On Lagrangians with Reduced-Order Euler-Lagrange Equations
If a Lagrangian defining a variational problem has order k, then its Euler-Lagrange equations generically have order 2k. This paper considers the case where the Euler-Lagrange equations have order strictly less than 2k, and shows that in such a case the Lagrangian must be a polynomial in the highest...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2018 |
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Інститут математики НАН України
2018
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| Цитувати: | On Lagrangians with Reduced-Order Euler-Lagrange Equations / D. Saunders // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860253985945944064 |
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| author | Saunders, D. |
| author_facet | Saunders, D. |
| citation_txt | On Lagrangians with Reduced-Order Euler-Lagrange Equations / D. Saunders // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | If a Lagrangian defining a variational problem has order k, then its Euler-Lagrange equations generically have order 2k. This paper considers the case where the Euler-Lagrange equations have order strictly less than 2k, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivative variables, with a specific upper bound on the degree of the polynomial. The paper also provides an explicit formulation, derived from a geometrical construction, of a family of such k-th order Lagrangians, and it is conjectured that all such Lagrangians arise in this way.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 089, 13 pages
On Lagrangians with Reduced-Order
Euler–Lagrange Equations
David SAUNDERS
Department of Mathematics, Faculty of Science, The University of Ostrava,
30. dubna 22, 701 03 Ostrava, Czech Republic
E-mail: david@symplectic.demon.co.uk
Received January 26, 2018, in final form August 23, 2018; Published online August 25, 2018
https://doi.org/10.3842/SIGMA.2018.089
Abstract. If a Lagrangian defining a variational problem has order k then its Euler–
Lagrange equations generically have order 2k. This paper considers the case where the
Euler–Lagrange equations have order strictly less than 2k, and shows that in such a case the
Lagrangian must be a polynomial in the highest-order derivative variables, with a specific
upper bound on the degree of the polynomial. The paper also provides an explicit formu-
lation, derived from a geometrical construction, of a family of such k-th order Lagrangians,
and it is conjectured that all such Lagrangians arise in this way.
Key words: Euler–Lagrange equations; reduced-order; projectable
2010 Mathematics Subject Classification: 58E30
1 Introduction
If L is a Lagrangian function depending on some independent variables xi, some dependent
variables uα, and some first derivative variables, then the resulting Euler–Lagrange equations
∂L
∂uα
− d
dxi
∂L
∂uαi
= 0
are generically of second order: the total derivative operator d/dxi maps first-order variables
to second-order variables. For a Lagrangian depending also on higher-order derivative variables
uαij , u
α
ijh, . . . (of order up to k) the Euler–Lagrange equations, written in a multi-index notation as
k∑
|I|=0
(−1)|I|
d|I|
dxI
∂L
∂uαI
= 0,
are generically of order 2k. This paper considers the case of k-th order Lagrangians whose
Euler–Lagrange equations have order strictly less than 2k.
The existence of Lagrangians with reduced-order Euler–Lagrange equations has been known
for a long time. The Einstein–Hilbert Lagrangian from general relativity (see, for example, [2])
is second-order but its Euler–Lagrange equations, the Einstein field equations, are again second-
order rather than fourth-order. In this case, though, the Lagrangian may be written (although
not invariantly) as the sum of a first-order Lagrangian and a total divergence [2, Sections 3.3.1
and 3.3.2], so that these second-order Euler–Lagrange equations may in fact be derived from
a first-order Lagrangian.
Some examples with more substance may be found in [4, 5] in the context of Lagrangians
involving a single independent variable. Any such k-th order Lagrangian which is linear in
the derivative variables of highest order k will give rise to Euler–Lagrange equations of order
strictly less than 2k. Of course any total derivative L = df/dx will satisfy this condition trivially,
mailto:david@symplectic.demon.co.uk
https://doi.org/10.3842/SIGMA.2018.089
2 D. Saunders
because its Euler–Lagrange equations will vanish identically. But not every Lagrangian linear
in the highest order derivatives is a total derivative (or a total derivative plus a lower-order
function); the simplest such example is the first-order Lagrangian L1 = uvx − vux, giving rise
to the first-order Euler–Lagrange equations ux = vx = 0. On the other hand, a Lagrangian
involving a single independent variable which is not linear in the derivative variables of highest
order k will necessarily give rise to Euler–Lagrange equations of order 2k.
With more than one independent variable, the situation becomes more complicated. It re-
mains the case that linearity in the highest order derivatives is a sufficient condition for reduced-
order Euler–Lagrange equations; but now the condition is no longer necessary. For example,
in [8] a class of ‘special Lagrangians’ is defined. These are differential forms (integrands of the
variational problem) rather than functions, and they are constructed using a procedure of ho-
rizontalization. The coefficient function (the Lagrangian function) is a polynomial of degree m
in the derivative variables of highest order k, where m is the number of independent variables,
and indeed the polynomial is a linear combination of determinants in those variables. Once
again, the Euler–Lagrange equations have order strictly less than 2k. A simple example is the
Lagrangian L2 = ux
(
uxxuyy − u2xy
)
which gives rise to a third order Euler–Lagrange equation;
the 2-form L2dx ∧ dy is the horizontalization of uxdux ∧ duy.
These special Lagrangians do not, though, exhaust the possibilities. Consider a problem
with two independent variables x, y and three dependent variables u, v, w. The second order
Lagrangian
L3 = uxxvxywyy − uxxvyywxy + uxyvyywxx − uxyvxxwyy + uyyvxxwxy − uyyvxywxx
gives rise to third order Euler–Lagrange equations but is not a special Lagrangian in the sense
of [8] because it is cubic rather than quadratic in the second derivative variables (although it is
again a determinant). The same comment applies to the fourth order Lagrangian
L4 = uxxxxuxxyyuyyyy + 2uxxxyuxxyyuxyyy − uxxxxu2xyyy − u2xxxyuyyyy − u3xxyy,
which gives rise to a sixth order Euler–Lagrange equation. This suggests that some more general
alternating structure might be needed.
We obtain such a structure by using a version of the theory of differential hyperforms [6]1.
These are tensors with symmetry properties corresponding to Young diagrams; if the diagram
contains a single column then the tensor is purely alternating and so corresponds to an ordinary
differential form. We shall make particular use of hyperforms which are alternating combinations
of symmetric tensors. It is known that hyperforms give rise to particular types of determinant
known as hyperjacobians (see [7]), but the determinants used in the present paper appear to be
of a somewhat different nature.
The structure of this paper is as follows. In Section 2 we summarize, for the reader’s conve-
nience, the notation and conventions we shall adopt, and in Section 3 we prove the polynomial
property for Lagrangians with reduced-order Euler–Lagrange equations; this proof makes use
(in Lemma 3.1) of a surprising geometrical interpretation of the space of multi-indices. Next in
Section 4 we give a formal definition of the specific types of hyperform we shall use to construct
Lagrangians, and in Section 5 we show that any Lagrangian obtained from such a hyperform
has reduced-order Euler–Lagrange equations.
Our conjecture is that every Lagrangian with reduced-order Euler–Lagrange equations may
be constructed as a sum of Lagrangians obtained in this way, and finally in Section 6 we present
some evidence in support of that conjecture. It would also be interesting, for future work, to
attempt to extend this approach to give geometrical constructions of Lagrangians of order k
whose Euler–Lagrange equations have order less than 2k − 1, 2k − 2, and so on, using the
hyperjacobian structure of those equations [1, Proposition 4.52].
1I am grateful to Peter Olver for providing me with information about this reference.
On Lagrangians with Reduced-Order Euler–Lagrange Equations 3
2 Notation and conventions
We consider a fibred manifold π : E → M with dimM = m and dimE = m + n. Local
coordinates on M will be (xi), and adapted local coordinates on E will be (xi, uα). We adopt
the convention for wedge products (and also symmetric products) that no fractional factorial
coefficient is used, so that for instance
dxi ∧ dxj = dxi ⊗ dxj − dxj ⊗ dxi, dxi � dxj = dxi ⊗ dxj + dxj ⊗ dxi
without, in these cases, any factor of one-half.
For any order k ≥ 1 we consider the fibred manifold πk : Jkπ →M of k-th order jets (of local
sections of π) with adapted local coordinates (xi, uαI ) where I ∈ Nm is a multi-index indicating
that, for 1 ≤ i ≤ m, I(i) derivatives have been taken with respect to the variable xi; by default
J0π = E. We note that πk,k−1 : Jkπ → Jk−1π is an affine bundle with model vector bundle
V π ⊗ SkT ∗M → Jk−1π, so that it makes sense to say that a function on Jkπ is polynomial
in the ‘highest derivatives’, the fibre coordinates uαI where |I| =
m∑
i=1
I(i) = k. In general our
notation will follow that of [9] except where indicated.
For any k ≥ 0 we shall let Ωp
(
Jkπ
)
denote the module of p-forms on Jkπ. A p-form ω ∈
Ωp
(
Jkπ
)
is called horizontal if the contraction iXω = 0 for any vector field X on Jkπ vertical
over M ; if instead the pullback (jkφ)∗ω = 0 for any local section φ of π, where jkφ denotes the
prolonged local section of πk, then we say that ω is a contact form. Any p-form π∗k,k−1$, where
$ ∈ Ωp
(
Jk−1π
)
, may be written uniquely as h($) + c($) where h($) is horizontal and c($) is
contact; we say that h($) is the horizontalization of $.
A Lagrangian density of order k is a horizontal m-form λ ∈ Ωm
(
Jkπ
)
, and it is special in the
sense of [8] if λ = h($) for some $ ∈ Ωm
(
Jk−1π
)
. Any Lagrangian density λ, special or not,
may be written in coordinates λ = Ldx1 ∧ dx2 ∧ · · · ∧ dxm where L is the corresponding local
Lagrangian function. The Euler–Lagrange form of λ is the (m+ 1)-form ε on J2kπ obtained by
a canonical procedure from λ (essentially taking the exterior derivative and then integrating by
parts k times) and incorporates the Euler–Lagrange equations for λ; in coordinates
ε = εαduα ∧ dx1 ∧ dx2 ∧ · · · ∧ dxm, εα =
k∑
|I|=0
(−1)|I|
d|I|
dxI
∂L
∂uαI
.
The underlying structures involved in these constructions are either the infinite-order variational
bicomplex [1] or the finite-order variational sequence [3]. We shall say that the Euler–Lagrange
form ε and the associated Euler–Lagrange equations εα = 0 are projectable if the form ε,
generically defined on J2kπ, is projectable to J2k−1π; the order of the Euler–Lagrange equations
will then be strictly less than 2k.
Any p-form Ωp
(
Jkπ
)
is a section of the bundle
∧p T ∗Jkπ → Jkπ, and any horizontal p-form is
a section of the pull-back bundle
∧p T ∗M → Jkπ. We shall use the terminology horizontal (p, q)
hyperform of order k to denote a section of the pullback bundle
∧p SqT ∗M → Jkπ, so that in
a chart on Uk ⊂ Jkπ such a hyperform looks like∑
|Ir|=q
1≤r≤p
ωI1I2···Ipdx
I1 ∧ dxI2 ∧ · · · ∧ dxIp
with ωI1I2···Ip ∈ C∞
(
Uk
)
, where if the multi-index I corresponds to the list of ordinary indices
i1i2 · · · iq then
dxI = dxi1 � dxi2 � · · · � dxiq .
4 D. Saunders
The collection Ωp,q
h
(
Jkπ
)
of all such hyperforms of given degree (p, q) is module over both
C∞
(
Jkπ
)
and, significantly, over C∞
(
Jk−1π
)
.
We now fix the order k > 0, and restrict attention to the case 1 ≤ q ≤ k. We shall use
italic capitals such as I to denote multi-indices with length |I| = k − q (or, sometimes, with
length |H| = 2k), calligraphic letters such as J to denote multi-indices with length q, and
roman capitals such as K to denote multi-indices with length k. Except where stated otherwise
we shall adopt the usual summation convention for such multi-indices, as well as for ordinary
indices i, j, . . . and α, β, . . ., but readers should be aware that the contraction of symmetric
tensors using this multi-index notation introduces numerical constants.
3 The polynomial property
Our first result is that if a Lagrangian density of order k gives rise to a projectable Euler–
Lagrange form then in any coordinate system the corresponding Lagrangian function must be
a polynomial of order at most pk in the highest order derivative variables uαK, where pk is the
number of distinct multi-indices K of length k.
To illustrate our approach, we describe the procedure for two special cases.
In the first special case we take k = 2, so that we want to show that L must be a polynomial
of degree at most p2 = 1
2m(m + 1) in the second order derivative variables. We know, as
a consequence of projectability, that the expression
∑
|K|=2
d2
dxK
∂L
∂uβK
has order strictly less than 4, so if we expand the second order total derivatives we obtain
∑
|J|=|K|=2
uαJ+K
∂2L
∂uαJ∂u
β
K
+ · · · ,
where the dots indicate terms whose order is less than 4. It follows that, for each multi-index H
with |H| = 4, we must have
∑
J+K=H
∂2L
∂uαJ∂u
β
K
= 0. (3.1)
Fix an index i with 1 ≤ i ≤ m, and let Hi be the multi-index satisfying Hi(i) = 4, Hi(j) = 0
for j 6= i; here and more generally we call a multi-index with only a single nonzero entry a pure
multi-index. We then see immediately from equation (3.1) that if Ki is the pure multi-index
satisfying Ki(i) = 2, Ki(j) = 0 for j 6= i then
∂2L
∂uαKi∂u
β
Ki
= 0. (3.2)
Now fix indices i, j with j 6= i, and let Hij be the multi-index satisfying Hij(i) = Hij(j) = 2,
Hij(h) = 0 for h 6= i, j; we call this a mixed multi-index. If Kij is the mixed multi-index satisfying
Kij(i) = Kij(j) = 1, Kij(h) = 0 for h 6= i, j then we see from (3.1) that
∂2L
∂uαKij∂u
β
Kij
= − ∂2L
∂uαKi∂u
β
Kj
− ∂2L
∂uαKj∂u
β
Ki
,
On Lagrangians with Reduced-Order Euler–Lagrange Equations 5
so that
∂4L
∂uαKij∂u
β
Kij
∂uγKih∂u
δ
Kih
=
∂4L
∂uαKi∂u
β
Kj
∂uγKi∂u
δ
Kh
+
∂4L
∂uαKi∂u
β
Kj
∂uγKh∂u
δ
Ki
+
∂4L
∂uαKj∂u
β
Ki
∂uγKi∂u
δ
Kh
+
∂4L
∂uαKj∂u
β
Ki
∂uγKh∂u
δ
Ki
= 0. (3.3)
Thus if the expression
∂rL
∂uα1
J1
∂uα2
J2
· · · ∂uαrJr
does not vanish then in the list (J1, J2, . . . , Jr) each distinct pure multi-index Ki can appear at
most once (from (3.2)), and each distinct mixed multi-index Kij can appear at most three times
(from (3.3) with h = j). Furthermore if both Kij and Kih appear then either one or the other
must appear only once (from (3.3)).
Let a be the number of pure multi-indices Ki in the list, and let b, c and d be the number of
mixed multi-indices Kij with, respectively, multiplicities 1, 2 and 3. Clearly b+c+d ≤ 1
2m(m−1).
On the other hand, a ≤ m− 2(c+ d) because if Kij appears with multiplicity 2 or 3 (so that if
h 6= i, j then Kih and Kjh can have multiplicity at most 1) then neither Ki nor Kj can appear
at all. We therefore see that
r = a+ b+ 2c+ 3d ≤ m− 2(c+ d) + 1
2m(m− 1) + c+ 2d ≤ 1
2m(m+ 1) = p2.
In that first special case with k = 2 we were able to see explicitly the polynomial structure of
the Lagrangian, but for higher orders this detailed investigation rapidly becomes unmanageable,
so we need to adopt a more abstract approach. For our second special case we therefore let k
be arbitrary, but take m = 2. There are now pk = k + 1 multi-indices of length k, and now the
consequence of projectability is that, for 0 ≤ h ≤ k,
k−h∑
l=0
∂2L
∂uα(k−l,l)∂u
β
(h+l,k−h−l)
= 0. (3.4)
As before we use this relation to manipulate the repeated partial derivatives of L, but now we
need a mechanism to keep track of what we are doing and help us avoid going round in circles.
We do this by defining the weight2 of a multi-index J of length k to be the squared Euclidean
norm, ‖J‖2 =
m∑
j=1
(J(j))2. We also define the weight of a list of multi-indices (J1, J2, . . . , Jr) to
be the sum of the individual weights ‖J1‖2 + ‖J2‖2 + · · · + ‖Jr‖2. It is important to note that
the maximum weight of a multi-index is k2, and that this maximum is achieved if, and only if,
the multi-index is pure. In our special case with m = 2 we have ‖(h, l)‖2 = h2 + l2, and the
maximum weight is achieved by (k, 0) and (0, k).
We now use this to show that every repeated derivative of order pk + 1 = k + 2,
∂k+2L
∂uα1
J1
∂uα2
J2
· · · ∂uαk+2
Jk+2
(3.5)
must vanish. Here we use the fact that, as in the case k = 2, we have∑
J+K=H
∂2L
∂uαJ∂u
β
K
= 0,
where |H| = 2k, so that if J is a pure multi-index then ∂2L/∂uαJ∂u
β
J = 0.
2Note that this type of weight is different from the system of weights defined in a similar context in [1,
Definition 4.18].
6 D. Saunders
Let (J1, J2, . . . , Jk+2) be the list of multi-indices in the repeated derivative, so that necessarily
at least two of these multi-indices must be equal. If they are both equal to (k, 0), or both equal
to (0, k), so that they are pure, then the repeated derivative must vanish. So suppose this is
not the case, and assume without loss of generality that J1 = J2 = (k − g, g). From (3.4) with
h = k − 2g we see that
∂2L
∂uα1
(k−g,g)∂u
α2
(k−g,g)
=
∑
K1+K2=(2k−2g,2g)
K1,K2 6=(k−g,g)
− ∂2L
∂uα1
K1
∂uα2
K2
.
On the left-hand side the weight of the two multi-indices (J1, J2) is 2
(
(k− g)2 + g2
)
, whereas in
a term on the right-hand side with l 6= 0 and K1 = (k − g − l, g + l), K2 = (k − g + l, g − l) the
weight of (K1,K2) is(
(k − g − l)2 + (g + l)2
)
+
(
(k − g + l)2 + (g − l)2
)
= 2
(
(k − g)2 + g2
)
+ 4l2.
We may therefore write our original repeated derivative (3.5) as (apart from an overall sign)
a sum of similar repeated derivatives where, in each term, the weight of the multi-index list has
increased. Furthermore, each new term also has the property that two of its multi-indices must
be equal, so by repeating the process we must eventually be able to write (3.5) as a sum or
difference of terms, each of which has three pure multi-indices of maximum weight k2 (that is,
either (k, 0) or (0, k)), so that two of the pure multi-indices must be equal, and therefore each
such term must vanish.
The proof of the general result with k and m both arbitrary follows essentially the same
approach as in the second special case. We first confirm the relationship between the weights of
multi-indices of length k.
Lemma 3.1. If |J| = |K1| = |K2| = k and 2J = K1 + K2 then the weight ‖J‖ satisfies 2‖J‖2 ≤
‖K1‖2 + ‖K2‖2, with equality only when K1 = K2 = J.
Proof. This is just the parallelogram rule for any Euclidean norm, that
2‖x‖2 ≤ 2‖x‖2 + 2‖y‖2 = ‖x+ y‖2 + ‖x− y‖2
with equality only when y = 0. �
Theorem 3.2. If the Lagrangian density λ on Jkπ has projectable Euler–Lagrange equations
then in any coordinate system λ = Ldx1 ∧ dx2 ∧ · · · ∧ dxm where the function L, defined locally
on Jkπ, is a polynomial of order at most pk in the highest order derivative variables uαJ , where pk
is the number of distinct multi-indices of length k.
Proof. The consequence of projectability is now that the expression∑
|K|=k
d|K|
dxK
∂L
∂uβK
has order strictly less than 2k. Expanding the k-th order total derivatives gives∑
|J|=|K|=k
uαJ+K
∂2L
∂uαJ∂u
β
K
+ · · · ,
where the dots indicate terms whose order is less than 2k. It follows that, for each multi-index H
with |H| = 2k, we must have∑
J+K=H
∂2L
∂uαJ∂u
β
K
= 0. (3.6)
On Lagrangians with Reduced-Order Euler–Lagrange Equations 7
Now consider the repeated derivative of order p+ 1
∂rL
∂uα1
J1
∂uα2
J2
· · · ∂uαp+1
Jp+1
, (3.7)
where |J1| = |J2| = · · · = |Jp+1| = k, and suppose that in the list of multi-indices (J1, J2, . . ., Jp+1)
we have Jr = Js = J. Use (3.6) to write
∂2L
∂uαrJ ∂uαsJ
=
∑
K1+K2=2J
K1,K2 6=J
− ∂2L
∂uαrK1
∂uαsK2
,
so that by Lemma 3.1 we have in each term on the right-hand side ‖K1‖2 + ‖K2‖2 > 2‖J‖2.
By repeating this process we must eventually be able to write (3.7) as a sum or difference of
terms, each of which has m + 1 pure multi-indices of maximum weight k2 (so that two of its
pure multi-indices must be equal) and therefore each of which must vanish. �
4 Hyperforms
The necessary condition given above for a Lagrangian density λ on Jkπ to have projectable
Euler–Lagrange equations is obviously not sufficient; but the requirement that λ be the hori-
zontalization of some m-form on Jk−1π is, as noted in the Introduction, too strong. We shall
define a weaker condition on λ which will still be sufficient to ensure that the Euler–Lagrange
equations are projectable, using the idea of a horizontal (p, q) hyperform introduced in Section 2.
We first consider horizontal (1, q) hyperforms; any such hyperform θ may be written in
coordinates on Uk as θJ dxJ where θJ ∈ C∞
(
Uk
)
. We have mentioned that πk,k−1 : Jkπ →
Jk−1π is an affine bundle so that for any point of Jk−1π the fibre over that point is an affine
space. The restriction of θ to that fibre takes its values in the appropriate fibre of SqT ∗M ,
a vector space, so it makes sense to ask whether that restriction is an affine map. If this is the
case for every fibre of πk−1,k then we say that θ is an affine (1, q) hyperform; the coordinate
representation of θ then becomes(
θKαJ u
α
K + θJ
)
dxJ ,
where now θKαJ , θJ ∈ C∞(Uk−1), Uk−1 ⊂ Jk−1π. (Recall here that the roman multi-indices K
have length k, whereas the calligraphic multi-indices J have length q.)
The collection of affine hyperforms is, however, too large for our purposes. To obtain a suitable
restriction, we note that the map θ : Jkπ → SqT ∗M is affine precisely when the associated
difference map θ̄ : V π ⊗Jk−1π S
kT ∗M → SqT ∗M is linear on each fibre over Jk−1π. We say
that θ is a special affine hyperform if there is a tensor θ̃ ∈ V π∗ ⊗Jk−1π S
k−qTM such that the
fibre-linear map θ̄ is given by the contraction of elements of V π ⊗Jk−1π S
kT ∗M with θ̃. We
shall let Ω1,q
sa
(
Jkπ
)
denote the collection of such special affine hyperforms; it is a module over
C∞
(
Jk−1π
)
, though not of course over C∞
(
Jkπ
)
. A special affine hyperform may therefore be
written in coordinates as(
θIαu
α
I+J + θJ
)
dxJ ,
where θIα, θJ ∈ C∞
(
Uk−1
)
, Uk−1 ⊂ Jk−1π. (Here the italic multi-indices I have length k − q.)
As examples of affine and special affine hyperforms, consider the case where m = 2 and n = 1,
with coordinates x, y, u, and where q = k = 2, so that in this case each multi-index I is zero.
An affine hyperform will have a coordinate representation(
θxxxxuxx + θxyxxuxy + θyyxxuyy + θxx
)
dx� dx+
(
θxxxyuxx + θxyxyuxy + θyyxyuyy + θxy
)
dx� dy
+
(
θxxyyuyy + θxyyyuxy + θyyyyuyy + θyy
)
dy � dy,
8 D. Saunders
where the functions θxxxx, θ
xy
xx, . . . are at most first order, whereas a special affine hyperform will
have a coordinate representation(
θ0uxx + θxx
)
dx� dx+
(
θ0uxy + θxy
)
dx� dy +
(
θ0uyy + θyy
)
dy � dy,
where each term involves only a single second order coordinate, and where the (at most first
order) function θ0 is the same for all three terms.
We may see the relationship between this definition and the operation of horizontalization
on ordinary 1-forms by considering the special case where q = 1. In this case a special affine
hyperform θ (now just a horizontal 1-form) may be written in coordinates as
(
θIαu
α
I+1i
+ θi
)
dxi
(where 1i denotes the multi-index with a single 1 in position i) and is the horizontalization
of, for instance, the 1-form θIαduαI + θidx
i. There is, however, no well-defined horizontalization
operator mapping forms to hyperforms when q ≥ 2.
We now consider horizontal (p, q) hyperforms, where p is fixed to equal the number pq of
distinct multi-indices I ∈ Nm of length q, so that
pq =
(
m+ q − 1
q
)
=
(m+ q − 1)!
q!(m− 1)!
;
the fibre dimension of SqT ∗M is then equal to pq, so that
∧pq SqT ∗M → Jkπ is a line bundle.
We shall say that such a section of this bundle, a horizontal (pq, q) hyperform ω, is hyperaffine if
it can be written as a sum of wedge products of special affine (1, q) hyperforms, and we shall let
Ω
pq ,q
ha
(
Jkπ
)
denote the collection of such hyperforms; again this is a module over C∞
(
Jk−1π
)
.
If for a single wedge product
ω = θ1 ∧ θ2 ∧ · · · ∧ θpq ,
where
θr ∈ Ω1,q
ha
(
Jkπ
)
, θr =
(
θIr,αu
α
I+J + θr,J
)
dxJ ,
then in the coordinate expression for ω the coefficient of the single local basis element dxJ1 ∧
dxJ2 ∧ · · · ∧ dxJpq will be given as a linear combination (by functions projectable to Jk−1π) of
determinants in the highest order derivative variables uαI+J , ranging in size up to (pq × pq); for
instance the largest determinant will take the form∣∣∣∣∣∣∣∣∣∣
uα1
I1+J1 uα1
I1+J2 · · · uα1
I1+Jpq
uα2
I2+J1 uα2
I2+J2 · · · uα2
I2+Jpq
...
...
. . .
...
u
αpq
Ipq+J1 u
αpq
Ipq+J2 · · · u
αpq
Ipq+Jpq
∣∣∣∣∣∣∣∣∣∣
, (4.1)
where uα1
I1
, uα2
I2
, . . . , u
αpq
Ipq
are derivative variables of order k−q, and smaller determinants, arising
when one or more of the (1, q) hyperforms θr is projectable to J lπ with l < k, will be obtained as
suitably-sized minors. (If those derivative variables are not distinct then the largest determinant
will vanish, and this always happens when n < pq. It is nevertheless the case that sufficiently
small minors will be nonzero.)
As examples, we may see that the four Lagrangian functions mentioned in the Introduction
all arise as such coefficients. For L1 we take q = pq = k = 1 and for L2 we take q = 1, pq = 2,
k = 2; both the corresponding Lagrangian densities arise from conventional horizontalization.
For L3 we take q = k = 2, pq = 3, the variables u, v, w and the hyperform obtained from the
wedge product(
uxxdx� dx+ uxydx� dy + uyydy � dy
)
∧
(
vxxdx� dx+ vxydx� dy + vyydy � dy
)
∧
(
wxxdx� dx+ wxydx� dy + wyydy � dy
)
,
On Lagrangians with Reduced-Order Euler–Lagrange Equations 9
so that
L3 =
∣∣∣∣∣∣
uxx uxy uyy
vxx vxy vyy
wxx wxy wyy
∣∣∣∣∣∣ ;
for L4 we take q = 2, pq = 3, k = 4, the variables uxx, uxy, uyy and the hyperform obtained
from the wedge product(
uxxxxdx� dx+ uxxxydx� dy + uxxyydy � dy
)
∧
(
uxxxydx� dx+ uxxyydx� dy + uxyyydy � dy
)
∧
(
uxxyydx� dx+ uxyyydx� dy + uyyyydy � dy
)
,
so that
L4 =
∣∣∣∣∣∣
uxxxx uxxxy uxxyy
uxxxy uxxyy uxyyy
uxxyy uxyyy uyyyy
∣∣∣∣∣∣ .
5 Hyperaffine Lagrangians
The examples at the end of the previous section suggest how we might relate the construction
of hyperaffine (pq, q) hyperforms to Lagrangian densities. We note that such a hyperform may
be written in coordinates as
ω = ωqdx
J1 ∧ dxJ2 ∧ · · · ∧ dxJpq ,
and when q > 1 then this is obviously different from an ordinary horizontal m-form such as λ.
The two types of object are, nevertheless, related: they are both sections of line bundles, and
in coordinates each has a single coefficient function, L or ωq. Furthermore, under a change of
coordinates (xi, uα) 7→ (x̃i, ũα), L is altered by the Jacobian of the transformation xi 7→ x̃i,
whereas ωq is altered by a power of that Jacobian. As the condition for ω to be hyperaffine
may be expressed in terms of ωq in a way which is independent of transformations of the base
coordinates xi, it makes sense to say that a Lagrangian density λ is hyperaffine if, in any
coordinate system, the corresponding local Lagrangian function L may be written as a sum
L =
k∑
q=1
ωq where each ωq is the coefficient in that coordinate system of a hyperaffine (pq, q)
hyperform with pq = (m+ q − 1)!/q!(m− 1)!.
For example, in the case where m = 2 and n = 1 with coordinates x, y, u, we might consider
the third-order Lagrangian function
L = uxxxuyyy − uxxyuxyy.
We may write L as ω2, the scalar coefficient of ω = θ1 ∧ θ2 ∧ θ3, where
θ1 = uxxxdx� dx+ uxxydx� dy + uxyydy � dy,
θ2 = uxxydx� dx+ uxyydx� dy + uyyydy � dy,
θ3 = dx� dy,
so that ω2 is a (non-vanishing) 2× 2 minor of the (vanishing) determinant∣∣∣∣∣∣
uxxx uxxy uxyy
uxxy uxyy uyyy
uxxy uxyy uyyy
∣∣∣∣∣∣ .
We see that L is a null Lagrangian, so its Euler–Lagrange equations are trivially projectable.
Indeed the significance of our definition comes from the following result.
10 D. Saunders
Theorem 5.1. If λ is a hyperaffine Lagrangian density on Jkπ then λ has projectable Euler–
Lagrange equations.
Proof. It is sufficient to prove that a function L given in coordinates as an h× h minor of the
determinant (4.1) gives rise to Euler–Lagrange equations of order strictly less than 2k. As terms
in those equations of order 2k can arise only when considering
∑
|K|=k
d|K|
dxK
∂L
∂uβK
,
where we have written the sum over the multi-indices K explicitly, it is sufficient to show that
each such term (for a given index β) vanishes when L is such a determinant.
Write L in the form
L =
∑
σ∈Sh
εσu
α1
I1+Jσ(1)u
α2
I2+Jσ(2) · · ·u
αh
Ih+Jσ(h) ,
where Sh is the permutation group and εσ = ±1 is the parity of the permutation σ; then for
any given multi-index K we have
d|K|
dxK
∂L
∂uβK
=
∑
1≤r,s≤h
s 6=r
∑
σ∈Sh
δαrβ δKIr+Jσ(r)εσΦrsσu
αs
Ir+Is+Jσ(r)+Jσ(s) ,
where the coefficient functions Φrsσ are given by
Φrsσ = uα1
I1+Jσ(1)u
α2
I2+Jσ(2) · · · r̂ · · · ŝ · · ·u
αh
Ih+Jσ(h)
with the circumflex denoting the omission of a factor in the product. As the multi-indices
I1, I2, . . . , Ih and J1,J2, . . . ,Jh are given, it follows that
∑
|K|=k
d|K|
dxK
∂L
∂uβK
=
∑
1≤r,s≤h
s 6=r
∑
σ∈Sh
δαrβ εσΦrsσu
αs
Ir+Is+Jσ(r)+Jσ(s) ,
where the factor δKIr+Jσ(r) on the right-hand side is omitted.
Fix values for r and s; we shall show that∑
σ∈Sh
δαrβ εσΦrsσu
αs
Ir+Is+Jσ(r)+Jσ(s) = 0.
To see this, for each σ ∈ Sh let σ̃ be the permutation given by
σ̃(r) = σ(s), σ̃(s) = σ(r), σ̃(t) = σ(t) for 1 ≤ t ≤ h, t 6= r, s.
We obtain in this way a partition of Sh into equivalence classes of the form {σ, σ̃}, where each
equivalence class contains exactly two elements because r 6= s. As Φrsσ̃ = Φrsσ, Jσ̃(r) + Jσ̃(s) =
Jσ(r) + Jσ(s) and εσ + εσ̃ = 0, the result follows. �
Corollary 5.2. The upper bound pk given in Theorem 3.2, for the polynomial degree of a Lag-
rangian in the highest order derivatives, is sharp if the number of independent variables n satisfies
n ≥ pk.
On Lagrangians with Reduced-Order Euler–Lagrange Equations 11
Proof. Take q = k, and let J1,J2, . . . ,Jpk be the distinct multi-indices of length k. If
L =
∣∣∣∣∣∣∣∣∣∣
u1J1 u1J2 · · · u1Jpk
u2J1 u2J2 · · · u2Jpk
... · · · . . .
...
upkJ1 upkJ2 · · · upkJpk
∣∣∣∣∣∣∣∣∣∣
(so that all the multi-indices Ir are zero) then L is the coefficient of a hyperaffine hyperform, so
that it gives rise to projectable Euler–Lagrange equations by Theorem 5.1. �
6 Discussion
The arguments above show that there is a correspondence between hyperaffine (pq, q) hyperforms
and Lagrangian densities with projectable Euler–Lagrange equations. The correspondence is
certainly not injective, even locally in a fixed coordinate system. For instance, with m = k = 2
and n = 3 the Lagrangian functions
L3 =
∣∣∣∣∣∣
uxx uxy uyy
vxx vxy vyy
wxx wxy wyy
∣∣∣∣∣∣, L5 = w
∣∣∣∣uxx uxy
vxy vyy
∣∣∣∣, L6 = w
∣∣∣∣uxx uxy
vxx vxy
∣∣∣∣ = w
∣∣∣∣∣∣
uxx uxy uyy
vxx vxy vyy
0 0 1
∣∣∣∣∣∣
are all hyperaffine; we have (temporarily omitting the symmetric product symbol �)
L3dxdx ∧ dxdy ∧ dydy =
(
uxxdxdx+ uxydxdy + uyydydy
)
∧
(
vxxdxdx+ vxydxdy + vyydydy
)
∧
(
wxxdxdx+ wxydxdy + wyydydy
)
,
L5dx ∧ dy =
(
w(uxxdx+ uxydy)
)
∧
(
vxydx+ vyydy
)
,
but
L6dx ∧ dy =
(
w(uxxdx+ uxydy
)
∧
(
vxxdx+ vxydy
)
,
L6dxdx ∧ dxdy ∧ dydy =
(
uxxdxdx+ uxydxdy + uyydydy
)
∧
(
vxxdxdx+ vxydxdy + vyydydy
)
∧ dydy.
We do, however, make the conjecture that the correspondence is surjective: that is, that if
a Lagrangian density has projectable Euler–Lagrange equations then, in any coordinate system,
its Lagrangian function must be a sum of determinants of the form (4.1), or of minors of such
determinants with essentially the same format. One might clearly attempt to establish such
a conjecture by considering the homogeneous components of the Lagrangian, and it is certainly
the case that the quadratic component satisfies the condition.
Proposition 6.1. If the Lagrangian density λ on Jkπ has projectable Euler–Lagrange equa-
tions then the quadratic terms of the polynomial Lagrangian function L are determinants with
a hyperaffine structure.
Proof. Suppose the quadratic terms of L are AH1H2
α1α2
uα1
H1
uα2
H2
where |H1| = |H2| = k. Partition
the set of quadratic terms according to the multi-index H = H1 + H2, and consider the terms
ΨH =
∑
H1+H2=H
AH1H2
α1α2
uα1
H1
uα2
H2
12 D. Saunders
in a single component of the partition. There must be at least two distinct terms; for if there
were only a single term then there would have to be an index i such that H1(i) = H2(i) = k,
and then (3.6) would imply AH1H2
α1α2
= 0.
Choose, arbitrarily, one term AK1K2
α1α2
uα1
K1
uα2
K2
, so that (3.6) now gives
AK1K2
α1α2
=
∑
H1+H2=H
(H1,H2) 6=(K1,K2)
−AH1H2
α1α2
,
and hence
ΨH =
∑
H1+H2=H
AH1H2
α1α2
(
uα1
H1
uα2
H2
− uα1
K1
uα2
K2
)
,
so that each ΨH is a sum of determinants. (The restriction (H1,H2) 6= (K1,K2) is omitted from
the latter sum because if (H1,H2) = (K1,K2) then the term vanishes.)
To see that each determinant has a hyperaffine structure (that is, can be written in the
form (4.1)) consider a single expression
uα1
H1
uα2
H2
− uα1
K1
uα2
K2
(6.1)
and let I1, I2 be the multi-indices defined by
I1(i) = min{H1(i),K1(i)}, I2(i) = min{H2(i),K2(i)}, 1 ≤ i ≤ m.
Consider any index i. Suppose I1(i) = H1(i), so that H1(i) ≤ K1(i); then I2(i) = K2(i), for if
not we would have H2(i) = I2(i) < K2(i), contradicting H1(i)+H2(i) = K1(i)+K2(i). If instead
I1(i) < H1(i) then I1(i) = K1(i), and a similar argument shows that I2(i) = H2(i).
Now let J1, J2 be the multi-indices defined by I1 + J1 = H1, I2 + J2 = H2. Consider any
index i. If I1(i) = H1(i) and I2(i) = K2(i) then
I1(i) + J2(i) = H1(i) + H2(i)− I2(i) = K1(i) + K2(i)− I2(i) = K1(i)
and
I2(i) + J1(i) = K2(i) + H1(i)− I1(i) = K2(i),
and a similar argument shows that these relations also hold when I1(i) = K1(i) and I2(i) = H2(i).
We therefore see that I1 + J2 = K1 and I2 + J1 = K2, so that expression (6.1) can be written
as the determinant∣∣∣∣uα1
I1+J1 uα1
I1+J2
uα2
I2+J1 uα2
I2+J2
∣∣∣∣ .
To see that this is indeed an instance of determinant (4.1) (or one of its minors), we must finally
check that |I1| = |I2|. Let P = {i : I1(i) = H1(i)} and Q = {i : I1(i) < H1(i)}; then
|I1| − |I2| =
∑
i∈P
(
I1(i)− I2(i)
)
+
∑
i∈Q
(
I1(i)− I2(i)
)
=
∑
i∈P
(
H1(i)−K2(i)
)
+
∑
i∈Q
(
K1(i)−H2(i)
)
=
∑
i∈P
(
H1(i)−K2(i)
)
−
∑
i∈P
(
K1(i)−H2(i)
)
,
and for any index i we have H1(i)+H2(i) = K1(i)+K2(i). We may therefore set q = |J1| = |J2|
so that |I1| = |I2| = k − q. �
A similar result for an arbitrary homogeneous component of L does, however, seem to be
significantly more complicated to prove, and so work continues on the project.
On Lagrangians with Reduced-Order Euler–Lagrange Equations 13
Acknowledgements
The author would like to thank the referees for their helpful suggestions regarding the presen-
tation of some technical aspects of this work.
References
[1] Anderson I.M., The variational bicomplex, Technical report, Utah State University, 1989.
[2] Carmeli M., Classical fields: general relativity and gauge theory, A Wiley-Interscience Publication, John
Wiley & Sons, Inc., New York, 1982.
[3] Krupka D., Variational sequences on finite order jet spaces, in Differential Geometry and its Applications
(Brno, 1989), World Sci. Publ., Teaneck, NJ, 1990, 236–254.
[4] Krupková O., Lepagean 2-forms in higher order Hamiltonian mechanics. I. Regularity, Arch. Math. (Brno)
22 (1986), 97–120.
[5] Krupková O., The geometry of ordinary variational equations, Lecture Notes in Mathematics, Vol. 1678,
Springer-Verlag, Berlin, 1997.
[6] Olver P.J., Differential hyperforms I, University of Minnesota Mathematics Report 82-101, 1982, available
at http://www-users.math.umn.edu/~olver/a_/hyper.pdf.
[7] Olver P.J., Hyper-Jacobians, determinantal ideals and weak solutions to variational problems, Proc. Roy.
Soc. Edinburgh Sect. A 95 (1983), 317–340.
[8] Palese M., Vitolo R., On a class of polynomial Lagrangians, Rend. Circ. Mat. Palermo Suppl. (2001),
147–159, math-ph/0111019.
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Cambridge University Press, Cambridge, 1989.
https://doi.org/10.1007/BFb0093438
http://www-users.math.umn.edu/~olver/a_/hyper.pdf
https://doi.org/10.1017/S0308210500013020
https://doi.org/10.1017/S0308210500013020
https://arxiv.org/abs/math-ph/0111019
https://doi.org/10.1017/CBO9780511526411
1 Introduction
2 Notation and conventions
3 The polynomial property
4 Hyperforms
5 Hyperaffine Lagrangians
6 Discussion
References
|
| id | nasplib_isofts_kiev_ua-123456789-209761 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:46:52Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Saunders, D. 2025-11-26T11:22:56Z 2018 On Lagrangians with Reduced-Order Euler-Lagrange Equations / D. Saunders // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 9 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58E30 arXiv: 1801.06888 https://nasplib.isofts.kiev.ua/handle/123456789/209761 https://doi.org/10.3842/SIGMA.2018.089 If a Lagrangian defining a variational problem has order k, then its Euler-Lagrange equations generically have order 2k. This paper considers the case where the Euler-Lagrange equations have order strictly less than 2k, and shows that in such a case the Lagrangian must be a polynomial in the highest-order derivative variables, with a specific upper bound on the degree of the polynomial. The paper also provides an explicit formulation, derived from a geometrical construction, of a family of such k-th order Lagrangians, and it is conjectured that all such Lagrangians arise in this way. The author would like to thank the referees for their helpful suggestions regarding the presentation of some technical aspects of this work. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Lagrangians with Reduced-Order Euler-Lagrange Equations Article published earlier |
| spellingShingle | On Lagrangians with Reduced-Order Euler-Lagrange Equations Saunders, D. |
| title | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_full | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_fullStr | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_full_unstemmed | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_short | On Lagrangians with Reduced-Order Euler-Lagrange Equations |
| title_sort | on lagrangians with reduced-order euler-lagrange equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209761 |
| work_keys_str_mv | AT saundersd onlagrangianswithreducedordereulerlagrangeequations |