1}.

The dependent variables are the complex-valued scalar ψ (which isthe wavefunction for the particle), its complex conjugate, ψ∗, andthe real-valued electromagnetic four-potential Aµ, µ = 0, 1, 2, 3.



The Lagrangian is

L =1

4FµνFµν + (∇µψ) (∇µψ)∗ + m2ψψ∗ (11)

whereFµν = Aν,µ − Aµ,ν , ∇µ = Dµ + ieAµ.

Therefore the Euler–Lagrange equations are

0 = Eψ(L) ≡ − (∇µ∇µψ)∗ + m2ψ∗,

0 = Eψ∗(L) ≡ −∇µ∇µψ + m2ψ,

0 = Eσ(L) ≡ ieψ (∇σψ)∗ − ieψ∗∇σψ + ησαFαβ,β ,

where Eσ is obtained by varying Aσ.



The variational symmetries include the gauge symmetries

ψ 7→ exp(−ieξ)ψ, ψ∗ 7→ exp(ieξ)ψ∗, Aσ 7→ Aσ + ησαξ,α,

where ξ is an arbitrary real-valued function of x; the correspondinggauge symmetry characteristics have components

Qψ = −ieψg, Qψ∗= ieψ∗g, Qσ = ησαg,α,

where g is an arbitrary real-valued function of x. Consequently,

Eg

{QψEψ(L) + Qψ∗

Eψ∗(L) + QσEσ(L)}

= −ieψEψ(L) + ieψ∗Eψ∗(L)− Dα (ησαEσ(L)) ,

and so Noether’s Second Theorem yields the differential relation

−ieψEψ(L) + ieψ∗Eψ∗(L)− Dα (ησαEσ(L)) ≡ 0.


A discretization that preserves gauge symmetries


Christiansen & Halvorsen (2011) discovered a finite differenceapproximation to the last example that preserves all variationalgauge symmetries. The mesh x(n) is uniformly-spaced in eachdirection, with step lengths hµ. It is helpful to define scaledforward difference operators and their adjoints,

Dµ =Sµ − I

hµ, D

†µ = −

id− S−1µ

hµ, µ = 0, . . . , 3,

to obtain the correct limiting behaviour as hµ tends to zero.

Let ψ(n) denote the approximation to the wavefunction ψ at x(n).Christiansen & Halvorsen used a Yee discretization (see their paperfor details) to approximate Aµ on the edge that connects the

points x(n) and Sµx(n); we denote this approximation by Aµ(n).



Up to a sign, the Lagrangian for the scheme is

L =1

4Fµν Fµν +

(∇µψ

)(∇µψ

)∗+ m2ψψ∗ (12)

whereFµν = DµAν − DνAµ,

∇µ =1

hµ

{Sµ − exp

(− iehµAµ

)I}

= Dµ+1

hµ

{1− exp

(− iehµAµ

)}I.

Hence the Euler–Lagrange equations for the difference scheme are

0 = Eψ

(L) ≡ ∇†µ

(∇µψ

)∗+ m2ψ∗,

0 = Eψ∗(L) ≡

(∇†µ

)∗∇µψ + m2ψ,

0 = Eσ(L) ≡ ie exp(− iehσAσ

)ψ(∇σψ

)∗− ie exp

(iehσAσ

)ψ∗∇σψ − ησαD

†β Fαβ ;

here Eσ is obtained by varying Aσ.



The above discretization preserves the gauge symmetries asvariational symmetries. Explicitly, these are

ψ 7→ exp(−ieξ)ψ, ψ∗ 7→ exp(ieξ)ψ∗, Aσ 7→ Aσ + ησαDαξ,

where now ξ is an arbitrary real-valued function of n. Thecharacteristics of the gauge symmetries have components

Qψ = −ieψ g, Qψ∗= ieψ∗g, Qσ = ησαDαg,

where g is an arbitrary real-valued function of n. Consequently, thediscrete version of Noether’s Second Theorem immediately yields

−ieψEψ

(L) + ieψ∗Eψ∗(L) + D

†α (ησαEσ(L)) ≡ 0,

which Christiansen & Halvorsen obtained with the aid of Noether’sfirst theorem.


Why does everything transfer to P∆Es?


Q. Can’t one use equivalent formulations? This one is useful:

exp{Div(F)

}≡∏N

i=1 Si exp{

F i (n, [u])}∏N

i=1 exp{

F i (n, [u])} = 1 when A = 0.

A. For PDEs, the Euler operator E annihilates all divergences;indeed, the variational complex over RN is exact.

· · ·ΛN−2,0 dh−→ ΛN−1,0 dh−→ ΛN,0 E−→ ΛN,1∗ · · ·

Divergences are the elements of ker(E).



Difference operators also produce a variational complex; ker(E) isthe set of all (difference) divergence expressions in standard form.

Key features of the difference complex

dh is replaced by a difference operator;

the algebraic structure is unchanged, even though continuityis lost (ordering replaces continuity);

cohomology groups are preserved if “sufficiently many” pointsare used.

For details of difference forms and the variational complex, see H.& Mansfield (2004) and Mansfield & H. (2008).

Linyu Peng has developed the difference version of the freevariational bicomplex in his PhD thesis (U. of Surrey, 2013).


Summary: the main results in Lecture 5

Summary: the main results in Lecture 5

Noether’s theorems on variational symmetries apply equally toPDEs and P∆Es.

There is a result that bridges the gap between Noether’stheorems.

For at least some gauge theories, there exist discretizationsthat preserve the relations produced by Noether’s SecondTheorem.

A key reason for the close analogy between PDE and P∆Emethods is the existence of identical cohomological structures.

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Lecture 5: Noether-type theorems - University of Kent...Lecture 5: Noether-type theorems Constrained variational symmetries Example The interaction of a scalar particle of mass m and