The Helmholtz Conditions for the Difference Equations Systems

Dumitru Crăciun and Dumitru Opriş

Abstract

This paper present the analogous of the Helmholtz conditions for first and second order difference equations systems. We obtain variational implicit algoritms used in numerical analysis and numerical integration schemes for Hamilton systems. We give some representative examples.

Mathematics Subject Classification: 70H05, 39A12 Key words: Helmholtz conditions, Lagrangians, variation formulae

1 Introduction

The inverse problem of the calculus of variations is a subject which has been studied over several decades and an important research field. For first and second order ordinary differential equations there are numerous contributions and we can mention the following papers [3], [4], [8].

The purpose of the present paper is to extend this analysis and to derive the Helmholtz conditions for difference equations systems too. We derive the first and second order variational relations for Lagrangians that depend on a set of points (q_{k})_{k∈Z} on a differentiable submanifold Γ ⊂ Q×B. In these conditions we formulate the d’Alembert–Lagrange principle which leads us for L(k) = tr(q_{k}Jq_{k+1}^{T}), with q_{k} ∈ O(n), to the discrete Euler equations for the rigid body [9].

For a function system \{F_{i}(k)\}_{i=\overline{1,n}} with F_{i}(k) = F_{i}(q_{k−1}, q_{k}, q_{k+1}) we deduce the discrete Helmholtz conditions.

These conditions are satisfied for \{e_{i}(k)\}_{i=\overline{1,n}} where e_{i}(k) come from the variational principle for L(k). We are now able to write the discrete Hamilton equations used in numerical algorithms for Hamiltonian systems.

In the final section we give the Helmholtz conditions for some examples of difference equations systems.

Balkan Journal of Geometry and Its Applications, Vol.1, No.2, 1996, pp. 21-30 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

2 The first and second variation formulae

Let Q be a n-dimensional differentiable manifold and B a m -dimensional differentiable manifold.

Let (q_{k})_{k∈Z} be a set of points from Q and S : Q × Q → B a differentiable function denote S(k) = S(q_{k}, q_{k+1}).

We consider L : Q × B → R where

L(k) = L(q_{k}, S(k))

The functional

A(q) = ∑_{k∈Z} L(k) (1)

is called the action of L.

For q_{k} ∈ Q let λ : Q × I → Q where I = (−a, a) ∈ R, denote with λ(q_{k}, ε) = q_{k}(ε) and let be

η_{k}(ε) = \frac{∂λ(q_{k}, ε)}{∂ε} ξ_{k}(ε) = \frac{∂^{2}λ(q_{k}, ε)}{∂ε^{2}}

η_{k} = η_{k}(ε)|_{ε=0} ξ_{k} = ξ_{k}(ε)|_{ε=0}

Proposition 1. The first variation of the functional A(q) with respect to λ is

δA(q, η) = ∑_{k∈Z} e_{i}(k)η_{k}^{i} (2)

The second variation of A(q) with respect to λ is

δ^{2}A(q, η, ξ) = ∑_{k∈Z} e_{i}(k)ξ_{k}^{i} + ∑_{k∈Z} [Hess^{(1)}L(k)(η_{k}, η_{k}) + (3)

+ 2Hess^{(2)}L(k)(η_{k}, η_{k+1})]

where

e_{i}(k) = \frac{∂L(k)}{∂q_{k}^{i}} + \frac{∂L(k)}{∂S^{α}(k)} \frac{∂S^{α}(k)}{∂q_{k}^{i}} + \frac{∂L(k − 1)}{∂S^{α}(k − 1)} \frac{∂S^{α}(k − 1)}{∂q_{k}^{i}} (4)

Hess^{(1)}L(k)(η_{k}, η_{k}) = \frac{∂e_{i}(k)}{∂q_{k}^{j}}η_{k}^{i}η_{k}^{j} (5)

Hess^{(2)}L(k)(η_{k}, η_{k+1}) = \frac{∂}{∂q_{k}^{i}} (\frac{∂L(k)}{∂S^{α}(k)} \frac{∂S^{α}(k)}{∂q_{k+1}^{j}}) η_{k}^{i}η_{k+1}^{j} (6)

We can prove this proposition if we compute the first and second derivative with respect to ε for

A(q(ε)) = ∑_{k∈Z} L(q_{k}(ε), S(q_{k}(ε), q_{k+1}(ε)))

For L(k) = L(q_{k}, q_{k+1}) we obtain

e_{i}(k) = \frac{∂L(k)}{∂q_{k}^{i}} + \frac{∂L(k − 1)}{∂q_{k}^{i}} (7)

Hess^{(1)}L(k)(η_{k}, η_{k}) = (\frac{∂^{2}L(k)}{∂q_{k}^{i}∂q_{k}^{j}} + \frac{∂^{2}L(k − 1)}{∂q_{k}^{i}∂q_{k}^{j}}) η_{k}^{i}η_{k}^{j} (8)

Hess^{(2)}L(k)(η_{k}, η_{k+1}) = \frac{∂^{2}L(k)}{∂q_{k}^{i}∂q_{k+1}^{j}} η_{k}^{i}η_{k+1}^{j} (9)

For L : Q × R^{n} → R with S(q_{k}, q_{k+1}) = q_{k}^{1} = q_{k+1}^{i} − q_{k}^{i} results [1], [2], [5], [7]

e_{i}(k) = \frac{∂L(k)}{∂q_{k}^{i}} − \frac{∂L(k)}{∂q_{k}^{1i}} + \frac{∂L(k − 1)}{∂q_{k−1}^{1i}} (10)

Hess^{(1)}L(k)(η_{k}, η_{k}) = (\frac{∂^{2}L(k)}{∂q_{k}^{i}∂q_{k}^{j}} + \frac{∂^{2}L(k − 1)}{∂q_{k−1}^{1i}∂q_{k−1}^{1j}} ) η_{k}^{i}η_{k}^{j} (11)

Hess^{(2)}L(k)(η_{k}, η_{k}^{1}) = (\frac{∂^{2}L(k)}{∂q_{k}^{i}∂q_{k}^{1j}} − \frac{∂^{2}L(k)}{∂q_{k−1}^{1i}∂q_{k}^{1j}} ) η_{k}^{i}η_{k}^{1j} (12)

We define the one-forms β associated to L(k)

β(k) = \frac{∂L(k − 1)}{∂S^{α}(k − 1)} \frac{∂S^{α}(k − 1)}{∂q_{k}^{i}}dq_{k}^{i}

and the submanifold Γ ⊂ Q × B

Γ = \{(q_{k}, S(q_{k}, q_{k+1})), e_{i}(k) = 0 i = \overline{1, n}\}

Proposition 2. The submanifold Γ is an isotopic submanifold with respect to the two–form

Ω(k, k + 1) = dβ(k + 1) − dβ(k)

From this proposition follows that the Lagrangian L is a generating function of Γ in the domain where Hess^{(2)}L(k)(η_{k}, η_{k+1}) is nondegenerate.

3 The Lagrange–d’Alembert principle

We consider a set of functions f_{a} : Q × B → R, a = \overline{1, p}, where f_{a}(k) = f_{a}(q_{k}, S(q_{k}, q_{k+1})) with

rang ||\frac{∂f_{a}(k)}{∂S^{α(k)}} || = p < m

and the restrictions R ⊂ Q × B given by

R = \{(q_{k}, S(q_{k}, q_{k+1})) , f_{a}(k) = 0 , a = \overline{1, p}\}

We define the virtual variation for R, λ(q_{k}, ε) = q_{k}(ε) that satisfy

\frac{∂f_{a}(k)}{∂S^{α}(k)} \frac{∂S^{α}(k)}{∂q_{k}^{i}}η_{k}^{i} = 0 a = \overline{1, p} (13)

The Lagrange–d’Alembert principle for the system (Q, L, R) system is:

The elements q_{k} ∈ Q represents the motion of the (Q, L, R) system if

e_{i}(k)η_{k}^{i} = 0 (14)

for all virtual variations η_{k} for R.

From §3. (1) and (2) we deduce that the equations of the system described above (Q, L, R) satisfies

e_{i}(k) = μ^{a} \frac{∂f_{a}(k)}{∂S^{α}(k)} \frac{∂S^{α}(k)}{∂q_{k}^{i}} i = \overline{1, n} (15)

f_{a}(k) = 0

Using the Lagrange–d’Alembert principle for L : O(n) × O(n) → R with

L(k) = tr(q_{k}Jq_{k+1}^{T}) (16)

where J is a symmetric positive matrix, we obtain Arnold equations [9]

M_{k+1} = ω_{k}M_{k}ω_{k}^{−1} (17)

M_{k} = ω_{k}^{T} J − Jω_{k}

where ω_{k} = q_{k}^{T} q_{k−1} [5].

If q_{k} = q(t_{k}) with t_{k} = t_{0} + kε, §3. (5) leads to the Euler equations of the rigid body

4 The Helmholtz conditions for discrete second order equations

The equations system that describe the motion e_{i}(k) = 0 is caracterized by the functions F_{i}(k) = F_{i}(q_{k−1}, q_{k}, q_{k+1}), i = \overline{1, n}.

The inverse problem consist of finding a set of conditions for \{F_{i}(k)\}_{i=\overline{1,n}} in order to get F_{i}(k) = e_{i}(k) where e_{i}(k) is the first variation of the Lagrangian L(k). These conditions constitute the discrete variant of the Helmholtz conditions for differential equations (that means the continuous case).

For λ : Q × I → Q with λ(q_{k}, ε) = q_{k}(ε), ε ∈ (−a, a) = I we define the Fréchet derivative of F_{i}(k)

DF_{i}(k)(η(k)) = \frac{d}{dε}F_{i}(q_{k−1}(ε), q_{k}(ε), q_{k+1}(ε))|_{ε=0} (18)

where η(k) = (η_{k−1}, η_{k}, η_{k+1}). From §3 (1) we obtain

DF_{i}(k)(η(k)) = \frac{∂F_{i}(k)}{∂q_{k−1}^{j}}η_{k−1}^{j} + \frac{∂F_{i}(k)}{∂q_{k}^{j}}η_{k}^{j} + \frac{∂F_{i}(k)}{∂q_{k+1}^{j}}η_{k+1}^{j} i = \overline{1, n} (19)

The adjoint function of DF_{i}(k)η(k) is:

D^{∗}F_{i}(k)(η(k)) = \frac{∂F_{j}(k − 1)}{∂q_{k}^{i}}η_{k−1}^{j} + \frac{∂F_{j}(k)}{∂q_{k}^{i}}η_{k}^{j} + \frac{∂F_{i}(k + 1)}{∂q_{k}^{i}}η_{k+1}^{j} (20)

From §4. (2) and (3) we have

η̃_{k}^{i}DF_{i}(k)(η(k)) − η_{k}^{i}D^{∗}F_{i}(k)(η̃(k)) = Δ(η(k + 1), η̃(k + 1)) − Δ(η(k), η̃(k))

where

Δ(η(k), η̃(k)) = \frac{∂F_{i}(k − 1)}{∂q_{k}^{j}}η_{k−1}^{i}η̃_{k}^{j} − \frac{∂F_{i}(k)}{∂q_{k−1}^{j}}η_{k}^{i}η̃_{k−1}^{j}

The system is called selfadjoint if DF_{i}(k) = D^{∗}F_{i}(k), i = \overline{1, n}. Proposition 3. The functions \{F_{i}(k)\}_{i=\overline{1,n}} are selfadjoint iff a)

\frac{∂F_{i}(k)}{∂q_{k}^{j}} = \frac{∂F_{j}(k)}{∂q_{k}^{i}}; (21)

b)

\frac{∂F_{i}(k)}{∂q_{k+1}^{j}} = \frac{∂F_{j}(k + 1)}{∂q_{k}^{i}}

These conditions are the Helmholtz conditions for discrete systems. Proposition 4. The functions e_{i}(k), i = \overline{1, n}, given by §2. (4) satisfied §4. (4). Proposition 5. A solution of the system §4. (4) is

F_{i}(k) = \frac{∂φ(k)}{∂q^{i}(k)} + \frac{∂φ(k − 1)}{∂q^{i}(k)} (22)

with φ(k) = φ(q_{k−1}, q_{k}).

Assuming a functions set \{F_{i}(k)\}_{i=\overline{1,n}} and F̃_{i}(k) = c_{i}^{j}(k)F_{j}(k) with det(c_{i}^{j}(k)) ≠ 0, ∀k ∈ Z. Proposition 6. The function system \{F̃_{i}(k)\}_{i=\overline{1,n}} satisfies the selfadjoint con-

ditions iff

(\frac{∂c_{i}^{h}(k)}{∂q_{k}^{j}} − \frac{∂c_{j}^{h}(k)}{∂q_{k}^{i}}) F_{h}(k) + (c_{i}^{h}(k)δ_{j}^{l} − c_{j}^{h}(k)δ_{i}^{l}) \frac{∂F_{h}(k)}{∂q_{k}^{l}} = 0 (23)

(\frac{∂c_{i}^{h}(k)}{∂q_{k+1}^{j}} − \frac{∂c_{j}^{h}(k + 1)}{∂q_{k}^{i}}) F_{h}(k + 1) + c_{i}^{h}(k)\frac{∂F_{h}(k)}{∂q_{k+1}^{j}} − c_{j}^{h}(k + 1)\frac{∂F_{h}(k + 1)}{∂q_{k}^{i}} = 0

The matrix (c_{j}^{i}(k)) is called an integrant factor. In order to determine this integrant factor it must be consider the special cases with functions of the following type c_{j}^{i}(k, q_{k}, q_{k+1}).

5 Hamilton equations

Let S : Q × Q → R^{n}, L : Q × R^{n} → R and L(k) = L(q_{k}, S(q_{k}, q_{k+1})). We introduce

p_{α}(k) = \frac{∂L(k)}{∂S^{α}(k)} α = \overline{1, m} (24)

L(k) is regular with respect to S(k) if

det (\frac{∂^{2}L(k)}{∂S^{α}(k)∂S^{β}(k)}) ≠ 0 ∀k ∈ Z (25)

Under that assumptions we can see that

S^{α}(k) = l^{α}(q_{k}, p(k))

If H(k) = H(p(k), q(k)) = p_{α}(k)l^{α}(q_{k}, p(k)) − L(q_{k}, l(q_{k}, p(k))) (26) we construct his action

H(k) = ∑_{k∈Z} [p_{α}(k)l^{α}(q_{k}, p(k)) − L(q_{k}, l(q_{k}, p(k)))] (27)

Proposition 7. The first variation of H(k) is

δH(k)(ξ, η) = ∑_{k∈Z} [h_{i}(k)η_{k}^{i} + m^{α}(k)ξ_{α}(k)] (28)

where

m^{α}(k) = l^{α}(k) − p_{β}(k)\frac{∂e^{β}(k)}{∂p_{α}(k)} (29)

h_{i}(k) = p_{α}(k − 1)\frac{∂e^{α}(k)}{∂q_{k}^{i}} − p_{α}(k)\frac{∂e^{α}(k + 1)}{∂q_{k}^{i}}

Let Γ^{∗} ⊂ Q × R^{m} × R^{m∗} given by

Γ^{∗} = {(q_{k}, S(k), p(k)), h_{i}(k) = 0, m^{α}(k) = 0} (30)

Using §5. (3) and (6) we see that Γ^{∗} is characterized by a set of discrete equations – the discrete Hamilton equations

\frac{∂H(k)}{∂q_{k}^{i}} = p_{α}(k)\frac{∂S^{α}(k)}{∂q_{k}^{i}} + p_{α}(k − 1)\frac{∂S^{α}(k − 1)}{∂q_{k}^{i}} (31)

\frac{∂H(k)}{∂p_{α}(k)} = S^{α}(k)

If S(k) = q_{k}^{1} ∈ R^{n}, §5. (8) can be written in the following form [5]

p_{i}(k − 1) − p_{i}(k) = \frac{∂H(k)}{∂q_{k}^{i}} (32)

q_{k+1}^{i} − q_{k}^{i} = \frac{∂H(k)}{∂p^{i}(k)}

Such equations are used in numerical algorithms for solving first order differential equations systems.

6 The Helmholtz conditions for first order difference equations

The equations §5. (9) are in this case

F (k) = F^{i}(k)ξ_{i}(k) G(k) = G_{i}(k)η_{k}^{i} (33)

where F^{i}(k) = F^{i}(q_{k}, p_{k}, q_{k+1}) G_{i}(k) = G_{i}(p_{k−1}, q_{k}, p_{k}) (34)

We define the action of the pair (F (k), G(k))

A(F, G)(η, ξ) = ∑_{k∈Z} [F^{i}(k)ξ_{i}(k) + G_{i}(k)η_{k}^{i}] (35)

The first variation for §6. (3) is

δA(F, G)(η, ξ, η̄, ξ̄) = ∑_{k∈bfZ} [DF^{i}(k)ξ_{i}(k) + DG_{i}(k)η_{k}^{i} + F^{i}(k)ξ̄_{i}(k) + G_{i}(k)η_{k}^{i}]

(36) where

DF^{i}(k) = (\frac{∂F^{i}(k)}{∂q_{k}^{j}} + \frac{∂F^{i}(k − 1)}{∂q_{k}^{j}}) η_{k}^{j} + \frac{∂F^{i}(k)}{∂p_{j}(k)}ξ_{j}(k)

DG_{i}(k) = \frac{∂G_{i}(k)}{∂q_{k}^{j}}η_{k}^{j} + (\frac{∂G_{i}(k)}{∂p_{j}(k)} + \frac{∂G_{i}(k + 1)}{∂p_{j}(k)}) ξ_{j}(k) (37)

ξ̄_{i}(k) = \frac{∂^{2}θ_{i}(p_{k}, ε)}{∂ε∂ε}|_{ε=0} η̄_{k}^{i} = \frac{∂^{2}λ^{i}(q_{k}, ε)}{∂ε∂ε}|_{ε=0}

with θ(p(k), 0) = p(k), λ(q_{k}, 0) = q_{k}.

We call the adjoint pair of (F (k), G(k)) the functions pair (F^{∗}(k), G^{∗}(k)) given by F^{∗}(k) = F^{j}(k)ξ_{j}(k) − (G_{j}(k) + G_{j}(k + 1))η_{k}^{j} (38)

G^{∗}(k) = −(F^{j}(k) + F^{j}(k − 1))ξ_{j}(k) + G_{j}(k)η_{k}^{j}

The first variation for F^{∗}(k) with respect to p(k) and for G^{∗}(k) with respect to q_{k} are

δA_{p}(F^{∗})(ξ, ξ̄, η̄) = ∑_{k∈Z} [D^{∗}F^{i}(k)ξ_{i}(k) + F^{∗}(k)(ξ̄, η̄)] (39)

δA_{q}(G^{∗})(η, ξ̄, η̄) = ∑_{k∈Z} [D^{∗}G^{i}(k)η_{k}^{i} + G^{∗}(k)(ξ̄, η̄)]

where

D^{∗}F^{i}(k) = \frac{∂F^{j}(k)}{∂p_{i}(k)}ξ_{j}(k) + (\frac{∂G_{j}(k)}{∂p_{i}(k)} + \frac{∂G_{j}(k + 1)}{∂p_{i}(k)}) η_{k}^{j} (40)

D^{∗}G_{i}(k) = − (\frac{∂F^{j}(k)}{∂q_{k}^{i}} + \frac{∂F^{j}(k − 1)}{∂q_{k}^{i}}) ξ_{j}(k) + \frac{∂G_{j}(k)}{∂q_{k}^{i}}η_{k}^{j}

28 D.Crăciun and D.Opriş

The pair (F (k), G(k)) is selfadjoint if D^{∗}F^{i}(k) = DF^{i}(k) D^{∗}G_{i}(k) = DG_{i}(k) i = \overline{1, n}, k ∈ Z (41)

The relations §6. (5) and (8) implies (F (k), G(k)) to be self adjoint iff

\frac{∂F^{j}(k)}{∂p_{i}(k)} = \frac{∂F^{i}(k)}{∂p_{j}(k)} \frac{∂G_{j}(k)}{∂q_{k}^{i}} = \frac{∂G_{i}(k)}{∂q_{k}^{j}} (42)

\frac{∂F^{i}(k)}{∂q_{k}^{j}} + \frac{∂F^{i}(k − 1)}{∂q_{k}^{j}} + \frac{∂G_{j}(k)}{∂p_{i}(k)} + \frac{∂G_{j}(k + 1)}{∂p_{i}(k)} = 0

i, j = \overline{1, n}, ∀k. Proposition 8. The functions system

F^{i}(k) = q_{k+1}^{i} − q_{k}^{i} − h\frac{∂H(k)}{∂p_{i}(k)} (43)

G_{i}(k) = p_{i}(k) − p_{i}(k − 1) + h\frac{∂H(k)}{∂q_{k}^{i}}

is selfadjoint.

This system represents the Euler implicit algorithm for numerical integration of an Hamiltonian system.

Identically with the functions §6. (1) we can deduce that the functions system F^{i}(k) = F^{i}(q_{k}, p_{k}, q_{k+1}), G_{i}(k) = G_{i}(q_{k}, p_{k}, p_{k+1}) (44) is selfadjoint iff

\frac{∂F^{i}(k)}{∂p_{j}(k)} = \frac{∂F^{j}(k)}{∂p_{i}(k)}, \frac{∂G_{i}(k)}{∂q_{k}^{j}} = \frac{∂G_{j}(k)}{∂q_{k}^{i}}, (45)

\frac{∂F^{i}(k)}{∂q_{k}^{j}} + \frac{∂F^{i}(k − 1)}{∂q_{k}^{j}} + \frac{∂G_{j}(k)}{∂p_{i}(k)} + \frac{∂G_{j}(k − 1)}{∂p_{i}(k)} = 0

i, j = \overline{1, n}, ∀k. Proposition 9. The system a). F^{i}(k) = q_{k+1}^{i} − f^{i}(k) (46) b).

G_{i}(k) = p_{i}(k + 1) − g_{i}(k)

where f^{i}(k) = f^{i}(q_{k}) is selfadjoint iff

g_{i}(k) = (2δ_{i}^{j} − \frac{∂f^{j}(k)}{∂q_{k}^{i}}) p_{j}(k) + \frac{∂ψ(k)}{∂q_{k}^{j}} (47)

where ψ(k) = ψ(q_{k}).

It can be profed using §6. (12). The system §6. (14)_{b).} is called the adjoint of §6. (14)_{a).}.

We can give some examples. For the logistic function

F (k) = q_{k+1} − aq_{k}^{n}(1 − q_{k}^{r}) (48)

the adjoint function is

G(k) = p_{k+1} − [2 − naq_{k}^{n−1} − a(n + r)q_{k}^{n+r−1}]p_{k} + \frac{∂ψ(k)}{∂q_{k}} (49)

For the functions system F^{i}(k) = q_{k+1}^{i} − a_{j}^{i}q_{k}^{j} (50)

the adjoint functions are

G_{i}(k) = p_{i}(k + 1) − (2δ_{i}^{j} − a_{i}^{j})p_{j}(k) + \frac{∂ψ}{∂q_{k}^{i}} (51)

Examples

1. Let be the function F(k) = aq_{k−1}+bq_{k+1}+f(q_{k}). The selfadjoint condition §3. (7) leads to a = b. Let be p_{k} = q_{k} − q_{k−1}. We replace it in F(k) and obtain

F (k) = q_{k} − q_{k−1} − p_{k} (52)

G(k) = bp_{k+1} − ap_{k} + (a + b)q_{k} + F (q_{k})

The selfadjoint condition §6. (12) for the new functions system leads us to a = b. These two conditions in this case are equivalent.

2. Let be the functions system

F_{i}(k) = A_{ij}q_{k+1}^{j} − f_{i}(q_{k}, q_{k−1}) i = \overline{1, n} (53)

with A_{ij} = A_{ji}, det(A_{ij}) ≠ 0. The system §6. (14) is selfadjoint iff

f_{i}(k) = −A_{ij}q_{k−1}^{j} + \frac{∂ψ(k)}{∂q_{k}^{j}} (54)

From §6. (14) and (15) we obtain

F_{i}(k) = A_{ij}q_{k+1}^{j} + A_{ij}q_{k−1}^{j} − \frac{∂ψ(k)}{∂q_{k}^{i}} (55)

Let be p_{i}(k) = A_{ij}(q_{k−1}^{j} − q_{k}^{j}). We replace it in F_{i}(k) and obtain the functions system F^{i}(k) = q_{k}^{i} − q_{k−1}^{i} + Ã^{ij}p_{j}(k) (56)

G_{i}(k) = p_{i}(k + 1) − p_{i}(k) − 2A_{ij}q_{k}^{j} + \frac{∂ψ(k)}{∂q_{k}^{i}}

The last functions system is selfadjoint. Conclusions

All the principal results of this paper have been discussed throughout the text.

We studied the general inverse problem of a difference equation system. We established the selfadjoint conditions and deduced the discrete Helmholtz conditions. Finally we wrote the discrete Hamilton equations used in numerical schemes and gave some examples.

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Department of Theoretical Physics West University of Timişoara

Department of Mathematics West University of Timişoara