Intelligent Computational Methods for Financial Engineering

PII. S0161171204402282 http://ijmms.hindawi.com

© Hindawi Publishing Corp.

HAMILTONIAN APPROACHES OF FIELD THEORY

CONSTANTIN UDRI¸STE and ANA-MARIA TELEMAN Received 18 February 2004

We extend some results and concepts of single-time covariant Hamiltonian field theory to the new context of multitime covariant Hamiltonian theory. In this sense, we point out the role of the polysymplectic structureδ⊗J, we prove that the dual action is indefinite, we find the eigenvalues and the eigenfunctions of the operator(δ⊗J)(∂/∂t)with periodic boundary conditions, and we obtain interesting inequalities relating functionals created by the new context. As an important example for physics and differential geometry, we study the mul- titime Yang-Mills-Witten Hamiltonian, extending the Legendre transformation in a suitable way. Our original results are accompanied by well-known relations between Lagrangian and Hamiltonian, and by geometrical explanations regarding the Yang-Mills-Witten Lagrangian.

2000 Mathematics Subject Classification: 53C07, 46E20, 55R10.

1. Introduction. As it is well known, the fields theories have a Lagrangian and Hamil- tonian structure, since the interactions are mathematically modeled with energetic Lagrangians-Hamiltonians (related by Legendre transformations) and their associated action functionals.

The single-time Hamilton equations appeared for the first time in a paper of Lagrange (1809) on perturbation theory, but it was Cauchy (1831) who first gave the true signif- icance of those equations. In 1835, Hamilton put those equations at the basis of his analytical mechanics and gave the first exact formulation of the least action principle.

The theory of single-time Hamilton equations with periodic boundary conditions was initiated by Birkhoff (1917) who proved that the dual action is indefinite. Since then, there were produced a lot of valuable papers that are now summarized in the book of Mawhin and Willem [6].

The multitime Hamilton equations were first written in a book by de Donder [2].

But, to the authors’ knowledge, there is no prior work aimed at developing a theory of multitime Hamilton equations with periodic boundary conditions. This is the first subject of the present paper.

Some of the most important Lagrangians are those defined by Yang and Mills (1953) and Witten [18] to produce modern explanations in quantum field theory (Maxwell or Dirac equations). The second subject of the present paper is to introduce and study the (single-time and multitime) Hamiltonian for a Yang-Mills-Witten functional.

Section 1of this paper contains historical and bibliographical notes.Section 2(Sec- tion 3) recalls the relations between the equations of single-time (multitime) first-order Lagrangian field theory and the covariant Hamilton equations on the finite-dimen- sional symplectic (polysymplectic) phase space of covariant Hamiltonian field theory.

Section 4develops a theory of multitime Hamilton equations with boundary conditions.

Section 5studies the Yang-Mills-Witten Hamiltonian.

2. Single-time Hamilton canonical equations. IfL:R×R²ⁿ→R,(t, xⁱ,x˙ⁱ)→L(t, xⁱ,

˙

xⁱ)is a given Lagrangian, then the associated Euler-Lagrange equations are d

dt

∂L

∂x˙ⁱ= ∂L

∂xⁱ, i=1, . . . , n. (2.1) Hamilton (1834) simplified the structure of the Euler-Lagrange equations and turned them into a form that has remarkable symmetry by

(1) introducing the conjugate momenta

pi= ∂L

∂x˙ⁱ; (2.2)

(2) considering the Hamiltonian

H=pix˙ⁱ−L (2.3)

as a function oft,x,p.

We suppose that (2.2) defines, for everyx, a diffeomorphism ˙xⁱ→pi. This map is called theLegendre transform.

Theorem 2.1. The Euler-Lagrange equations (2.1) are equivalent to the Hamilton equations

dxⁱ dt = ∂H

∂pi

, dpi

dt = −∂H

∂xⁱ, i=1, . . . , n. (2.4) Proof. The definitions (2.2) and (2.3) imply

∂H

∂xⁱ=pj∂x˙^j

∂xⁱ− ∂L

∂xⁱ− ∂L

∂x˙^j

∂x˙^j

∂xⁱ = −∂L

∂xⁱ = −d dtpi,

∂H

∂pi=x˙ⁱ+pj

∂x˙^j

∂pi− ∂L

∂x˙^j

∂x˙^j

∂pi =x˙ⁱ.

(2.5)

Consequently, the (second-order) ODEs (2.1) are equivalent to (first-order) ODEs (2.4).

2.1. Conservation of the Hamiltonian. Since d

dtH= d dt

pix˙ⁱ−L

=dpi

dt x˙ⁱ+pi

d˙xⁱ dt −∂L

∂t− ∂L

∂xⁱ dxⁱ

dt − ∂L

∂x˙ⁱ dx˙ⁱ

dt = −∂L

∂t, (2.6) it follows thatH is a first integral for the Hamilton equations if and only if the La- grangian does not depend explicitly ont(autonomous Lagrangian).

The Hamilton equations can be written in a compact form:

0 δⁱ_j

−δⁱj 0 



 dx^j

dt dpi

dt





+







∂H

∂xⁱ

∂H

∂pi





= 0

0

, (2.7)

whereJ= 0 δⁱ_j

−δⁱ_j 0

is the symplectic (complex) structure onR²ⁿ.

The minimization of the dual single-time action, the eigenvalues and the eigenfunc- tions ofJ(d/dt)with periodic boundary conditions, and existence of periodic solutions of single-time Hamilton equations were discussed in [6].

3. Multitime Hamilton canonical equations. LetL:R^p×Rⁿ×R^np→R,(t^α, xⁱ, x_αⁱ)→ L(t^α, xⁱ, x_αⁱ),x_αⁱ =∂xⁱ/∂t^α, be a given Lagrangian. The associated Euler-Lagrange equa- tions are

∂

∂t^α

∂L

∂xⁱα= ∂L

∂xⁱ, i=1, . . . , n, α=1, . . . , p. (3.1) In [2], de Donder turned these equations into a form that has remarkable symmetry.

He used

(1) the conjugate momenta

p^α_k= ∂L

∂x^kα

; (3.2)

(2) the Hamiltonian

H=p_k^αx^k_α−L (3.3)

as a function oft^α,xⁱ,p^α_i.

Here it is, of course, required that (3.2) defines, for everyx=(xⁱ), a continuously differentiable bijectionx_αⁱ →p^α_i. This map is called theLegendre transform.

Theorem 3.1. The Euler-Lagrange equations (3.1) are equivalent to the Hamilton equations

∂xⁱ

∂t^α = ∂H

∂p^α_i , ∂p^α_i

∂t^α = −∂H

∂xⁱ, i=1, . . . , n, α=1, . . . , p (3.4) (summation overα).

Proof. The definitions (3.2) and (3.3) imply

∂H

∂xⁱ=p_k^α∂x_α^k

∂xⁱ− ∂L

∂xⁱ− ∂L

∂xα^k

∂x^k_α

∂xⁱ = − ∂L

∂xⁱ= −∂p_i^α

∂t^α,

∂H

∂p_i^α=x_αⁱ+p^β_k∂x_β^k

∂p^α_i − ∂L

∂x^k_β

∂x^k_β

∂p_i^α=x_αⁱ.

(3.5)

The Euler-Lagrange (second-order) PDEs (3.1) are equivalent to Hamilton (first-order) PDEs (3.4).

3.1. Conservation of the energy-momentum tensor field. The energy-momentum tensor field is defined by

T_β^α=x_βⁱ ∂L

∂xⁱα−δ^α_βL. (3.6)

We compute the divergence of this tensor field. We find

∂

∂t^αT_β^α= ∂

∂t^α xⁱ_βp^α_i

−δ^α_β ∂L

∂t^α+ ∂L

∂xⁱxⁱ_α+ ∂L

∂xγⁱ

∂xⁱ_γ

∂t^α

=∂x_βⁱ

∂t^αp_i^α+x_βⁱ∂p^α_i

∂t^α − ∂L

∂t^β− ∂L

∂xⁱx_βⁱ− ∂L

∂xγⁱ

∂xⁱ_γ

∂t^β

=x_βⁱ ∂p_i^α

∂t^α − ∂L

∂xⁱ

− ∂L

∂t^β = −∂L

∂t^β.

(3.7)

Consequently, the energy-momentum tensor field is conserved if and only if the La- grangianLdoes not depend explicitly ont^α(autonomous Lagrangian).

4. Minimization of dual multitime action. The Hamilton equations can be written in the form

δ^α_βδⁱ_j∂p^β_i

∂t^α+ ∂H

∂x^j =0, −δ^α_βδⁱ_j∂x^j

∂t^α+ ∂H

∂p^β_i =0. (4.1)

We will use also thepolysymplectic structure δ⊗J=

0 δ^α_βδⁱ_j

−δ^α_βδⁱ_j 0

, δ= δ^α_β

, J=

0 δⁱ_j

−δⁱ_j 0

(4.2) discovered by the first author [13,16,17]. In this sense, we introduce the notations

X=Rⁿ, X^∗=dual ofX, Y=R^p, Y^∗=dual ofY , δ=id_X⊗Y, δ⊗J=

0 ϕ(δ)

−ϑ(δ) 0

, (4.3)

whereϕ,ϑare the canonical isomorphisms:

ϕ: End(X⊗Y ) →Hom

X^∗⊗EndY , X^∗ ϑ: End(X⊗Y ) →Hom

X⊗Y^∗, X⊗Y^∗

. (4.4)

Then

δ⊗J∈Hom X⊗Y^∗

⊕

X^∗⊗EndY , X^∗⊕

X⊗Y^∗

(4.5) and we can give the Hamilton equations the following more compact form:

(δ⊗J)







∂x

∂t

∂p

∂t





= −







∂H

∂x

∂H

∂p





. (4.6)

According to the standard Riemannian metric onR^n+np, which is given by the matrix G=δ_ij 0

0 δ^αβδ_ij

, the associated Hamilton multitime actionψis given by

ψ(u)=

Ωᏸt, u, u

dv, Ω=[0, T ]^p⊂R^p, T >0, (4.7) whereu=(x, p),u=∂u/∂t,dv=dt¹∧···∧dt^p, and

ᏸt, u, u

=1 2

∂p_i^α

∂t^αxⁱ−∂xⁱ

∂t^αp^α_i

+H(t, x, p)

=1 2

∂p_j^α

∂t^α,−∂x^j

∂t^β

δij 0 0 δ^βαδij

xⁱ p_i^α

+H

t, x(t), p(t) .

(4.8)

As it is well known, the Euler-Lagrange equations associated with the actionψare equivalent to (3.4). The actionψis indefinite. This is easily shown by substituting the functionuwith the sequence of functions

uk= xk, pk

, xⁱ_k(t)=cⁱcos λkt^β

, p_k,i^α(t)=c_i^αsin λkt^β

, uk(t)=cos

λkt^β

(δ⊗I)c−sin λkt^β

(δ⊗J)c, β= fixed, (4.9) into the previous integral; we chooseλk=2kπ /T,k∈Z,c=(cⁱ, c^α_i)∈R^n+np, c =1.

It follows that

uk(t)=1, ∂p_i^α

∂t^α,−∂xⁱ

∂t^µ

δij 0 0 δ^µγδij

x^j p^γ_j

=λkc^β_jc^j ∀k∈Z. (4.10)

Consequently, ψ

uk

=kπ

T T^pc^β_icⁱ+

Ω

t, uk(t)

dv → −∞or +∞ (4.11)

according tok→ −∞ or k→ +∞. That is why the direct method of the calculus of variations cannot be applied directly and more sophisticated approaches like minimax methods, isoperimetric natural constraints, or dual least action principles have to be used. In a forthcoming paper, we will concentrate on the case where the Hamiltonian H(t, u)is convex inu, in which case the dual least action principle seems to provide the best results in the simplest way. Here we discuss the eigenvalues and the eigenfunctions of the operator(δ⊗J)(∂/∂t)with periodic boundary conditions:

(δ⊗J) ∂u

∂t

=λu(t), u(0, . . . ,0)=u

k1T , . . . , kpT

, (4.12)

whereλ∈R, ∂u/∂tis the matrix of elements∂u/∂t^α, andkα∈Z. The PDEs system (4.12) is equivalent to

∂xⁱ

∂t^α= −λp^α_i, ∂p_i^α

∂t^α =λxⁱ. (4.13)

We get particular solutions by taking xⁱ(t)=cⁱcos

λⁱ_αt^α

, p^β_i =c_i^βsin µiαt^α

, (4.14)

whereλⁱ_α,µiα,cⁱ,c_i^βare real numbers such that T λⁱ_α

2π ∈Z, λⁱ_α=µiα, cⁱλⁱ_α=λc_i^α, c_i^αµiα=λcⁱ. (4.15) Supposing thatcⁱ≠0, we get

λ²=δ^αβλⁱ_αλⁱ_β, i=1, . . . , n. (4.16) We have the following theorem.

Theorem4.1. The system (4.12) has a nontrivial solution of the form (4.14) if and only if the equationT²λ²=4π²δ^αβkαkβhas solutions(k1, . . . , kp)∈Z^p.

Note4.2. The PDEs system (4.12) is associated with the Lagrangian L1

x,∂x

∂t

=λ 2

δijxⁱx^j+δ^αβδijx_αⁱx^j_β

. (4.17)

We could consider the Hamiltonian H2(x, p)=1

2

gijxⁱx^j+G^ij_αβp^α_ip_j^β

, (4.18)

with real and constant coefficientsgij,G^ij_αβ. This generalizes the Hamiltonian associated with the linear oscillator. The associated PDEs system will be linear and with constant coefficients:

∂xⁱ

∂t^α=G^ij_αβp^β_j, ∂p^α_i

∂t^α = −gijx^j. (4.19)

When we look for solutions of the form xⁱ(t)=cⁱexp

λαt^α

, p^β_i(t)=c_i^βexp λαt^α

(4.20) with constantscⁱ,c^β_i, we obtain the algebraic equations

λαcⁱ=G^ij_αβc^β_j, λαc^α_i = −gijc^j. (4.21) Supposing that det(gij)≠0, we get further

c^j= −g^ijc_i^αλα,

G_αβ^ij +g^ijλαλβ

c_i^α=0. (4.22)

Remark4.3. Since the set of eigenvaluesλof the differential operatorδ⊗J is un- bounded (from below and from above), the quadratic form

u →1 2

Ω

(δ⊗J)∂u

∂t(t), u(t)

dv (4.23)

is indefinite on the space H_Ω¹=

u:R^p →R^n+np|u=absolutely continuous, kα∈Z ⇒u

t¹, . . . , t^p

=u

t¹+k1T , . . . , t^p+kpT , ∂u

∂t ∈L² R^p

.

(4.24)

Theorem4.4. Ifu∈H_Ω¹, then

Ω

u(t)²dv≤ T² 4π²

Ω

∂u

∂t(t)

²dv. (4.25)

Proof. By assumption, the functionuhas a Fourier expansion

u(t)=

(k₁,...,k_p)∈Z^p

c

k1, . . . , kp

exp

2π ik1t1+···+kptp

T

. (4.26)

If we denotek=(k1, . . . , kp), then k ²=k²₁+···+k²_p. The Perseval equality implies

Ω

∂u

∂t

²dv=

k₁,...,kp

T^p4π² k ² T² c

k1, . . . , kp²

≥4π² T²

k₁,...,k_p

T^pk1, . . . , kp²=4π² T²

Ω

u(t)²dv.

(4.27)

Theorem4.5. Everyu∈H_Ω¹ satisfies the inequality

Ω

(δ⊗J)∂u

∂t, u(t)

dv≥ −

√pT 2π

Ω

∂u

∂t(t)

dv. (4.28)

Proof. We use the deviation

˜

u(t)=u(t)− 1 T^p

Ωu(t)dv. (4.29)

The Cauchy-Schwarz inequality and inequality (4.25) imply

Ω

(δ⊗J)∂u

∂t, u(t)

dv=

Ω

(δ⊗J)∂u

∂t,u(t)˜

dv

≥ −

Ω

(δ⊗J)∂u

∂t(t) ²dv

1/2

Ω

u(t)˜ ²dv 1/2

≥ − T 2π

Ω

(δ⊗J)∂u

∂t(t) ²dv

1/2

Ω

∂u˜

∂t(t) ²dv

1/2

≥ −

√pT 2π

Ω

∂u

∂t(t) ²dv.

(4.30)

5. Witten-type Hamiltonians. The study of the Yang-Mills-Witten-type functionals [1,3,4,5,7,8,9,10,11,12,14,15], which was initially related to a modern approach of the Maxwell or Dirac equations, gave extremely important results for the development of the differential geometry. This theory which is both experimentally and theoreti- cally grounded, became very interesting and with many possible applications and open

problems for research. In this section, we will apply the Hamiltonian formalism to a functional of this type.

Let Spin^c(4)=Spin(4)×Z2S¹=[SU(2)×SU(2)]×Z2S¹. There is a natural identifica- tion

Spin^c(4)≡ a₊, a₋

∈U (2)×U (2); det a₊

=det a₋

, (5.1)

which induces the projections

_±: Spin^c(4)→U (2), (5.2)

and also the morphism

det : Spin^c(4) →S¹ (5.3)

defined by det(a₊, a₋)=det(a₊)=det(a₋)for any(a₊, a₋)∈Spin^c(4). Let(X, g)be a Riemannian, compact, and oriented 4-manifold. Let(P^c, γ)be a Spin^c(4)-structure in (X, g)andΣ^±=P^c×_±C², its associated spinor-vector bundles.

We will consider as well the complex line bundle detΣ⁺ ≡detΣ⁻ ≡P^c×detC and also the space of the unitary connections in detΣ⁺, denoted byᏭ(detΣ⁺). The space Ꮽ(detΣ⁺)has a natural structure of affine space modeled over the vector spaceiA¹(X), whereA¹(X)is the space of theC^∞−1-forms defined on the manifoldX. Let

W:ᏭdetΣ⁺

×A⁰ Σ⁺

→R, (5.4)

W (A, ψ)=∇A^ˆψ²+ 1

16FA²+1

4 ψ ⁴_L4+1 4

Xs|ψ|² (5.5) bethe Witten-type functional, where(A, ψ)∈Ꮿ=Ꮽ(detΣ⁺)×A⁰(Σ⁺),|ψ|²=ψψ, the¯ norms · areL²-norms, and ˆAis the connection naturally induced inΣ⁺byAand by the Levi-Civita connection of(X, g).

We assume that the restriction of the Riemannian structuregto a coordinate neigh- borhood is Euclidean and we make our computations only over this neighborhood. It follows that the Levi-Civita connection associated toghas its coefficients, the curvature, and its scalar curvature all equal to 0. Under such conditions, we apply the single-time or multitime Hamiltonian formalism, and we will use the symbol sigma with indices for summation.

If the connectionAhas its local associated formia, then∇Aˆψ=dψ+(1/2)iaψfor anyψ∈A⁰(Σ⁺). Of course,ψ=(ψ¹, ψ²)=(ϕ¹+iη¹, ϕ²+iη²)anda=aαdt^α.

Lemma5.1. The Lagrangian associated toW is given by the formula

L(A, ψ)= 3 α=0

2 j=1

1 4

aα2 ϕ^j2

+ η^j2

+ ϕ^jα

2

+ η^jα

2

+

ϕ^jη^jα−η^jϕ^jα

aα

+ 1 32

3 α,β=0

aα,β−aβ,α2

+1 4

₂

j=1

ϕ^j2

+

η^j2² (5.6)

for anyψ=(ϕ¹+iη¹, ϕ²+iη²)∈A⁰(Σ⁺),a=aαdt^α.

5.1. Single-time Hamilton equations. We can assume, under a suitable gauge trans- formation, thata0=0.

Lemma5.2. The momenta associated toLare p^α= ∂L

∂aα,0=1

8aα,0, qj= ∂L

∂ϕ₀^j=2ϕ^j₀, rj= ∂L

∂η^j₀=2η^j₀, (5.7) withα=1,2,3andj=1,2.

Lemma5.3. The Hamilton density associated toWis

H=4 3 s=1

p^s2

+1 4

2 j=1

q_j²+r_j²

− 3 s=1

2 j=1

1 4

as

2 ϕ^j2

+ η^j2

+ ϕ^js2

+ η^js2

+

ϕ^jη^js−η^jϕ^js as

− 1 16

3 s,t=1

as,t

2

−as,tat,s

−1 4

₂

j=1

ϕ^j2

+ η^j22

.

(5.8)

Theorem5.4. The Hamilton generalized equations, associated toW, are

˙

as=8p^s, ϕ˙^j=1

2qj, η˙^j=1 2rj,

˙ p^s=1

2as|ψ|²+ 2 j=1

ϕ^jη^js−η^jϕ^js +1

8 3 t=1

as,tt−at,st

,

˙

qj= |ψ|²ϕ^j+ 3 t=1

1 2

at2

ϕ^j+2ϕ^j_tt−at,tη^j

,

r˙j= |ψ|²η^j+ 3 t=1

1 2

at2

η^j+2η^j_tt+at,tϕ^j

, j=1,2; s=1,2,3.

(5.9)

These equations follow from direct computations. Also, the first coordinatet⁰has been privileged and considered as a time variable.

5.2. Multitime Hamilton equations. There are a lot of problems in which there is no reason to prefer one variable to the other by choosing it as time. In such cases, we use those field theories which involve many time variables (multitime or multiparameter).

In this section, we study the same functional, but instead of a single-time one, we will consider it as a multitime functional, with no privileged variablet^α. We use for our com- putations the appropriate Hamiltonian formalism which has been given inSection 3.

This formalism has been also studied and developed in [3,4,13,16,17]. The momenta associated toLwill be denoted by

p^αβ= ∂L

∂aα,β

, q^α_j = ∂L

∂ϕ^jα

, r_j^α= ∂L

∂η^jα

, (5.10)

withα, β=0,1,2,3 andj=1,2.

Lemma5.5. The momenta associated toLare p^αβ=1

8

aα,β−aβ,α

, q_j^α=2ϕ^jα−aαη^j, r_j^α=2η^jα+aαϕ^j, (5.11)

withα, β=0,1,2,3andj=1,2.

Remark5.6. Since the first 16 relations ofLemma 5.5(p^αβ=···, forα, β=0,1,2,3) are not independent, the general conditions of the Hamiltonian formalism are not ful- filled. It is yet possible to extend the Hamiltonian formalism so that such cases become workable, in various ways. One way is the following. Suppose we have a Lagrangian L(t, y, y)such that the generalized momenta

p_i^α= ∂L

∂yαⁱ

(5.12)

are not independent. It is possible to find functionsf_αⁱ(y, p)such that (5.12) be verified when we writey_αⁱ =f_αⁱ(y, p). Suppose we made such a choice and define the formal Hamiltonian

Hf(y, p)=f_αⁱp^α_i−L

y, f (y, p)

(5.13) by treating the variablesp_i^αas if they were functionally independent. Then we get

∂Hf

∂yⁱ =∂fα^j

∂yⁱp^α_j− ∂L

∂yⁱ− ∂L

∂y_β^j

∂f_β^j

∂yⁱ= − ∂L

∂yⁱ= −∂p_i^α

∂t^α,

∂Hf

∂p_i^α =f_αⁱ+∂f_β^j

∂p^α_i p_j^β− ∂L

∂y_β^j

∂f_β^j

∂p^α_i =f_αⁱ=∂yⁱ

∂t^α.

(5.14)

The equations thus obtained,

∂yⁱ

∂t^α =∂Hf

∂p_i^α, ∂p_i^α

∂t^α = −∂Hf

∂yⁱ, (5.15)

will be named thegeneralized canonical equations. Note thatHf, as a function of the variablesp_i^α, depends on the choice of the functionsf_αⁱ, but it becomes a well-defined function on the momentap_i^αby taking into consideration (5.12).

We apply this generalized Hamiltonian formalism to the functional (5.5) under the same restrictions introduced at the beginning ofSection 5. The corresponding Lagrang- ian has been written inLemma 5.1. The relationsp^αβ=(1/8)(aα,β−aβ,α)= −p^βαshow that the momenta are not independent. In this case, we can choose the functionsfαβ

such that

aα,β=fαβ(u, p)=4p^αβ+gαβ(u, p), (5.16) wheregαβ=gβα.

Lemma5.7. The Hamilton density associated toWand to the choice (5.16) is

Hf=2 3 α,β=0

p^αβ2

+ 3 α,β=0

p^αβgαβ+1 4

3 α=0

2 j=1

q^α_j2

+ r_j^α2

+1 2

3 α=0

2 j=1

η^jq_j^α−ϕ^jr_j^α aα−1

4 ₂

j=1

ϕ^j2

+ η^j)²2

.

(5.17)

Theorem5.8. The Hamilton generalized equations associated toHf are

∂aα

∂t^β =4p^αβ+gαβ, ∂ϕ^j

∂t^α = −1 2

q^α_j+aαη^j ,

∂η^j

∂t^α =1 2

r_j^α−aαϕ^j

, ∂p^αβ

∂t^β =1 2

η^jq_j^α−ϕ^jr_j^α ,

∂q_j^β

∂t^β =1 2aαr_j^α+

2 k=1

ϕ^k2

+ η^k2

ϕ^j,

∂r_j^β

∂t^β = −aαq_j^α+ 2

k=1

ϕ^k2

+ η^k2

η^j, j=1,2; α, β=0,1,2,3.

(5.18)

These equations follow from direct computations.

Remarks5.9. (1) In the particular casegαβ=0, the solutions of the previous equa- tions are subject to the conditions∂aα/∂t^β= −∂aβ/∂t^α.

(2) FromLemma 5.5, it follows that p^αβ = −p^βα. When we take into consideration these relations, any choice of the functionsf^αβgives the same Hamilton density

H=2 3 α,β=0

p^αβ2

+1 4

3 α=0

2 j=1

q_j^α2

+ r_j^α2

+1 2

3 α=0

2 j=1

η^jq^α_j−ϕ^jr_j^α aα−1

4 ₂

j=1

ϕ^j2

+ η^j22

.

(5.19)

Suitable computations give theformal Hamilton equations

∂aα

∂t^β =4p^αβ, ∂ϕ^j

∂t^α =1 2

q_j^α+aαη^j

, ∂η^j

∂t^α =1 2

r_j^α−aαϕ^j ,

∂p^αβ

∂t^β = −1 2

η^jq_j^α−ϕ^jr_j^α

, ∂q^β_j

∂t^β =1 2aαr_j^α+

₂

k=1

ϕ^k2

+ η^k2

ϕ^j,

∂r_j^β

∂t^β = −1 2aαq^α_j+

₂

k=1

ϕ^k2

+ η^k2

η^j, j=1,2; α, β=0,1,2,3.

(5.20)

These are the equations fromTheorem 5.8in the particular casegαβ=0.

(3) The first formula ofTheorem 5.8requires the integrability conditions

∂²aα

∂t^β∂t^γ− ∂²aα

∂t^γ∂t^β=4 ∂p^αβ

∂t^γ −∂p^αγ

∂t^β

+∂gαβ

∂t^γ −∂gαγ

∂t^β =0. (5.21)

References

[1] J. P. Bourguignon,An introduction to geometric variational problems, Lectures on Geomet- ric Variational Problems (Sendai, 1993) (S. Nishikawa and R. Schoen, eds.), Springer, Tokyo, 1996, pp. 1–41.

[2] T. de Donder,Theorie Invariantive du Calcul des Variations, Gauthier-Villars, Paris, 1935.

[3] G. Giachetta, L. Mangiarotti, and G. Sardanashvily,Covariant Hamilton equations for field theory, J. Phys. A32(1999), no. 38, 6629–6642.

[4] M. J. Gotay, J. Isenberg, J. E. Marsden, and R. Montgomery,Momentum maps and classical relativistic fields. Part I: covariant field theory, preprint, 1997,http://xxx.lanl.gov/

physics/9801019.

[5] J. Jost, X. Peng, and G. Wang,Variational aspects of the Seiberg-Witten functional, Calc. Var.

Partial Differential Equations4(1996), no. 3, 205–218.

[6] J. Mawhin and M. Willem,Critical Point Theory and Hamiltonian Systems, Applied Mathe- matical Sciences, vol. 74, Springer, New York, 1989.

[7] Ch. Okonek and A. Teleman,Seiberg-Witten invariants and rationality of complex surfaces, Math. Z.225(1997), no. 1, 139–149.

[8] A. Teleman,Seiberg-Witten Theorie und komplexe Geometrie, Vorlesung, Phylosophische Fakultät II, Universität Zürich, Zurich, 1996-1997.

[9] ,Non-abelian Seiberg-Witten theory, Habilitationsschrift, Universität Zürich, Zurich, 1996.

[10] ,Non-abelian Seiberg-Witten theory and stable oriented pairs, Internat. J. Math.8 (1997), no. 4, 507–535.

[11] A.-M. Teleman,First variation formulas for a Witten-type functional. I, An. Univ. Bucure¸sti Mat. Inform.50(2001), no. 2, 247–260.

[12] ,First variation formulas for a Witten-type functional. II., An. Univ. Bucure¸sti Mat.

Inform.52(2003), no. 1, 87–104.

[13] A.-M. Teleman and C. Udri¸ste,On Hamiltonian formalisms in mathematical physics, 4th International Conference of Balkan Society of Geometers, Aristotle University of Thessaloniki, Greece, June 2002.

[14] K. Teleman,Teorie Seiberg-Witten, Curs-Anul 4, Universitatea Bucure¸sti, Bucharest, 1995- 1996.

[15] H. Triebel,Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing, Amsterdam, 1978.

[16] C. Udri¸ste,Solutions of DEs and PDEs as potential maps using first order Lagrangians, Balkan J. Geom. Appl.6(2001), no. 1, 93–108.

[17] C. Udri¸ste and M. Postolache,Atlas of Magnetic Geometric Dynamics, Balkan Society of Geometers Monographs and Textbooks, vol. 3, Geometry Balkan Press, Bucharest, 2001.

[18] E. Witten,Monopoles and four-manifolds, Math. Res. Lett.1(1994), no. 6, 769–796.

Constantin Udri¸ste: Department of Mathematics I, University Politehnica of Bucharest, Splaiul Independen¸tei 313, Bucharest 060042, Romania

E-mail address:[email protected]

Ana-Maria Teleman: Faculty of Mathematics and Informatics, University of Bucharest, Acade- miei 14, Bucharest 70109, Romania

E-mail address:[email protected]

Journal of Applied Mathematics and Decision Sciences

Special Issue on

Intelligent Computational Methods for Financial Engineering

Call for Papers

As a multidisciplinary field, financial engineering is becom- ing increasingly important in today’s economic and financial world, especially in areas such as portfolio management, as- set valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the re- cently approved Basel II guidelines advise financial institu- tions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to im- prove the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.

However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used e

ectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)

This special issue will include (but not be limited to) the following topics:

• Computational methods

: artificial intelligence, neu- ral networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learn- ing, multiagent learning

• Application fields

: asset valuation and prediction, as- set allocation and portfolio selection, bankruptcy pre- diction, fraud detection, credit risk management

• Implementation aspects

: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, imple- mentation

Authors should follow the Journal of Applied Mathemat- ics and Decision Sciences manuscript format described at the journal site

Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Track- ing System at

lowing timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Lean Yu,

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China;

Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;

[email protected]

Shouyang Wang,

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China; [email protected]

K. K. Lai,

Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong; [email protected]

Hindawi Publishing Corporation http://www.hindawi.com