Conservation laws for invariant functionals containing compositions
Conservation laws for invariant functionals
containing compositions∗
Gastão S. F. Frederico†
gfrederico@mat.ua.pt
Higher Institute of Education
University of Cabo Verde
Praia, Santiago – Cape Verde
Delfim F. M. Torres‡
delfim@mat.ua.pt
Department of Mathematics
University of Aveiro
3810-193 Aveiro, Portugal
Abstract
The study of problems of the calculus of variations with composi-
tions is a quite recent subject with origin in dynamical systems governed
by chaotic maps. Available results are reduced to a generalized Euler-
Lagrange equation that contains a new term involving inverse images of
the minimizing trajectories. In this work we prove a generalization of the
necessary optimality condition of DuBois-Reymond for variational prob-
lems with compositions. With the help of the new obtained condition,
a Noether-type theorem is proved. An application of our main result is
given to a problem appearing in the chaotic setting when one consider
maps that are ergodic.
Mathematics Subject Classification 2000: 49K05, 49J05.
Keywords: variational calculus, functionals containing compositions,
symmetries, DuBois-Reymond condition, Noether’s theorem.
1 Introduction and motivation
The theory of variational calculus for problems with compositions has been re-
cently initiated in [5]. The new theory considers integral functionals that depend
not only on functions q(·) and their derivatives q̇(·), but also on compositions
(q ◦ q)(·) of q(·) with q(·). As far as chaos is often a byproduct of iteration
∗Accepted for an oral presentation at the 7th IFAC Symposium on Nonlinear Control
Systems (NOLCOS 2007), to be held in Pretoria, South Africa, 22–24 August, 2007.
†This work is part of the author’s PhD project. Supported by the Portuguese Institute for
Development (IPAD).
‡Supported by the Centre for Research on Optimization and Control (CEOC) through
the Portuguese Foundation for Science and Technology (FCT), cofinanced by the European
Community fund FEDER/POCTI.
http://arxiv.org/abs/0704.0949v1
of nonlinear maps [2], such problems serve as an interesting model for chaotic
dynamical systems. Let us briefly review this relation (for more details, we refer
the interested reader to [3, 4, 5]). Let q : [0, 1] → [0, 1] be a piecewise mono-
tonic map with probability density function fq(·), which captures the long term
statistical behavior of a nonlinear dynamical system. It is natural (see [2]) to
consider the problem of minimizing or maximizing the functional
I[q(·), fq(·)] =
(q(t) − t)
fq(t)dt , (1)
which depends on q(·) and its probability density function fq(·) (usually a com-
plicated function of q(·)). It turns out that fq(·) is the fixed point of the
Frobenius-Perron operator Pq[·] associated with q(·). For a piecewise mono-
tonic map q : [0, 1] → [0, 1] with r pieces, Pq[·] has the representation
Pq[f ](t) =
v∈{q−1(t)}
|q̇(v)|
where for any point t ∈ [0, 1] the set {q−1(t)} consists of at most r points. The
fixed point fq(·) associated with an ergodic map q(·) can be expressed as the
limit
fq = lim
P iq [1] , (2)
where 1 is the constant function 1 on [0, 1]. Substituting (2) into (1), and using
the adjoint property [2, Prop. 4.2.6], one eliminates the probability density
function fq(·), obtaining (1) in the form
I[q(·)] =
t, q(t), q(2)(t), q(3)(t), . . .
where we are using the notation q(i)(·) to denote the i-th composition of q(·)
with itself: q(1)(t) = q(t), q(2)(t) = (q ◦ q)(t), q(3)(t) = (q ◦ q ◦ q)(t), etc. In [5] a
generalized Euler-Lagrange equation, which involves the inverse images of the
extremizing function q(·) (cf. (11)), was proved for such functionals in the cases
t, q(t), q(2)(t)
t, q(t), q̇(t), q(2)(t)
t, q(t), q(2)(t), q(3)(t)
To the best of our knowledge, these generalized Euler-Lagrange equations com-
prise all the available results on the subject. Thus, one concludes that the
theory of variational calculus with compositions is in its childhood: much re-
mains to be done. Here we go a step further in the theory of functionals con-
taining compositions. We are mainly interested in Noether’s classical theo-
rem, which is one of the most beautiful results of the calculus of variations
and optimal control, with many important applications in Physics (see e.g.
[6, 13, 14]), Economics (see e.g. [1, 17]), and Control Engineering (see e.g.
[11, 15, 18, 20, 22]), and source of many recent extensions and developments
(see e.g. [7, 8, 9, 10, 16, 19, 21]). Noether’s symmetry theorem describes the
universal fact that invariance with respect to some family of parameter transfor-
mations gives rise to the existence of certain conservation laws, i.e. expressions
preserved along the Euler-Lagrange or Pontryagin extremals of the problem.
Our results are a generalized DuBois-Reymond necessary optimality condition
(Theorem 7), and a generalized Noether’s theorem (Theorem 13) for function-
als of the form
t, q(t), q̇(t), q(2)(t)
dt. In §4 an illustrative example is
presented.
2 Preliminaries – review of classical results of
the calculus of variations
There exist many different ways to prove the classical Noether’s theorem (cf.
e.g. [6, 12, 13, 17]). We review here one of those proofs, which is based on the
DuBois-Reymond necessary condition. Although this proof is not so common
in the literature of Noether’s theorem, it turns out to be the most suitable
approach when dealing with functionals containing compositions.
Let us consider the fundamental problem of the calculus of variations:
I[q(·)] =
L (t, q(t), q̇(t)) dt −→ min (P)
under the boundary conditions q(a) = qa and q(b) = qb, where q̇ =
, with q(·)
a piecewise-smooth function, and the Lagrangian L : [a, b] × Rn × Rn → R is a
C2 function with respect to all its arguments.
The concept of symmetry has a very important role in mathematics and its
applications. Symmetries are defined through transformations of the system
that leave the problem invariant.
Definition 1 (Invariance of (P)). The integral functional (P) it said to be
invariant under the ε-parameter infinitesimal transformations
t̄ = t + ετ(t, q) + o(ε) ,
q̄(t) = q(t) + εξ(t, q) + o(ε) ,
where τ and ξ are piecewise-smooth, if
L (t, q(t), q̇(t)) dt =
∫ t̄(tb)
t̄(ta)
L (t̄, q̄(t̄), ˙̄q(t̄)) dt̄ (4)
for any subinterval [ta, tb] ⊆ [a, b].
Along the work we denote by ∂iL the partial derivative of L with respect to
its i-th argument.
Theorem 2 (Necessary condition of invariance). If functional (P) is invariant
under the infinitesimal transformations (3), then
∂1L (t, q, q̇) τ + ∂2L (t, q, q̇) · ξ + ∂3L (t, q, q̇) ·
ξ̇ − q̇τ̇
+ L (t, q, q̇) τ̇ = 0 . (5)
Proof. Since (4) is to be satisfied for any subinterval [ta, tb] ⊆ [a, b], equality (4)
is equivalent to
t + ετ + o(ε), q + εξ + o(ε),
q̇ + εξ̇ + o(ε)
1 + ετ̇ + o(ε)
= L (t, q, q̇) . (6)
We obtain (5) differentiating both sides of (6) with respect to ε, and then setting
ε = 0.
Another very important notion in mathematics and its applications is the
concept of conservation law. One of the most important conservation laws was
proved by Leonhard Euler in 1744: when the Lagrangian L(q, q̇) corresponds to
a system of conservative points, then
− L (q(t), q̇(t)) +
(q(t), q̇(t)) · q̇(t) ≡ constant , (7)
t ∈ [a, b], holds along the solutions of the Euler-Lagrange equations.
Definition 3 (Conservation law). A quantity C(t, q, q̇) defines a conservation
law if
C(t, q(t), q̇(t)) = 0 , t ∈ [a, b] ,
along all the solutions q(·) of the Euler-Lagrange equation
∂3L (t, q, q̇) = ∂2L (t, q, q̇) . (8)
Conservation laws can be used to lower the order of the Euler-Lagrange equa-
tions (8) and simplify the resolution of the respective problems of the calculus of
variations and optimal control [16]. Emmy Amalie Noether formulated in 1918
a very general principle on conservation laws, with many important implications
in modern physics, economics and engineering. Noether’s principle asserts that
“the invariance of the functional
L (t, q(t), q̇(t)) dt under one-parameter in-
finitesimal transformations (3), imply the existence of a conservation law”. One
particular example of application of Noether’s theorem gives (7), which corre-
sponds to conservation of energy in classical mechanics or to the income-wealth
law of economics.
Theorem 4 (Noether’s theorem). If functional (P) is invariant, in the sense
of the Definition 1, then
C(t, q, q̇) = ∂3L (t, q, q̇) · ξ(t, q) + (L(t, q, q̇) − ∂3L (t, q, q̇) · q̇) τ(t, q) (9)
defines a conservation law.
We recall here the proof of Theorem 4 by means of the classical necessary
optimality condition of DuBois-Reymond.
Theorem 5 (DuBois-Reymond condition). If q(·) is a solution of problem (P),
∂1L (t, q, q̇) =
[L (t, q, q̇) − ∂3L (t, q, q̇) · q̇] . (10)
Proof. The DuBois-Reymond necessary optimality condition is easily proved
using the Euler-Lagrange equation (8):
[L (t, q, q̇) − ∂3L (t, q, q̇) · q̇]
= ∂1L (t, q, q̇) + ∂2L (t, q, q̇) · q̇ + ∂3L (t, q, q̇) · q̈
∂3L (t, q, q̇) · q̇ − ∂3L (t, q, q̇) · q̈
= ∂1L (t, q, q̇) + q̇ ·
∂2L (t, q, q̇) −
∂3L (t, q, q̇)
= ∂1L (t, q, q̇) .
Proof. (of Theorem 4) To prove the Noether’s theorem, we use the Euler-
Lagrange equation (8) and the DuBois-Reymond condition (10) into the neces-
sary condition of invariance (5):
0 = ∂1L (t, q, q̇) τ + ∂2L (t, q, q̇) · ξ
+ ∂3L (t, q, q̇) ·
ξ̇ − q̇τ̇
+ L (t, q, q̇) τ̇
= ∂2L (t, q, q̇) · ξ + ∂3L (t, q, q̇) · ξ̇ + ∂1L (t, q, q̇) τ
+ τ̇ (L (t, q, q̇) − ∂3L (t, q, q̇) · q̇)
∂3L (t, q, q̇) · ξ + ∂3L (t, q, q̇) · ξ̇
(L (t, q, q̇) − ∂3L (t, q, q̇) · q̇) τ
+ τ̇ (L (t, q, q̇) − ∂3L (t, q, q̇) · q̇)
∂3L (t, q, q̇) · ξ +
L(t, q, q̇) − ∂3L (t, q, q̇) · q̇
3 Main results
We consider the following problem of the calculus of variations with composition
of functions:
I[q(·)] =
L (t, q(t), q̇(t), z(t)) dt −→ min (Pc)
subject to given boundary conditions q(a) = qa, q(b) = qb, z(a) = za, and
z(b) = zb, where q̇ =
and z(t) = (q ◦ q)(t). We assume that the Lagrangian
L : [a, b] × R × R × R → R is a function of class C2 with respect to all the
arguments, and that admissible functions q(·) are piecewise-smooth. The main
result of [5] is an extension of the Euler-Lagrange equation (8) for problems of
the calculus of variations (Pc).
Theorem 6 ([5]). If q(·) is a weak minimizer of problem (Pc), then q(·) satisfies
the Euler-Lagrange equation
∂2L (x, q(x), q̇(x), z(x)) −
∂3L (x, q(x), q̇(x), z(x))
+ ∂4L (x, q(x), q̇(x), z(x)) q̇(q(x)) +
t=q−1(x)
∂4L (t, q(t), q̇(t), z(t))
|q̇(t)|
= 0 (11)
for any x ∈ (a, b).
3.1 Generalized DuBois-Reymond condition
We begin by proving an extension of the DuBois-Reymond necessary optimality
condition (10) for problems of the calculus of variations (Pc).
Theorem 7 (cf. Theorem 5). If q(·) is a weak minimizer of problem (Pc), then
q(·) satisfies the DuBois-Reymond condition
L (x, q(x), q̇(x), z(x)) − ∂3L (x, q(x), q̇(x), z(x)) q̇(x)
= ∂1L (x, q(x), q̇(x), z(x)) − q̇(x)
t=q−1(x)
∂4L (t, q(t), q̇(t), z(t))
|q̇(t)|
for any x ∈ (a, b).
Remark 8. If L (t, q, q̇, z) = L (t, q, q̇), then (12) coincides with the classical
DuBois-Reymond condition (10).
Proof. To prove Theorem 7 we use the Euler-Lagrange equation (11):
L (x, q, q̇, z) − ∂3L (x, q, q̇, z) q̇
= ∂1L (x, q, q̇, z) + ∂2L (x, q, q̇, z) q̇
+ ∂3L (x, q, q̇, z) q̈ + ∂4L (x, q, q̇, z) q̇(q(x))q̇
∂3L (x, q, q̇, z) − ∂3L (x, q, q̇, z) q̈
= ∂1L (x, q, q̇, z) + q̇
∂2L (x, q, q̇, z)
+ ∂4L (x, q, q̇, z) q̇(q(x)) −
∂3L (x, q, q̇, z)
= ∂1L (x, q, q̇, z) − q̇(x)
t=q−1(x)
∂4L (t, q(t), q̇(t), z(t))
|q̇(t)|
3.2 Noether’s theorem for functionals containing compo-
sitions
We introduce now the definition of invariance for the functional (Pc). As done
in the proof of Theorem 2 (see (6)), we get rid off of the integral signs in (4).
Definition 9 (cf. Definition 1). We say that functional (Pc) is invariant under
the infinitesimal transformations (3) if
L (t̄, q̄(t̄), q̄′(t̄), z̄(t̄))
= L (t, q(t), q̇(t), z(t)) + o(ε) , (13)
where q̄′ = dq̄/dt̄.
Along the work, in order to simplify the presentation, we sometimes omit
the arguments of the functions.
Theorem 10 (cf. Theorem 2). If functional (Pc) is invariant under the in-
finitesimal transformations (3), then
∂1L (t, q, q̇, z) τ + ∂2L (t, q, q̇, z) ξ + ∂3L (t, q, q̇, z)
ξ̇ − q̇τ̇
+ ∂4L (t, q, q̇, z) q̇(q(t))ξ + ∂4L (t, q, q̇, z) ξ(q(t)) + Lτ̇ = 0 . (14)
Proof. Equation (13) is equivalent to
t + ετ + o(ε), q + εξ + o(ε),
q̇ + εξ̇ + o(ε)
1 + ετ̇ + o(ε)
q(q + εξ + o(ε)) + εξ(q + εξ + o(ε))
× (1 + ετ̇ + o(ε))
= L (t, q, q̇, z) + o(ε) . (15)
We obtain equation (14) differentiating both sides of equality (15) with respect
to the parameter ε, and then setting ε = 0.
Remark 11. Using the Frobenius-Perron operator (see [2, Chap. 4]) and the
Euler-Lagrange equation (11), we can write (14) in the following form:
∂1L (x, q, q̇, z) τ + ∂2L (x, q, q̇, z) ξ
+ ∂3L (x, q, q̇, z)
ξ̇ − q̇τ̇
+ ∂4L (x, q, q̇, z) q̇(q(x))ξ
t=q−1(x)
∂4L (t, q(t), q̇(t), z(t))
|q̇(t)|
ξ + Lτ̇
= ∂1L (x, q, q̇, z) τ +
∂3L (x, q, q̇, z) ξ
+ ∂3L (x, q, q̇, z)
ξ̇ − q̇τ̇
+ Lτ̇ = 0 . (16)
Definition 12 (Conservation law for (Pc)). We say that a quantity C (x, q, q̇, z)
defines a conservation law for functionals containing compositions if
C (x, q(x), q̇(x), z(x)) = 0
along all the solutions q(·) of the Euler-Lagrange equation (11).
Our main result is an extension of Noether’s theorem for problems of the
calculus of variations (Pc) containing compositions.
Theorem 13 (Noether’s theorem for (Pc)). If functional (Pc) is invariant, in
the sense of the Definition 9, and there exists a function f = f(x, q, q̇, z) such
(x, q(x), q̇(x), z(x)) = τ q̇(x)
t=q−1(x)
∂4L (t, q(t), q̇(t), z(t))
|q̇(t)|
, (17)
C (x, q(x), q̇(x), z(x))
L(x, q(x), q̇(x), z(x)) − ∂3L (x, q(x), q̇(x), z(x)) q̇
τ(x, q)
+ ∂3L (x, q(x), q̇(x), z(x)) ξ(x, q) + f(x, q(x), q̇(x), z(x)) (18)
defines a conservation law (Definition 12).
Remark 14. If L (x, q, q̇, z) = L (x, q, q̇), then f is a constant and expression
(18) is equivalent to the conserved quantity (9) given by the classical Noether’s
theorem.
Proof. To prove the theorem, we use conditions (12) and (17) in (16):
0 = ∂1L (x, q, q̇, z) τ +
∂3L (x, q, q̇, z) ξ
+ ∂3L (x, q, q̇, z)
ξ̇ − q̇τ̇
+ Lτ̇
L (x, q, q̇, z) − ∂3L (x, q, q̇), z)) q̇
+ τ̇ [L (x, q, q̇, z) − ∂3L (x, q, q̇), z)) q̇]
+ ξ̇∂3L (x, q, q̇, z) + ξ
∂3L (x, q, q̇, z)
+ τ q̇(x)
t=q−1(x)
∂4L (t, q(t), q̇(t), z(t))
|q̇(t)|
∂3L (x, q, q̇, z) ξ +
L (x, q, q̇, z)
− ∂3L (x, q, q̇, z) q̇
τ + f(x, q, q̇, z)
4 An example
Let us consider the problem
I[q(·)] =
[x + q(x) + q(q(x))] dx −→ min
q(0) = 1 , q(1) = 0 , (19)
q(q(0)) = 0 , q(q(1)) = 1 .
In [5, §3] it is proven that (19) has the extremal
q(x) =
q1(x) = −2x + 1 , x ∈
q2(x) = −2x + 2 , x ∈
that is, (20) satisfies the Euler-Lagrange equation (11) for L(x, q, q̇, z) = 1
(x + q + z).
We now illustrate the application of our Theorem 13 to this problem. First, we
need to determine the variational symmetries. Substituting the Lagrangian L
in (16) we obtain that
x + q + z
τ̇ = 0 . (21)
The differential equation (21) admits the solution
τ = ke−
x+q+z , (22)
where k is an arbitrary constant. From Theorem 13 we conclude that
(x + q1 + z1)τ +
τ q̇1
|q̇1(t)|
dx, x ∈
, (23)
(x + q2 + z2)τ +
τ q̇2
|q̇2(t)|
dx, x ∈
, (24)
defines a conservation law, where τ is obtained from (22):
τ = ke−
3x = kelnx
= kx−
3 , x ∈ [0, 1] . (25)
Since for this problem we know the extremal, we can verify the validity of the
obtained conservation law directly from Definition 12: substituting equalities
(20) and (25) in (23) and (24), we obtain, as expected, a constant (zero in this
case):
(x + q1 + z1)τ +
τ q̇1
|q̇1(t)|
= 3kxτ − 2
τdx = 3kx
3 − 3kx
3 = 0 ,
(x + q2 + z2)τ +
τ q̇2
|q̇2(t)|
= 3kxτ − 2
τdx = 3kx
3 − 3kx
3 = 0 .
5 Conclusions
We proved a generalization (i) of the necessary optimality condition of DuBois-
Reymond, (ii) of the celebrated Noether’s symmetry theorem, for problems of
the calculus of variations containing compositions (respectively Theorems 7 and
13). Our main result is illustrated with the example studied in [5].
The compositional variational theory is in its childhood, so that much re-
mains to be done. In particular, it would be interesting to obtain an Hamil-
tonian formulation and to study more general optimal control problems with
compositions.
Acknowledgements
The authors are grateful to Pawe l Góra who shared Chapter 4 of [2].
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	Introduction and motivation
	Preliminaries – review of classical results of the calculus of variations
	Main results
	Generalized DuBois-Reymond condition
	Noether's theorem for functionals containing compositions
	An example
	Conclusions
