Constants of motion for fractional action-like variational problems

Abstract

We extend Noether's symmetry theorem to the fractional Riemann-Liouville integral functionals of the calculus of variations recently introduced by El-Nabulsi.

Full text

arXiv:math/0607472v1 [math.OC] 19 Jul 2006
Constants of motion for fractional action-like
variational problems
∗
Gast˜ao S. F. Frederico
gfrederico@mat.ua.pt
Delfim F. M. Torres
delfim@mat.ua.pt
Department of Mathematics
University o f Aveiro
3810-193 Aveiro, Portugal
Abstract
We extend Noether’s symmetry theorem to the fractional Riemann-
Liouville integral functionals of the calculus of variations recently intro-
duced by El-Nabulsi.
AMS Subject Classification: 49K05, 49S05, 70H33.
Keywords and Phrases: fractional action-like variational approach,
symmetry, constants of motion, Noether’s theorem.
1 Introduction
The concept of symmetry plays an important role both in Physics and Math-
ematics. Symmetries are described by transformations of the system, which
result in the same object after the transformation is carried out. They are
described mathematically by parameter groups of transfo rmations. Their im-
portance ranges from fundamental and theoretical a spec ts to concrete applica-
tions, having profound implications in the dynamica l behavior of the systems,
and in their basic qualitative properties.
Another fundamental notion in Physics and Mathematics is the one of con-
stant of motion. Typical application o f the constants of motion in the calculus
of variations is to reduce the numbe r of degrees of freedom, thus reducing the
problems to a lower dimension, facilitating the integration of the differential
equations given by the necessary optimality conditions.
Emmy Noether was the first who proved, in 1918, that the notions of sym-
metry and cons tant of motion are connected: when a system exhibits a sy m-
metry, then a constant of motion can be obtained. One of the most impor tant
∗
Research Report CM06/I-24, Department of Mathematics, University of Aveiro.

and well known illustra tions of this deep and rich relation, is given by the
conservation of energy in Mechanics: the autonomous L agrangian L(q, ˙q), cor-
respondent to a mechanical system of conservative points, is invariant under
time-translations (time-homogeneity symmetry), and
L (q, ˙q) −
∂L
∂ ˙q
(q, ˙q) · ˙q ≡ constant (1)
follows from No e ther’s theorem, i.e., the total energy of a conservative system
always remain constant in time, “it cannot be created or destroyed, but only
transferred from one form into another”. Expression (1) is valid along all the
Euler-Lagrange extremals q of an autonomous problem of the calculus of vari-
ations. The co nstant of motion (1) is known in the calculus of variations as
the 2nd Erdmann necessary condition; in concrete applications, it gains differ-
ent interpre tations: conservation of energy in Mechanics; income-wealth law
in Economics; first law of Thermodynamics; etc. The literature on Noether’s
theorem is vast, and many extensions of the classical results of Emmy Noether
are now ava ilable in the liter ature (see e.g. [13, 14] and r e fere nce s therein).
Here we remark that constants of motion appear naturally in closed systems.
It turns out that in practical terms close d systems do not exis t: forces
that do not store energy, so -called nonconse rvative or dissipative forces, are
always present in real systems. Friction is an ex ample of a nonconser vative
force. Any friction-type force, like air resistance, is a nonconservative fo rce.
Nonconservative forces remove energy from the systems and, as a consequence,
the constant o f motion (1) is broken. This explains, for instance, why the
innumerable “perpetual motion machines” that have been proposed fail. In
presence of ex ternal nonconservative forces, Noether’s theorem and re spective
constants of motion cease to be valid. However, it is still possible to obtain
a Noether-type theorem which covers both conservative (closed system) and
nonconservative cases [3, 6]. Roughly sp eaking, one can prove that Noether’s
conservation laws are still valid if a new term, involving the nonconservative
forces, is added to the standard constants of motion.
The study of fractional problems of the calculus of variations and respec tive
Euler-Lagrange type equations is a subject of strong current research because of
its numerous applications: see e.g. [1, 2, 4, 5, 8, 9, 10, 1 1, 12]. F. Riewe [11, 12]
obtained a version of the Euler-Lagrange equations for problems of the calcu-
lus of variations with fractio nal derivatives, that combines the conservative and
non-conservative cas es. In 2002 O. Agrawal proved a formulation for variational
problems with right a nd left fractional derivatives in the Riemann-Lio uville
sense [1]. Then these Euler-Lagrange equations were used by D. Baleanu and
T. Avkar to investigate problems with Lagr angians which are linear on the ve-
locities [2]. In [8, 9] fractional problems of the calculus of variations with sym-
metric fractional derivatives are considered and correspondent Euler-Lagrange
equations obtained, using both Lagr angian and Hamiltonian formalisms. In
all the above mentioned studies, Euler-Lagrange equations depend on left and

right fractional derivatives, even when the problem depend only on one type
of them. In [10] problems depending on symmetric derivatives are considered
for which Euler-Lag range equations include only the derivatives that appear in
the formulation of the pro blem. In [4, 5] Riemann-Liouville fractional integral
functionals, depending on a parameter α but not on fr actional-order deriva-
tives of order α, are introduced and respective fractional Euler-La grange type
equations obtained.
A Noether-type theorem for problems of the calculus of var iations with
fractional-order derivatives of order α is given in [7]. Here we use the results
of E l-Nabulsi [4, 5] to prove a nonconservative Noether’s theorem in the new
fractional action-like framework.
2 Fractional action-like Noether’s theorem
We consider the fundamental problem of the calculus of variations with Riemann-
Liouville fractional integral, as cons idered by El-Nabulsi [4, 5]:
I[q(·)] =
1
Γ(α)
Z
b
a
L (θ, q(θ), ˙q(θ)) (t − θ)
α−1
dθ −→ min , (2)
under given boundary conditions q(a) = q
a
and q(b) = q
b
, where ˙q =
dq
dθ
, Γ is
the Euler gamma function, 0 < α ≤ 1, θ is the intrinsic time, t is the observer
time, t 6= θ, and the Lagrangian L : [a, b] × R
n
× R
n
→ R is a C
2
function
with respect to its ar guments. We will denote by ∂
i
L the partial derivative of
L with respect to the i-th argument, i = 1, 2, 3. Admissible functions q(·) are
assumed to be C
2
.
Theorem 1 (cf. [4]). if q is a minimizer of problem (2), then q satisfies
the following Euler-Lagrange eq uation:
∂
2
L (θ, q(θ), ˙q(θ)) −
d
dθ
∂
3
L (θ, q(θ), ˙q(θ)) =
1 − α
t − θ
∂
3
L (θ, q(θ), ˙q(θ)) . (3)
We now introduce the following definition of variational quasi-invariance up
to a gauge term (cf. [1 3]).
Definition 2 (quasi-invariance of (2) up to a gauge term Λ). Func-
tional (2) is said to be quasi-invariant under the infinitesimal ε-pa rameter trans-
formations
(
¯
θ = θ + ετ (θ, q) + o(ε)
¯q(
¯
θ) = q(θ) + εξ(θ, q) + o(ε)
(4)

up to the gauge term Λ if, and only if,
L

¯
θ, ¯q(
¯
θ), ¯q
′
(
¯
θ)

(t −
¯
θ)
α−1
d
¯
θ
dθ
= L (θ, q(θ), ˙q(θ)) (t − θ)
α−1
+ ε(t − θ)
α−1
dΛ
dθ
(θ, q(θ), ˙q(θ)) + o(ε) . (5)
Lemma 3 (necessary and sufficient condition for quasi-invariance).
If functional (2) is quasi-invariant up to Λ under the infinitesimal transforma-
tions (4), then
∂
1
L (θ, q, ˙q) τ + ∂
2
L (θ, q, ˙q) · ξ + ∂
3
L (θ, q, ˙q) ·

˙
ξ − ˙q ˙τ

+ L (θ, q, ˙q)

˙τ +
1 − α
t − θ
τ

=
˙
Λ (θ, q, ˙q) . (6)
Proof. Equality (5) is equivalent to
"
L
θ + ετ + o(ε), q + εξ + o(ε),
˙q + ε
˙
ξ + o(ε)
1 + ε ˙τ + o(ε)
!#
(t − θ − ετ − o(ε))
α−1
(1 + ε ˙τ + o(ε))
= L (θ, q, ˙q) (t − θ)
α−1
+ ε(t − θ)
α−1
d
dθ
Λ (θ, q, ˙q) + o(ε) . (7)
Equation (6) is obtained differentiating both sides o f equality (7) with respect
to ε and then putting ε = 0.
Definition 4 (constant of motion). A quantity C (θ, q(θ), ˙q(θ)), θ ∈
[a, b], is said to be a constant of motion if, and only if,
d
dθ
C (θ, q(θ), ˙q(θ )) = 0
for all the solutions q of the Euler-Lagrange equation (3).
Theorem 5 (Noether’s theorem). If the fractional integral (2) is quasi-
invariant up to Λ, in the sense of Definition 2, and functions τ(θ, q) and ξ(θ, q)
satisfy the condition
L (θ, q, ˙q) τ = −∂
3
L (θ, q, ˙q) · (ξ − ˙qτ) , (8)
then
∂
3
L (θ, q, ˙q) · ξ(θ, q) + [L(θ, q, ˙q) − ∂
3
L (θ, q, ˙q) · ˙q] τ(θ, q) − Λ (θ, q, ˙q) (9)
is a consta nt of motion.
Remark 6. Under our hypo thesis (8) the necess ary and sufficient condi-
tion of quasi-invariance (6) is reduced to
∂
1
L (θ, q, ˙q) τ + ∂
2
L (θ, q, ˙q) · ξ + ∂
3
L (θ, q, ˙q) ·

˙
ξ − ˙q ˙τ

+ L (θ, q, ˙q) ˙τ −
1 − α
t − θ
∂
3
L (θ, q, ˙q) · (ξ − ˙qτ) =
˙
Λ (θ, q, ˙q) . (10)

Conditions (8) and (10) correspond to the generalized equations of Noether-
Bessel-Hagen of a non-conservative mechanical system [3].
Proof. We can write (10) in the form

∂
1
L (θ, q, ˙q) +
1 − α
t − θ
∂
3
L (θ, q, ˙q) · ˙q

τ + [L (θ, q, ˙q) − ∂
3
L (θ, q, ˙q) · ˙q] ˙τ
+

∂
2
L (θ, q, ˙q) −
1 − α
t − θ
∂
3
L (θ, q, ˙q)

· ξ + ∂
3
L (θ, q, ˙q) ·
˙
ξ −
˙
Λ = 0 . (11)
Using the Euler-Lagr ange equation (3) equality (11) is e quivalent to
d
dθ
[L (θ, q, ˙q) − ∂
3
L (θ, q, ˙q) · ˙q] τ + [L (θ, q, ˙q) − ∂
3
L (θ, q, ˙q) · ˙q] ˙τ
+
d
dθ
[∂
3
L (θ, q, ˙q)] · ξ + ∂
3
L (θ, q, ˙q) ·
˙
ξ −
˙
Λ = 0
and the intended conclusion follows:
d
dθ
[∂
3
L (θ, q, ˙q) · ξ + (L(θ, q, ˙q) − ∂
3
L (θ, q, ˙q) · ˙q) τ − Λ (θ, q, ˙q)] = 0 .
3 Examples
In [5, §4] El-Nabulsi remarks that conservation of momentum when L is not a
function of q or conser vation of energy when L has no explicit dependence on
time θ are no more true for a fra ctional order of integration α, α 6= 1 . As we
shall see now, these facts are a trivial c onsequence o f our Theorem 5. Moreover,
our Noether’s theorem gives new explicit formulas for the fractional constants
of motion. For the particular case α = 1 we r e cover the classical constants of
motion of momentum and energy.
Let us first c onsider an arbitrary fractional action-like problem (2) with an
autonomous L: L (θ, q, ˙q) = L (q, ˙q). In this case ∂
1
L = 0, and it is a simple
exercise to check that (10) is satisfied with τ = 1, ξ = 0 and Λ given by
˙
Λ =
1 − α
t − θ
∂L
∂ ˙q
· ˙q .
It follows from our Noether’s theorem (Theore m 5) that
L (q, ˙q) −
∂L
∂ ˙q
(q, ˙q) · ˙q − (1 − α)
Z
1
t − θ
∂L
∂ ˙q
(q, ˙q) · ˙q dθ ≡ constant . (12)
In the classical framework α = 1 and we then get from our expression (12)
the well known constant of motion (1), which corresponds in mechanics to
conservation of energy.

When L is not a function of q one has
∂L
∂q
= 0 and (10) holds true with
τ = 0, ξ = 1 and Λ given by
˙
Λ = −
1 − α
t − θ
∂L
∂ ˙q
(θ, ˙q) .
The constant of motion (9) takes the form
∂L
∂ ˙q
(θ, ˙q) + (1 − α)
Z
1
t − θ
∂L
∂ ˙q
(θ, ˙q) dθ . (13)
For α = 1 (13) implies conservation of momentum:
∂L
∂ ˙q
= const.
Acknowledgments
The first author acknowledges the suppo rt of the Portuguese Institute for Devel-
opment (IPAD); the second author the support by the Centre for Research on
Optimization and Control (CEOC) from the “Funda˜ao para a Ciˆencia e a Tec-
nologia” (FCT), cofinanced by the European Community Fund FEDER/POCTI.
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