Multi-Time Euler-Lagrange Dynamics - wseas

Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 2007

Multi-Time Euler-Lagrange Dynamics∗ CONSTANTIN UDRISTE University Politehnica of Bucharest Department of Mathematics Splaiul Indpendentei 313 ROMANIA [email protected]

IONEL TEVY University Politehnica of Bucharest Department of Mathematics Splaiul Indpendentei 313 ROMANIA [email protected]

Abstract: This paper introduces new types of Euler-Lagrange PDEs required by optimal control problems with performance criteria involving curvilinear or multiple integrals subject to evolutions of multidimensional-flow type. Particularly, the anti-trace multi-time Euler-Lagrange PDEs are strongly connected to the multi-time maximum principle. Section 1 comments the limitations of classical multi-variable variational calculus. Sections 2-3 refer to variational calculus with gradient variations and curvilinear or multiple integral functionals. Section 4 is dedicated to the study of the properties of multi-time Euler-Lagrange operator (affine changing of the Lagrangian, anti-trace multi-time Euler-Lagrange PDEs and new conservation laws). Section 5 formulates an application to multi-time rheonomic dynamics. Section 6 underlines the importance of the anti-trace multi-time Euler-Lagrange PDEs. Key–Words: gradient variations, multi-time Euler-Lagrange PDEs, multi-time maximum principle. Typing manuscripts, LATEX

1

Overview of classical multivariable variational calculus

The foundations of variational calculus have been built using the classic Lagrange variation of an admissible function, but this determines some limitations that are not suitable for multi-time control theory. The most important limitation comes from the fact that the classical multi-variable variational calculus cannot be applied directly to create a multi-time maximum principle. In fact, the functionals given as multiple integrals, subject to general variation functions produce multi-variable Euler-Lagrange or Hamilton PDEs containing a trace (total divergence), which is not convenient for the conservation of the Hamiltonian. Indeed, the Hamiltonian is not a first integral for the multi-variable Hamilton PDEs, even in the autonomous case. Our multi-time control theory successfully overcomes the previous limitations [3]-[8]. This theory requires m-needle-shaped variations and complete integrability conditions as core issues. Adding new ideas in variational calculus via the gradient variations in curvilinear and multiple integrals and the anti-trace Euler-Lagrange or Hamilton PDEs, we have justified a multi-time maximum principle which is similar to the Pontryaguin maximum principle. The previous two types of functional variations ∗

WSEAS Transactions on Mathematics, ., . (2007), ...-....

can be considered as ”celebrities” of the optimization theory, although the use of classical variations is hardly compatible with amplitude constraints, while m-needle-shaped variations are barely used in smooth optimization problems.

2

Curvilinear integral functional and gradient variations

Let xi , i = 1, . . . , n denote the field variables on the target space Rn , let tα , α = 1, . . . , m be the multitime variables on the source space Rm , and let xiα = ∂xi be the partial velocities. In this context, the jet ∂tα bundle of order one is the manifold J 1 (Rm , Rn ) = {(tα , xi , xiα )}. Assume we are given a smooth completely integrable 1-form m L = Lβ (x(t), xγ (t))dtβ , β, γ = 1, ..., m, t ∈ R+ ,

called autonomous Lagrangian 1-form. The Lagrangian 1-form is determined by the Lagrange covector field Lβ (x(t), xγ (t)). The complete integrability conditions are ∂Lβ ∂xi ∂Lβ ∂xiγ ∂Lλ ∂xi ∂Lλ ∂xiγ + = + . ∂xi ∂tλ ∂xiγ ∂tλ ∂xi ∂tβ ∂xiγ ∂tβ Let Γ0,t0 be an arbitrary piecewise C 1 curve joinm , and let Ω ing the points 0 and t0 in R+ 0,t0 be a par-

66

Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 2007

allelepiped represented as the closed interval 0 ≤ t ≤ t0 , fixed by the diagonal opposite points 0(0, ..., 0) m and t0 = (t10 , ..., tm 0 ) in R+ . We fix the multi-time m t0 ∈ R+ , and two points x0 , x1 ∈ Rn and we introduce a new problem of the calculus of variations asking to find an m-sheet x∗ (·) : Ω0,t0 → Rn that minimizes the functional (curvilinear integral) Z

J(x(·)) = Γ0,t0

Z



J() = Γ0,t0

Lσ x(t) + λ yλ (t),

∂x ∂yλ (t) + λ γ (t) dtσ γ ∂t ∂t 

Lβ (x(t), xγ (t))dtβ , α, β = 1, ..., m, and we write

among all functions x(·) satisfying the conditions x(0) = x0 , x(t0 ) = x1 , using C 1 gradient variation functions constrained by boundary conditions. Fundamental question: how can we characterize that function x∗ (·) which is the solution of the previous variational problem? Theorem 1 (multi-time non-homogeneous Euler-Lagrange PDEs). If the m-sheet x∗ (·) minimizes the functional J(x(·)) in the previous sense, then x∗ (·) is a solution of the multi-time non-homogeneous Euler-Lagrange PDEs ∂Lβ ∂ ∂Lβ − = cβi , i = 1, ..., n, β, γ = 1, ..., m. ∂xi ∂tγ ∂xiγ (E − L)1 Here we have a system of nm second order PDEs with n unknown functions xi (·). Theorem 1 shows that if we can solve the (E − L)1 PDEs system, then the minimizer of the functional J α (assuming that it exists) will be among the solutions. Proof. Step 1. Select a smooth gradient variation yα : Ω0,t0 → Rnm , yαi (0) = 0, yαi (t0 ) = 0 with the primitive y : Ω0,t0 → Rn , y i (0) = 0, y i (t0 ) = 0. We add the conditions (complete integrability conditions of 1-forms L and of gradient variations) ∂yγi ∂Lβ i ∂Lβ ∂yγi ∂Lλ i ∂Lλ ∂yγi ∂yβi y + = y + , = . ∂xi λ ∂xiγ ∂tλ ∂xi β ∂xiγ ∂tβ ∂tγ ∂tβ (CI)1 Let us define the parameter  = (1 , ..., m ) and α

∂J Step 2. We compute the partial derivatives β ∂ of the function

α

β

J () = J (x(·) +  yβ (·)), for  ∈ Rm and we write x(·) = x∗ (·), i.e., we omit the superscript ∗. We remark that the perturbed function x(·) + β yβ (·) takes the same values as x(·) at the diagonal points (endpoints) 0 and t0 . Since x(·) is a minimizer, we can write J() ≥ J(x(·)) = J(0). In this way, the function J() has a minimum at the point  = 0, and consequently this must be a critical ∂J point, i.e., β (0) = 0. ∂

∂J 0 = β (0) = ∂



Z Γ0,t0

∂Lσ (x(t), xγ (t))yβi (t) ∂xi

∂yβi ∂Lσ (t) dtσ . + (x(t), x (t)) γ ∂xiγ ∂tγ !

To process this formula we use the conditions for variations, writing ∂Lσ i ∂Lσ ∂yγi y + ∂xi β ∂xiγ ∂tβ

Z

0= Γ0,t0

Z

= Γ0,t0

∂Lβ ∂yγi ∂Lβ i y + ∂xi σ ∂xiγ ∂tσ ∂Lβ γ ∂ ∂Lβ δσ − σ i ∂x ∂t ∂xiγ

Z

= Γ0,t0

Z

+ Γ0,t0

∂ ∂tσ

!

dtσ

!

dtσ

!

yγi dtσ

!

∂Lβ i y dtσ . ∂xiγ γ

Step 3. Since the 1-forms ∂Lβ γ ∂ ∂Lβ δ − ∂xi σ ∂tσ ∂xiγ

!

dtσ

must be pullbacks of some 1-forms d(Aγβi ), we can evaluate the first curvilinear integral using the formula ∂y i ∂ ∂ γ i (y d(A )) = d(Aγβi ) + y i γ d(Aγβi ). βi γ γ ∂t ∂t ∂t ∂ d(Aγβi ) = 0, for all variγ ∂t Γ0,t0 ations yγi with y i (0) = 0, y i (t0 ) = 0, yαi (0) = 0, yαi (t0 ) = 0 satisfying (CI)1 and for all curves Γ0,t0 in the curvilinear !integrals. Consequently, ∂Lβ γ ∂ ∂ ∂Lβ = 0. Therefore the (E − δ − ∂tγ ∂xi σ ∂tσ ∂xiγ L)1 PDEs hold for all multi-times t ∈ Ω0,t0 . Remark. There are two other ways to process the previous formula, but finally they are inconvenient because of the complete integrability conditions: Z

One obtains

yi

67

Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 2007

1) the creation of a divergence operator, used usually in the variational calculus for multiple integrals, ∂ ∂Lσ ∂Lσ − γ i ∂x ∂t ∂xiγ

Z Γ0,t0

Z

+ Γ0,t0

∂ ∂tγ

!

yβi dtσ

!

∂Lσ i y dtσ ; ∂xiγ β

2) the introduction a procedure suggested by our multi-time maximum principle theory, based on the ∂yβi ∂yγi idea γ = β , namely ∂t ∂t ∂Lσ γ i ∂Lσ ∂yβi δ y + ∂xi β γ ∂xiγ ∂tγ

Z

0= Γ0,t0

∂Lσ γ i ∂Lσ ∂yγi δ y + ∂xi β γ ∂xiγ ∂tβ

Z

= Γ0,t0

∂ ∂Lσ ∂Lσ γ δβ − β i ∂x ∂t ∂xiγ

Z

= Γ0,t0

Z

+ Γ0,t0

3

∂ ∂tβ

!

dtσ

sense, then x∗ (·) is a solution of the multi-time non-homogeneous Euler-Lagrange PDEs ∂ ∂L ∂L − γ i = ci , i = 1, ..., n, γ = 1, ..., m. i ∂x ∂t ∂xγ (E − L)2 Here we have a system of n second order PDEs with n unknown functions xi (·). Theorem 2 shows that if we can solve the (E − L)2 PDEs system, then the minimizer of the functional J (assuming it exists) will be among the solutions. Proof. Step 1. Select a smooth gradient variation yα : Ω0,t0 → Rnm , yαi |∂Ω0,t0 = 0 with the primitive y : Ω0,t0 → Rn , y i |∂Ω0,t0 = 0. We add the complete integrability conditions ∂yβi ∂yγi = . ∂tγ ∂tβ

!

dtσ

Define the parameter  = (1 , ..., m ) and

!

yγi dtσ J() = J(x(·) + β yβ (·)),

!

∂Lσ i y dtσ . ∂xiγ γ

Multiple integral functional and gradient variations

for  ∈ Rm and we write x(·) = x∗ (·), i.e., we omit the superscript ∗. We remark that the perturbed function x(·) + β yβ (·) takes the same values as x(·) at the boundary of Ω0,t0 . Since x(·) is a minimizer, we can write

Assume we are given a smooth Lagrangian

J() ≥ J(x(·)) = J(0).

m L(x(t), xγ (t)), t ∈ R+ . m , the parallelepiped We fix the multi-time t0 ∈ R+ m Ω0,t0 ⊂ R+ with the diagonal opposite points 0(0, ..., 0) and t0 = (t10 , ..., tm 0 ), and two points x0 , x1 ∈ Rn .. We introduce a new problem of the calculus of variations asking to find an m-sheet x∗ (·) : Ω0,t0 → Rn that minimizes the functional (multiple integral)

Z

J(x(·)) =

(CI)2

In this way, the function J() has a minimum at the point  = 0, and consequently this must be a critical ∂J point, i.e., β (0) = 0. ∂ ∂J Step 2. We compute the partial derivatives β ∂ of the function 

Z

J() = Ω0,t0

1

L x(t) + λ yλ (t),

∂x (t) ∂tγ

m

L(x(t), xγ (t))dt ...dt , Ω0,t0

among all functions x(·) satisfying the conditions x(0) = x0 , x(t0 ) = x1 , using gradient variation functions (satisfying not only usual boundary conditions but also the complete integrability conditions). Fundamental question: how can we characterize that function x∗ (·) which is the solution of the previous variational problem? Theorem 2 (multi-time non-homogeneous Euler-Lagrange PDEs). If the m-sheet x∗ (·) minimizes the functional J(x(·)) in the previous

∂yλ λ γ (t) ∂t



+ and we put

dt1 ...dtm ,

∂J (0) = 0 or ∂β Z

0= Ω0,t0



∂L (x(t), xγ (t))yβi (t) ∂xi

∂yβi ∂L + (x(t), xγ (t)) γ (t) dt1 ...dtm . ∂xiγ ∂t !

68

Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 2007

To process this formula we use the conditions (CI)2 for the variations, writing

together with the non-homogeneous Euler-Lagrange PDEs system

∂L i ∂L ∂yγi y + β ∂xi ∂xiγ ∂tβ

dt ...dt

∂L ∂ ∂L − γ i = ci , i = 1, ..., n, γ = 1, ..., m. i ∂x ∂t ∂xγ

yγi dt1 ...dtm

The non-homogeneous Euler-Lagrange PDEs system becomes homogeneous if the Lagrangian L is replaced by

Z

0= Ω0,t0

∂L γ ∂ ∂L δ − ∂xi β ∂tβ ∂xiγ

Z

= Ω0,t0

Z

+ Ω0,t0

∂ ∂tβ

! 1

m

!

!

∂L i y dt1 ...dtm . ∂xiγ γ

The last integral is evaluated by the Fubini formula. The first integral can be modified via Gauss formula since ∂ ∂y i γ ∂ γ i γ (y B ) = Bβi + y i γ Bβi βi γ γ ∂t ∂t ∂t γ Bβi =

∂ ∂L ∂L γ δβ − β i . i ∂x ∂t ∂xγ

Step 3. It remains Z

∂ y γ ∂t i

Ω0,t0

∂L γ ∂ ∂L δβ − β i i ∂x ∂t ∂xγ

∂L γ ∂ ∂L δβ − β i i ∂x ∂t ∂xγ

4.1

where ∂Aγj i ∂Aγj ∂Aγi ∂B ( j − )x − γ + j = cj . ∂x ∂xi γ ∂t ∂x 1 A particular solution is W = 0, Aαi = − m ci tα . Combining the previous remarks with some formulas in Sections 2-3, we obtain the following Theorem 4. The Euler-Lagrange operator EL has the property

ˆ h) = EL(L, ∇h), EL(L,

dt1 ...dtm .

!

=0

and consequently the multi-time (E − L)2 PDEs hold for t ∈ Ω0,t0 .

4

+Aαi (t, x(t))xiα ,

!

This equality holds for all differentiable variations y i with y i |∂Ω0,t0 = 0. Therefore ∂ ∂tγ

ˆ x(t), xγ (t)) = L(t, x(t), xγ (t)) + W (t, x(t)) L(t,

Properties of multi-time EulerLagrange operator Affine changing of the Lagrangian m R+

Suppose that L(t, x(t), xγ (t)), t ∈ is a smooth Lagrangian. The following Lemma is well-known. Lemma 3. The Euler-Lagrange derivative EL(L) is independent of the partial accelerations xαβ if and only if the Lagrangian L is an affine function in velocities, i.e., L(t, x(t), xγ (t)) = W (t, x(t)) + Aαi (t, x(t))xiα . Let us consider the homogeneous Euler-Lagrange PDEs system ∂ ∂L ∂L − γ i = 0, i = 1, ..., n, γ = 1, ..., m (1) i ∂x ∂t ∂xγ

where h stands for standard variations. Therefore, the non-homogeneous Euler-Lagrange PDEs system (which originally was obtained from L using gradient variations) is a homogeneous sysˆ and the standard tem for the modified Lagrangian L variations. From another point of view, the nonhomogeneous Euler-Lagrange PDEs system is a controlled Lagrangian system for a constant control [1]. Open problem. What is the sense of affine changing of classical Lagrangians? What is the sense of homographic changing of classical Lagrangians? When the gradient variations are adequate?

4.2

Anti-trace multi-time Euler-Lagrange PDEs

The statements in Sections 2-3 suggest to introduce the anti-trace multi-time Euler-Lagrange PDEs ∂L γ ∂ ∂L δβ − β i = 0, i ∂x ∂t ∂xγ as generalizations of classical homogeneous EulerLagrange PDEs. But, while the classical homogeneous Euler-Lagrange equations are invariant with re0 0 spect to changes of variables (t, x) → (t , x ), the anti-trace multi-time Euler-Lagrange PDEs are invari0 ant only with respect to (t, x) → (tα = aαβ tβ + 0 bα , x ).

69

Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 2007

Let us consider the anti-trace procedure as an algebraic operator applied to Euler-Lagrange operators. Looking for the inverse operator, we discovered two inverses: one algebraic and one differential. Theorem 5. 1) The trace followed by a fibre scaling is an algebraic inverse of the anti-trace procedure. 2) The divergence is a differential inverse of the anti-trace procedure. Proof. 1) If we start with anti-trace multi-time Euler-Lagrange PDEs, by the trace after β, γ and by scaling the partial velocities, in the sense of changing the Lagrangian after the rule L(t, x(t), xγ (t)) = L1 (t, x(t), mxγ (t)), we obtain the classical Euler-Lagrange PDEs associated to the new Lagrangian L1 . 2) Applying the divergence operator to anti-trace multi-time Euler-Lagrange PDEs, i.e., ∂ ∂tγ we find

∂ ∂L ∂L γ δβ − β i i ∂x ∂t ∂xγ

4.4

New conservation laws and anti-trace PDEs in case of curvilinear integrals

When we use path independent curvilinear integral functionals, we have similar properties. But, a smooth m produces two Lagrangian L(x(t), xγ (t)), t ∈ R+ smooth completely integrable 1-forms: - the differential dL =

of components (

(dxi , dxiγ ); - the restriction of dL to (x(t), xγ (t)), i.e., the pullback

= 0,

∂L ∂xi ∂L ∂xiγ + ∂xi ∂tβ ∂xiγ ∂tβ

!

dtβ ,

of components ∂ ∂L ∂L − γ i = ci . i ∂x ∂t ∂xγ

Lβ (x(t), xγ (t)) =

Consequently the divergence and the anti-trace procedure are related as ”the primitive with the derivative”. +

4.3

∂L ∂L , ), with respect to the basis ∂xi ∂xiγ

dL|(x(t),xγ (t)) =

!

∂L i ∂L dx + i dxiγ i ∂x ∂xγ

New conservation law in case of multiple integrals

We start with the autonomous Lagrangian L(x(t), xγ (t)) and the associated homogeneous Euler-Lagrange PDEs (1). Given the m-sheet x(·), let us introduce the multi-momentum p = (pαi ) by the re∂L lations pαi (t) = (x(t), xγ (t)). Suppose that these ∂xiα nm equations define nm functions xiγ = xiγ (x, p). Sometimes, the variables x and p are called canonical variables. We define two tensor fields: − anti-trace Euler-Lagrange tensor field, Aαβi (x, xγ ) =

∂L ∂ ∂L (x, xγ )δβα − β i (x, xγ ), i ∂x ∂t ∂xα

− energy-momentum tensor field Tβα (x, p) = pαi xiβ (x, p) − L(x, p)δβα . These tensor fields represent conservation laws for Euler-Lagrange PDEs (1) since ∂ α ∂ α A = 0, T =0 ∂tα βi ∂tα β along the solutions of (1). While the second law is well-known, the first is new and it appears from the anti-trace idea.

∂xi ∂L (x(t), x (t)) (t) γ ∂xi ∂tβ

∂xiγ ∂L (x(t), x (t)) (t), γ ∂xiγ ∂tβ

with respect to the basis (dtβ ). In this case, we must underline that we have two different anti-trace EulerLagrange tensor fields

Aγαβi (x, xλ ) =

∂Lα ∂ ∂Lα (x, xλ )δβγ − β (x, xλ ) ∂xi ∂t ∂xiγ

γ Bβαi (x, xλ ) =

∂Lβ ∂ ∂Lα (x, xλ )δαγ − β (x, xλ ), ∂xi ∂t ∂xiγ

since the following Theorem is true. Theorem 6. The relations ∂ ∂Lα ∂ ∂Lβ − α =0 ∂tβ ∂x ∂t ∂x ∂ ∂Lα ∂ ∂Lβ ∂ ∂L − α = (δαλ δβγ − δβλ δαγ ) λ ∂tβ ∂xγ ∂t ∂xγ ∂t ∂x ∂ γ ∂ γ Aαβi (x, xλ ) = γ Bβαi (x, xλ ) γ ∂t ∂t hold true. The first relation can be written as a conservation law ∂ ∂Lα λ ∂Lβ λ ( δ − δ )=0 ∂tλ ∂x β ∂x α

70

Proceedings of the 7th WSEAS International Conference on Systems Theory and Scientific Computation, Athens, Greece, August 24-26, 2007

where t1 = x, t2 = y, t3 = z, t4 = it. The extremals of L under gradient variations are described by the controlled wave PDEs

or as ∂Lα λ ∂Lβ λ ∂ ∂ ∂L δβ − δα = ( α δβλ − β δαλ ) . ∂x ∂x ∂t ∂t ∂x It implies the third relation between anti-trace multitime Euler-Lagrange operators. Furthermore, the tenγ sor fields Aγαβi (x, xλ ), Bβαi (x, xλ ) represent equivalent conservation laws for the homogeneous EulerLagrange PDEs ∂Lα ∂ ∂Lα − γ = 0. i ∂x ∂t ∂xiγ

(2)

Indeed, their difference is a tensor of ”curl” type and ∂ γ ∂ γ Aαβi (x, xλ ) = 0, γ Bβαi (x, xλ ) = 0 γ ∂t ∂t along the solutions of (2). Starting from the Euler-Lagrange PDEs (2), we introduce: 1) the first kind of anti-trace multi-time EulerLagrange PDEs ∂Lα ∂ ∂Lα (x, xλ )δβγ − β (x, xλ ) = 0; i ∂x ∂t ∂xiγ

∂Lβ ∂ ∂Lα (x, xλ )δαγ − β (x, xλ ) = 0. ∂xi ∂t ∂xiγ Both anti-trace procedures have the same differential inverse (divergence), but different algebraic inverses (the first, trace followed by a fibre scaling; the second, the trace). Consequently, applying the divergence, both imply the same controlled Euler-Lagrange PDEs system ∂Lα ∂ ∂Lα − γ = cαi . i ∂x ∂t ∂xiγ That is why, the solutions of anti-trace multi-time Euler-Lagrange PDEs are among the solutions of the controlled Euler-Lagrange PDEs.

Application in Multi-Time Rheonomic Dynamics

Our theory has applications in Relativistic and MultiTime Rheonomic Dynamics; the electromagnetic field E = (E i ), H = (H i ), i = 1, 2, 3 determines the density of electromagnetic deformation energy 1 L = δij δ αβ 2

∂E i ∂E j ∂H i ∂H j + ∂tα ∂tβ ∂tα ∂tβ

6

∂2Ei ∂2H i = bi , ∆H i − = ci , i = 1, 2, 3. 2 ∂t ∂t2

Conclusion

The present point of view regarding the multi-time Euler-Lagrange PDEs has the key ideas in the union between [3]-[8] and [1], [2]. Accepting that the evolution is m-dimensional, all the results confirm the possibility of passing from the single-time Pontryaguin’s maximum principle to a multi-time maximum principle [4]-[8]. Furthermore, the main results belong to PDEs-constrained optimal control theory. Acknowledgements: Partially supported by Grant CNCSIS 86/ 2007 and by 15-th ItalianRomanian Executive Programme of S&T Cooperation for 2006-2008, University Politehnica of Bucharest. References:

2) the second kind of anti-trace multi-time EulerLagrange PDEs

5

∆E i −

!

,

[1] D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden, C. A. Woolse, The euivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 393-422. [2] D. F. M. Torres, On the Noether theorem for optimal control, European Journal of Control (EJC), 8, 1 (2002), pp. 56-63. [3] C. Udris¸te, Geodesic motion in a gyroscopic field of forces, Tensor, N. S., 66, 3 (2005), 215228. [4] C. Udris¸te, Multi-time maximum principle, short communication at International Congress of Mathematicians, Madrid, August 22-30, 2006. [5] C. Udris¸te, Maxwell geometric dynamics, manuscript, 2007. [6] C. Udris¸te, M. Ferrara, Multi-time optimal economic growth, manuscript, 2007. [7] C. Udris¸te, A. M. Teleman, Hamiltonian approaches of field theory, IJMMS, 57 (2004), pp. 3045-3056. [8] C. Udris¸te, I. T¸evy, Multi-Time Euler-LagrangeHamilton Theory, WSEAS Transactions on Mathematics, 6, 6 (2007), 701-709.

71