Symmetries of differential equations
This is an overview of the basic method for the analysis of continuous symmetries of differential equations, mostly collecting some useful formulas rather than being self-contained in any sense. The key concept is the notion of a prolongation of the vector field, representing the generator of a symmetry, acting jointly on the independent variables (hereafter called “coordinates”) and the dependent variables (called “fields”). Let us denote the coordinates and fields asxµandφA, respectively. A generalized symmetry is represented by the infinitesimal transformation rules
x0µ−xµ= ωµ[φ], φ0A(x0)−φA(x) = ξA[φ], (1)
where is the infinitesimal parameter of the transformation and the square brackets indicate that the functions ωµ[φ] and ξA[φ] in general depend on the coordinates, fields as well as their derivatives. The same transformation can be equivalently encoded in a generalized vector field,
v ≡ωµ[φ] ∂ ∂xµ +ξ
A[φ] ∂
∂φA, (2)
acting on the geometric space of the independent and dependent variables. A set of differential equations can be thought of geometrically as algebraic equations on the jet space spanned on the coordinates xµ, the fieldsφA, and their derivativesφA
J, where J is a multi-index, collecting one or more spacetime indices µ, ν, . . .. The great advantage of working in the jet space is that we can treat all ofxµ, φA, φA
J as independent variables, as far as the symmetry analysis of differential equations is concerned. The information about the coordinate dependence of φA(x) and the fact that φAJ are actual derivatives ofφAwith respect to the coordinatesxµ, is encoded in the prolongation of the vector field (2) to the jet space, which can be evaluated separately beforehand. The (infinite) prolongation can be generally expressed in two equivalent forms,
prv =ωµ ∂ ∂xµ +
X
J
DJ(ξA−ωµ∂µφA) +ωµ∂µφAJ
∂
∂φA J
=ωµDµ+
X
J
DJ(ξA−ωµ∂µφA) ∂ ∂φA
J
≡ωµDµ+
X
J
DJQA ∂ ∂φA
J ,
(3)
where DJ is thetotal derivativewith respect to a set of coordinates determined by the multiindex J, taking into account the coordinate dependence of the fields and their derivatives. Moreover, QA≡ξA−ωµ∂
µφA is the characteristic of the field v.
With certain technical assumptions, the generalized vector field v in Eq. (2) generates a sym-metry of a set of differential equations, Fa[φ] = 0, if and only if the functions Fa[φ] are left unchanged by the prolongation of v for all fields φA satisfying the set of differential equations,
prv(Fa[φ]) = 0 whenever Fa[φ] = 0. (4)
In practice, working out the prolongation prv and its action on the differential equations is a straightforward, if daunting, task. Imposing the condition that the fields satisfy the differential equations (that is, that they are “on-shell”) consistently without solving these equations may, however, be nontrivial. In the special case that the differential equations are of first order in time derivatives, it is best to use the equations to eliminate all time derivatives from prv(Fa[φ]).
Owing to the presence of the total derivative Dµ on the right-hand side of Eq. (3), a set of differential equations is invariant under prv if and only if it is invariant under the prolongation of theevolutionary representative ofv,vQ≡QA[φ]∂φ∂A. In plain terms, any transformation (1)
of coordinates and fields is, at least locally, equivalent to a transformation of fields only, if we set ¯ωµ= 0 and ¯ξA ≡QA =ξA−ωµ∂
Explicit expressions for vector field prolongations
The sums in Eq. (3) include an infinite number of contributions. Explicit finite expressions can, however, be worked out on the assumption that the differential equations and the coefficient functions in the vector field v only include derivatives of the fields up to certain finite order.
One dependent and two independent variables
This special case corresponds to the field theory of a single scalar field in two spacetime di-mensions. In accord with the notation common in mathematical physics, we will denote the independent variables asx, tand the dependent variable asu(x, t). Derivatives will be indicated by subscripts. For the sake of simplicity, we will only include(Lie) point symmetries where the coefficient functions in v do not depend on the derivatives ofu. The generating vector field of a point symmetry then reads
v =ξ(x, t, u) ∂
∂x +τ(x, t, u) ∂
∂t+φ(x, t, u) ∂
∂u, (5)
and its prolongation to up to two derivatives of u has the generic form
pr(2)v =v+φx ∂ ∂ux
+φt ∂ ∂ut
+φxx ∂ ∂uxx
+φxt ∂ ∂uxt
+φtt ∂ ∂utt
. (6)
The coefficient functions φx, φt, φxx, φxt, φtt can be worked out with a little effort,
φx =φx+ux(φu−ξx)−utτx−u2xξu−uxutτu, φt=φt−uxξt+ut(φu−τt)−uxutξu−u2tτu,
φxx =φxx+ux(2φxu−ξxx)−utτxx+u2x(φuu−2ξxu)−2uxutτxu−u3xξuu−u2xutτuu +uxx(φu−2ξx)−2uxtτx−3uxxuxξu−uxxutτu−2uxtuxτu,
φxt=φxt+ux(φtu−ξxt) +ut(φxu−τxt)−u2xξtu+uxut(φuu−ξxu−τtu)−u2tτxu
−u2xutξuu−uxu2tτuu−uxxξt+uxt(φu −ξx−τt)−uttτx
−uxxutξu−2uxtuxξu−2uxtutτu−uttuxτu,
φtt =φtt−uxξtt+ut(2φtu−τtt)−2uxutξtu+u2t(φuu−2τtu)−uxu2tξuu−u3tτuu
−2uxtξt+utt(φu−2τt)−2uxtutξu−uttuxξu−3uttutτu.
(7)
General case
Let us now return to the general case of an arbitrary number of dependent and independent variables, and to the notation introduced in Eq. (1). We will still deal with point symmetries only though. The vector field v in Eq. (2) then takes the generic form
v =ωµ(x, φ) ∂ ∂xµ +ξ
A(x, φ) ∂
Its prolongation to up to two derivatives of φA becomes
pr(2)v =
ωµ ∂ ∂xµ +ξ
A ∂ ∂φA
+
∂µξA+ ∂ξ A
∂φB∂µφ B−
∂µων∂νφA− ∂ω
ν
∂φB∂µφ B
∂νφA
∂ ∂(∂µφA)
+
∂µ∂νξA+ ∂ 2ξA
∂xµ∂φB∂νφ B
+ ∂ 2ξA
∂xν∂φB∂µφ B−
∂µ∂νωλ∂λφA+ ∂ 2ξA
∂φB∂φC∂µφ B
∂νφC
− ∂
2ωλ
∂xµ∂φB∂λφ A
∂νφB− ∂
2ωλ
∂xν∂φB∂λφ A
∂µφB− ∂
2ωλ
∂φB∂φC∂λφ A
∂µφB∂νφC (9)
+ ∂ξ A
∂φB∂µ∂νφ B−∂
µωλ∂λ∂νφA−∂νωλ∂λ∂µφA
− ∂ω
λ
∂φB∂λ∂νφ A∂
µφB− ∂ωλ ∂φB∂λ∂µφ
A∂ νφB−
∂ωλ ∂φB∂λφ
A∂ µ∂νφB
∂ ∂(∂µ∂νφA)
.
Examples
Below, we work out three explicit examples, dealing with equations well-known in mathematical physics. The first two illustrate the prolongation formulas (7) and (9). The last one shows how to find higher-order generalized symmetries in their evolutionary form.
Point symmetries of the heat equation in one spatial dimension
In one spatial dimension and upon introducing dimensionless coordinates, the heat equation can be brought to the form
ut=uxx. (10)
This is invariant under the action of the vector field (5) ifφt =φxx for all on-shell fieldsu(x, t). We take the on-shell condition into account by replacing ut with uxx everywhere in Eq. (7). The relevant pieces thereof then become
φt →φt−uxξt+uxx(φu−τt)−uxxuxξu −u2xxτu,
φxx →φxx+ux(2φxu−ξxx) +u2x(φuu−2ξxu)−u3xξuu+uxx(φu−2ξx−τxx)
−uxxux(3ξu+ 2τxu)−uxxu2xτuu−u2xxτu−2uxxxτx−2uxxxuxτu.
(11)
Upon comparing the coefficients of the individual operators in these expressions, we obtain the set ofdetermining equations for the point symmetries of the heat equation,
φt−φxx = 0, ξt−ξxx+ 2φxu= 0, φuu−2ξxu= 0, ξuu= 0, τt−τxx−2ξx = 0, ξu+τxu= 0, τuu = 0, τx = 0, τu = 0.
(12)
It is easy to see that τ must be independent of x and u, whereas ξ must be independent of u. The remaining conditions thus simplify to
The last two of these imply that φ(x, t, u) must be linear in u and that ξ(x, t) must be linear in x. It is now a simple matter to find the general solution to the determining equations,
ξ(x, t) = 12a1+a2t
x+b0+b1t, τ(t) = a0+a1t+a2t2,
φ(x, t, u) = c− 1 2b1x−
1 4a2x
2− 1 2a2t
u+d(x, t),
(14)
where d(x, t) is an arbitrary solution to the heat equation, dt =dxx, and a0, a1, a2, b0, b1, c are constant coefficients. The geometric interpretation of the various independent symmetries en-coded in the generator (5) is as follows:
• a0: time translation,
• a1: (nonrelativistic) scaling of space and time,
• a2: a nontrivial symmetry without a clear geometric interpretation,
• b0: space translation,
• b1: Galilei boost,
• c: linearity of the heat equation under rescaling of u,
• d(x, t): linearity of the heat equation under addition.
Point symmetries of the Laplace/wave equation in an arbitrary number of dimensions
Consider the general second-order partial differential equation of the type
gµν∂µ∂νφ≡φ = 0, (15)
where gµν is the inverse of a metric g
µν assumed to be constant (coordinate-independent) and invertible but otherwise arbitrary. This class of equations includes as important special cases the Laplace equation and the wave equation in an arbitrary numberdof spacetime dimensions. Its point symmetries can be found with the help of Eq. (9). Removing the field indices, the action of pr(2)v on the equation reduces to
pr(2)v(φ) = ξ+ 2 ∂ 2ξ
∂xµ∂φ∂ µ
φ−ωµ∂µφ+ ∂2ξ ∂φ2(∂µφ)
2− 2 ∂
2ων
∂xµ∂φ∂ µ
φ∂νφ− ∂2ων
∂φ2 (∂µφ) 2
∂νφ
+ ∂ξ
∂φφ−2∂µω ν∂µ∂
νφ−2 ∂ων
∂φ ∂µ∂νφ∂
µφ− ∂ων
∂φ φ∂νφ. (16)
By applying the on-shell condition, the determining equations for the symmetries of the differ-ential equation (15) thus reduce to the following conditions,
ξ= 0, ωµ−2 ∂
2ξ
∂xµ∂φ
= 0, ∂ 2ωµ
∂xν∂φ + ∂
2ων
∂xµ∂φ
−gµν∂ 2ξ
∂φ2 = 0,
∂ωµ
∂φ = 0, (17)
along with the requirement that the traceless symmetric part of ωµν vanishes. [This last con-dition stems from projecting out the part of the operator ∂µ∂νφ that does not vanish as a consequence of Eq. (15).] These conditions can be equivalently cast as follows:
• ωµ is independent of φ,
• ωµ= 2∂µα,
• ∂µων+∂νωµ= 2dgµν∂·ω,
where we introduced the conventional dot notation for inner product induced by the metricgµν, ∂·ω≡gµν∂µων. The last of the above conditions represents the conformal Killing equation. The strategy to attack the problem is to first solve this equation, and then find the corresponding allowed functions α(x) and β(x).
Let us introduce the shorthand notation Ω ≡ 1
d∂ ·ω. By taking the alternating sum of the derivative of the Killing equation with respect to xλ with cyclically permuted indices λ, µ, ν, we arrive at an expression for the second partial derivative of ωµ in terms of Ω alone,
∂µ∂νωλ = (gνλ∂µ+gλµ∂ν −gµν∂λ)Ω. (18)
By further acting on this with ∂λ and using that Ω = 0, which follows from the conditions ωµ = 2∂µα and α = 0, we find that (d−2)∂µ∂νΩ = 0. The case of d = 2 is special in that the space of linearly independent solutions to the conformal Killing equation is infinite-dimensional. Assuming from now ond6= 2, we infer∂µ∂νΩ = 0, and Eq. (18) then immediately implies that all third partial derivatives of ωµ vanish. We can now use as an Ansatz the most general quadratic polynomial in coordinates,
ωµ(x) = aµνλxνxλ+bµνxν +cµ (19)
with aµνλ = aµλν. The constant piece, cµ, is not constrained by the Killing equation and can be chosen arbitrarily. The linear piece is forced by (the coordinate-independent part of) the conformal Killing equation to take the form
bµν =σgµν+τµν, (20)
whereσ is a constant andτµν is an arbitrary antisymmetric matrix. Finally, the quadratic part of ωµ(x) is constrained by Eq. (18), which implies
aµνλ = 1
d(gµνaααλ+gλµaααν−gνλaααµ). (21) The whole tensor coefficient aµνλ is thus uniquely determined by the vector nµ ≡ 1daααµ, in terms of which we have aµνλxνxλ = 2xµn·x−nµx2. Now that we have the quadratic part of ωµ(x), the conditionωµ= 2∂µα leads immediately to α(x) = n·x+γ with an undetermined constant γ.
Altogether, the general solution to the determining equations for symmetries of Eq. (15) reads
ξ(x, φ) = (n·x+γ)φ+β(x),
ωµ(x) = 2xµn·x−nµx2+σxµ+τµνxν +cµ, (22)
where β(x) is an arbitrary solution to β = 0, γ and σ are constant scalars, cµ and nµ are constant vectors, and τµν is a constant antisymmetric matrix. The geometric interpretation of the various independent symmetries included here is as follows:
• cµ: spacetime translation,
• τµν: spacetime rotation,
• σ: spacetime dilatation,
• nµ: special conformal transformation,
• γ: linearity of Eq. (15) under rescaling of φ,
Higher-order generalized symmetries of the Laplace/wave equation
According to Eq. (3), the prolongation of the evolutionary representative of a vector field, vQ, is determined completely by the total derivatives of the characteristic of the field, QA[φ]. The highest order of total derivative needed for a given set of differential equations corresponds to the order of the equations. Denoting the arguments ofQA[φ] collectively asχB ={x, φ, ∂φ, . . .} where, with a slight abuse of notation, B is now a multiindex that labels both the fields and their derivatives, the total derivatives can be worked out in a compact form using the chain rule. Up to the second order, they are given by
DµQA = (∂χBQA)∂µχB,
DµDνQA = (∂χBQA)∂µ∂νχB+ (∂χB∂χCQA)∂µχB∂νχC.
(23)
The explicit form of these depends on the assumed highest order of derivatives of φA that the characteristic depends on, that is, on the order of the generalized symmetry. For instance, for first-order symmetries where QA[φ] =QA(x, φ, ∂φ), Eq. (23) expands as
DµQA =∂µQA+ (∂φBQA)∂µφB+ (∂∂ νφBQ
A
)∂µ∂νφB, DµDνQA = (∂φBQA)∂µ∂νφB+ (∂∂
λφBQ
A
)∂µ∂ν∂λφB+∂µ∂νQA+ (∂φB∂φCQA)∂µφB∂νφC
+ (∂∂κφB∂∂λφCQ
A
)∂µ∂κφB∂ν∂λφC+ (∂µ∂φBQA)∂νφB+ (∂ν∂φBQA)∂µφB
+ (∂µ∂∂λφBQ
A
)∂ν∂λφB+ (∂ν∂∂λφBQ
A
)∂µ∂λφB + (∂φB∂∂
λφCQ
A
)(∂µφB∂ν∂λφC +∂νφB∂µ∂λφC).
(24)
As an illustration, let us now work out second-order generalized symmetries of the Laplace/wa-ve equation, where prvQ(φ) = DµDµQ. Assuming thus that Q[φ] =Q(x, φ, ∂φ, ∂∂φ), we find using Eq. (23) that
DµDµQ= (∂φQ)φ+ (∂∂µφQ)∂µφ+ (∂∂µ∂νφQ)∂µ∂νφ+Q+ (∂φ∂φQ)(∂µφ)
2
+ (∂∂νφ∂∂λφQ)∂µ∂νφ∂
µ
∂λφ+ (∂∂ν∂αφ∂∂λ∂βφQ)∂µ∂ν∂αφ∂
µ ∂λ∂βφ
+ 2(∂µ∂φQ)∂µφ+ 2(∂µ∂∂νφQ)∂
µ∂
νφ+ 2(∂µ∂∂ν∂λφQ)∂
µ∂ ν∂λφ + 2(∂φ∂∂νφQ)∂µφ∂
µ∂
νφ+ 2(∂φ∂∂ν∂λφQ)∂µφ∂
µ∂ ν∂λφ + 2(∂∂νφ∂∂κ∂λφQ)∂µ∂νφ∂
µ ∂κ∂λφ.
(25)
Given thatQ[φ] does not contain any third derivatives ofφ, all terms with explicitly indicated third derivatives in DµDµφ must cancel on-shell. This constrains the dependence of Q[φ] on the second derivatives of φ. One then proceeds in turn to analyze terms in DµDµφ containing second derivatives ofφ, and so on. At end of the day, we find that the most general second-order generalized symmetry of the Laplace/wave equation corresponds to
Q[φ] =Qµν2 (x)∂µ∂νφ+Qµ1(x)∂µφ+Q0(x)φ+ ¯Q(x), (26)
subject to the following conditions:
• Q0 =Q¯ = 0,
• Qµ1 + 2∂µQ0 = 0,
• (Qµν2 +∂µQν
1 +∂νQ µ
1)∂µ∂νφ = 0 for on-shell fieldsφ.
• ∂µQνλ
The fact that Qµ1 does not depend on φ indicates that there are no nontrivial first-order gen-eralized symmetries; all first-order symmetries are point symmetries, found above by applying Eq. (9). Any new symmetries are therefore encoded in the tensor function Qµν2 (x). Note that this function has an intrinsic ambiguity with respect to adding gµνQ(x) with an arbitrary func-¯¯ tion ¯Q(x). This does not affect the action of the symmetry on on-shell fields, and thus does not¯ change the symmetry as a mapping turning one solution of the differential equation to another.
Upon symmetrization in all the indices, the derivative ∂µQνλ
2 can be decomposed as
∂µQνλ2 +∂νQλµ2 +∂λQµν2 =gµνΩλ +gνλΩµ+gλµΩν+Sµνλ, (27)
where the vector Ωµ is fixed by taking the trace,
Ωµ= 2 d+ 2∂νQ
µν
2 , (28)
and the fully symmetric tensor Sµνλ is traceless, that is, it vanishes upon contracting any pair of its indices with the metric. The last of our determining equations above then holds if and only if Sµνλ vanishes. Setting Sµνλ = 0 in Eq. (27), taking a derivative with respect to xκ and then an alternating sum over cyclic permutations of κ, λ, µ, ν, we obtain the condition
2(∂µ∂νQκλ2 −∂κ∂λQµν2 ) = gκλ(∂µΩν +∂νΩµ)−gµν(∂κΩλ+∂λΩκ)
−gκµ(∂λΩν −∂νΩλ)−gλν(∂κΩµ−∂µΩκ)
−gκν(∂λΩµ−∂µΩλ)−gλµ(∂κΩν −∂νΩκ).
(29)
Contracting this further with gκλ, and using the fact that Qµν2 can without loss of generality be assumed to be traceless, leads to
Qµν2 =gµν∂·Ω−d
2(∂ µ
Ων +∂νΩµ). (30)
Contracting this once again, with ∂µ∂ν, finally gives the constraint(∂·Ω) = 0. Together with the third of our determining equations, these constraints imply that Qµ1 − d
2Ω
µ is a conformal Killing vector. Without adding further details, we can thus summarize the strategy that leads to the classification of all second-order generalized symmetries of the Laplace/wave equation:
• Find all conformal Killing tensors Qµν2 by solving the condition (27).
• Find Ωµfrom Eq. (28) and fixQµ
1 by adding to d 2Ω
µ an arbitrary conformal Killing vector.
• FixQ0 by integrating Qµ1 + 2∂µQ0 = 0.
• Add an arbitrary solution ¯Q of the Laplace/wave equation.
