Discretized Classical Model of the New Equation

In this post, I try to make n=∞ straight time version of Cartoon of the Equation.

Below I solve classically the equation of motion of the mechanical system defined by the following Hamiltonian.
H(p; x) = Qφ(x) + [1／(2M)] Σ_k=－∞^∞ [p^k － QA^k(x)]².
M = mεα², Q = m.
A^k(x) = α(x^k+1 － x^k), φ(x) = －[1／(2ε)] Σ_k=－∞^∞ (x^k+1 － x^k)².
Derivation of this Hamiltonian is the same as one of Cartoon of the Equation.

Hamilton's canonical equations of motion of this system is as follows.
dx^j／dt = ∂H／∂p^j = (p^j － QA^j)／M,
dp^j／dt = －∂H／∂x^j = (Q／M)Σ_k (p^k － QA^k)∂A^k／∂x^j － Q∂φ／∂x^j.

From the first of these equations, it is derived that
p^j = Mdx^j／dt + QA^j.
Substituting this into the second of the equations, it is derived that
M(d／dt)² x^j + QdA^j／dt
= (Q／M)Σ_k (p^k － QA^k)∂A^k／∂x^j － Q∂φ／∂x^j
= (Q／M)Σ_k (Mdx^k／dt)∂A^k／∂x^j － Q∂φ／∂x^j.
For the term dA^j／dt in the left hand side of this equation, substituting that
dA^j／dt = Σ_k (∂A^j／∂x^k)dx^k／dt,
it is derived that
M(d／dt)² x^j = (Q／M)Σ_k (Mdx^k／dt)∂A^k／∂x^j － QΣ_k (∂A^j／∂x^k)dx^k／dt － Q∂φ／∂x^j.
∴ M(d／dt)² x^j = QΣ_k (dx^k／dt)(∂A^k／∂x^j －∂A^j／∂x^k) － Q∂φ／∂x^j.

Here I use the concrete expressions of A and φ.
∂A^k／∂x^j = α(δ_k+1,j － δ_k,j), ∂A^j／∂x^k = α(δ_j+1,k － δ_j,k) = α(δ_k-1,j － δ_k,j)
∴ ∂A^k／∂x^j － ∂A^j／∂x^k = α(δ_k+1,j － δ_k-1,j) = α(δ_j-1,k － δ_j+1,k).
∂φ／∂x^j = －(1／ε)[(x^j － x^j-1) － (x^j+1 － x^j)] = －(2／ε)[x^j － (1／2)(x^j-1 + x^j+1)].

Substituting these expressions into the above result, we get the concrete form of the equation of motion of the main problem of this post as follows.
M(d／dt)² x^j = Qα(dx^j-1／dt － dx^j+1／dt) + Q(2／ε)[x^j － (1/2)(x^j-1 + x^j+1)].
Substituting mεα² and m for M and Q in this equation, we get the following expression as the final form of the equation of motion.
(d／dt)² x^j = [1／(εα)](dx^j-1／dt － dx^j+1／dt) + [2／(ε²α²)][x^j － (1/2)(x^j-1 + x^j+1)].

This equation is the opposite of the following equation representing, as it were, an automatic collision avoidance system.
M(d／dt)² x^j = Qα[(dx^j-1／dt － dx^j／dt) + (dx^j+1／dt － dx^j／dt)] － Q(2／ε)[x^j － (1/2)(x^j-1 + x^j+1)].

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Multiplying both sides of the equation by dx^j／dt and summing over j, we get the following equation representing the energy conservation law.
(d／dt)Σ_a≦j≦b(M／2)(dx^j／dt)² = Qα[(dx^a-1／dt)(dx^a／dt) － (dx^b／dt)(dx^b+1／dt)] + (Q／ε)(d／dt)Σ_a≦j≦b[(x^j)² － x^jx^j+1] － (Q／ε)(x^a-1 dx^a／dt － x^b dx^b+1／dt).
∴ (d／dt)Σ_a≦j≦b[(M／2)(dx^j／dt)² － (Q／ε)x^j(x^j － x^j+1)]
= Qα[(dx^a-1／dt)(dx^a／dt) － (dx^b／dt)(dx^b+1／dt)] － (Q／ε)(x^a-1 dx^a／dt － x^b dx^b+1／dt).
∴ (d／dt)Σ_a≦j≦b[(M／2)(dx^j／dt)² － (Q／ε)x^j(x^j － x^j+1)]
= Q[α(dx^a-1／dt)(dx^a／dt) － (1／ε)x^a-1(dx^a／dt)] － Q[α(dx^b／dt)(dx^b+1／dt) － (1／ε)x^b(dx^b+1／dt)].

The right hand side of this equation represents the time derivative of energy possessed by the part such that a≦j≦b, and the left hand side of this equation represents
(energy flow at j=a) - (energy flow at j=b).
Therefore this equation represents the energy conservation law.

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To investigate the character of this problem briefly, now I omit the scalar potential term and try solving the following equation.
(d／dt)² x^j = [1／(εα)](dx^j-1／dt － dx^j+1／dt).
Introducing a Matrix A defined as
A_jk = δ_j-1,k － δ_j+1,k,
the equation is rewritten as
(d／dt)² x^j = [1／(εα)]Σ_k A_jk dx^k／dt.

Then the solution is writen as follows.
dx／dt = a exp[tA/(εα)] v,
x = aεα exp[tA／(εα)] u + w,
where a is a constant real number, and u, v, w are infinite dimensional constant real column vectors such that v = Au.

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From the point of view of energy conservation, velocity does not increase or decrease as a whole.
This situation is expected to be represented well by
dx／dt = a exp(it × somematrix) v,
but the above solution does not include the imaginary unit at all.
Perhaps this superficial contradiction is cancelled by the fact that the matrix A has a property analogous to the property of a pure imaginary number.
As for the case of a 2×2 matrix, the matrix A defined by the equation:
A_jk = δ_j-1,k - δ_j+1,k
is actually a representation of the imaginary unit.
We can fully express this understanding by redefining the matrix A' as A'_jk = －iA_jk and rewriting as
dx／dt = a exp[itA'／(εα)] v.
Here A' is Hermitian.
We know well from quantum mechanics that this represents a rotation in an infinite dimensional space.

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If A^-1 exists, u = A^-1 v.
However, perhaps A^-1 does not exist, because there is an invariant vector under the transformation exp[tA／(εα)] so zero is an eigenvalue of A so the determinant of A is zero.
This is understood by considering about A' = -iA.
A' is Hermitian so it is diagonalized to the form whose all diagonal elements are eigenvalues of A' so the determinant of A' is the product of all its eigenvalues.
Therefore, if zero is an eigenvalue of A' then the determinant of A' is zero.
The eigenvalues of A are completely same as the eigenvalues of A'.
The determinant of A is different from one of A' only by a factor, so if zero is an eigenvalue of A then the determinant of A is zero.
It is shown below that there is an invariant vector under the transformation exp[tA／(εα)].

The space of all classical solutions of the main equation of this post which is an equation of motion is
{ x | ∃ξ,η; ∀j; x^j = jξ + η }.
Every vector in this space is written in the form:
a ∂x/∂ξ + b ∂x/∂η.
The j-th component of this vector is
(a ∂x/∂ξ + b ∂x/∂η)^j = aj + b.

This individual vector is not necessarily invariant under exp[tA/(εα)], but the space which consists of all such vectors is invariant under exp[tA/(εα)].
I prove it below.

[A(a ∂x/∂ξ + b ∂x/∂η)]_j
= Σ_k A_jk (a ∂x/∂ξ + b ∂x/∂η)^k
= Σ_k (δ_j-1,k - δ_j+1,k)(ak + b)
= [(j-1)a + b] - [(j+1)a + b]
= -2a.
(So, the vector is invariant if a = 0.)

[A² (a ∂x/∂ξ + b ∂x/∂η)]_j
= Σ_k A_jk (-2a)
= Σ_k (δ_j-1,k - δ_j+1,k)(-2a)
= 0.

Aⁿ (a ∂x/∂ξ + b ∂x/∂η) = A^n-2 A² (a ∂x/∂ξ + b ∂x/∂η) = A^n-2 0 = 0 (n≧3).

{exp[tA/(εα)](a ∂x/∂ξ + b ∂x/∂η)}_j
= {[1 + tA/(εα)](a ∂x/∂ξ + b ∂x/∂η)}_j
= (aj + b) + [t/(εα)](-2a)
= aj + [b - 2at/(εα)].

∴ exp[tA/(εα)](a ∂x/∂ξ + b ∂x/∂η)
　　= a ∂x/∂ξ + [b - 2at/(εα)]∂x/∂η
　　∈ the original space.

This justifies our thinking of exp[tA/(εα)] as a rotation around the space of all classical solutions as an invariant axis.

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To write down exp[tA/(εα)] explicitly, below I try to get explicit expressions of all powers of A.

(A²)_jk
= Σ_sA_j,sA_s,k
= Σ_s(δ_j-1,s - δ_j+1,s)(δ_s-1,k - δ_s+1,k)
= δ_j-2,k - 2δ_j,k + δ_j+2,k

(A³)_jk
= Σ_s(δ_j-2,s - 2δ_j,s + δ_j+2,s)(δ_s-1,k - δ_s+1,k)
= δ_j-3,k - 3δ_j-1,k + 3δ_j+1,k - δ_j+3,k

(A⁴)_jk
= Σ_s(δ_j-3,s - 3δ_j-1,s + 3δ_j+1,s - δ_j+3,s)(δ_s-1,k - δ_s+1,k)
= δ_j-4,k - 4δ_j-2,k + 6δ_j,k - 4δ_j+2,k + δ_j+4,k

(A⁵)_jk
= Σ_s(δ_j-4,s - 4δ_j-2,s + 6δ_j,s - 4δ_j+2,s + δ_j+4,s)(δ_s-1,k - δ_s+1,k)
= δ_j-5,k - 5δ_j-3,k + 10δ_j-1,k - 10δ_j+1,k + 5δ_j+3,k - δ_j+5,k

(Aⁿ)_jk
= Σ_0≦s≦n/2(-1)^sc₁(s,n)δ_j-n+2s,k + Σ_0≦s＜n/2(-1)^n+sc₂(s,n)δ_j+n-2s,k,
c_r(s,n+1) = c_r(s,n) + c_r(s-1,n),
c_r(0,n) = 1.

Using these results in the expression:
dx／dt = a exp[tA/(εα)] v,
it is derived that
dx^j／dt = Σ_n=0^∞ (1/n!)[t／(εα)]ⁿ[Σ_0≦s≦n/2 (-1)^s c₁(s,n) δ_j-n+2s,0 - Σ_0≦s＜n/2 (-1)^s c₂(s,n) δ_j+n-2s,0]
in the case that a = 1 and v_j = δ_j0.

I can not go further because the recurrence relation for c can not be solved fully.
It seems to be impossible to write down fully c₁(s,n) and c₂(s,n) as functions of s and n.

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To take the scalar potential term into account, I rewrite the main equation of this post to the following form.
(d／dt + F)(d／dt + G)x = 0.
By comparing this expression with the original equation, the condition to determine F and G is found to be as follows.
(F + G)_jk = [1／(εα)](δ_j+1,k － δ_j－1,k),
(FG)_jk = [1／(ε²α²)](δ_j－1,k + δ_j+1,k － 2δ_jk).

Then the solution is written as follows.
(d／dt + G)x = b exp(－tF) v₁,
x = exp(－tG)[v₁ + b ∫_c^tdt' exp(t'G)exp(－t'F)v₂].

Below I try to know F and G concretely.

F_jk = Σ_s=－∞^∞ f_s δ_j+s,k,
G_jk = Σ_s=－∞^∞ g_s δ_j+s,k.

(FG)_jk = Σ_r F_jr G_rk
= Σ_r,s1,s2 f_s1 δ_j+s1,r g_s2 δ_r+s2,k
= Σ_s1,s2 f_s1 g_s2 δ_j+s1+s2,k

The condition for FG means that (FG)_jk = 0 if |j－k| ≧ 2.
So f and g must satisfy the following condition.
|r| ≧ 2 ⇒ Σ_s1+s2=r f_s1 g_s2 = 0,
∴ |r| ≧ 2 ⇒ Σ_s f_s g_r－s = 0.
The condition for FG means that (FG)_jk = －2／(ε²α²) if j = k.
So f and g must satisfy the condition:
Σ_s f_s g_－s = －2／(ε²α²)
derived as the case that r = 0.
The condition for FG means that (FG)_jk = 1／(ε²α²) if j+1 = k.
So f and g must satisfy the condition:
Σ_s f_s g_1－s = 1／(ε²α²)
derived as the case that r = 1.
The condition for FG means that (FG)_jk = 1／(ε²α²) if j－1 = k.
So f and g must satisfy the condition:
Σ_s f_s g_－1－s = 1／(ε²α²)
derived as the case that r = －1.

From the condition for F + G, the following conditions for f and g are derived.
f₁ + g₁ = 1／(εα),
f_－1 + g_－1 = －1／(εα),
not(s = 1 or s = －1) ⇒ f_s + g_s = 0.

Using the condition derived from the condition for F + G, now I eliminate f in the condition derived from the condition for FG.

|r| ≧ 2 ⇒ f₁ g_r－1 + f_－1 g_r+1 + Σ_1≠s≠－1 f_s g_r－s = 0.
∴ |r| ≧ 2 ⇒ [1／(εα) － g₁]g_r－1 + [－1／(εα) － g_－1]g_r+1 + Σ_1≠s≠－1 (－g_s)g_r－s = 0.
∴ |r| ≧ 2 ⇒ [1／(εα)]g_r－1 － [1／(εα)]g_r+1 = Σ_s g_s g_r－s.

As for the part as r = 0,
f₁ g_－1 + f_－1 g₁ + Σ_1≠s≠－1 f_s g_－s = －2／(ε²α²).
∴ [1／(εα) － g₁]g_－1 + [－1／(εα) － g_－1]g₁ + Σ_1≠s≠－1 (－g_s)g_－s = －2／(ε²α²).
∴ [1／(εα)]g_－1 － [1／(εα)]g₁ + 2／(ε²α²) = Σ_s g_s g_－s.

As for the part as r = 1,
f₁ g_1－1 + f_－1 g₁₊₁ + Σ_1≠s≠－1 f_s g_1－s = 1／(ε²α²).
∴ [1／(εα) － g₁]g₀ + [－1／(εα) － g_－1]g₂ + Σ_1≠s≠－1 (－g_s)g_1－s = 1／(ε²α²).
∴ [1／(εα)]g₀ － [1／(εα)]g₂ － 1／(ε²α²) = Σ_s g_s g_1-s.

As for the part as r = －1,
f₁ g_－1－1 + f_－1 g_－1+1 + Σ_1≠s≠－1 f_s g_－1－s = 1／(ε²α²).
[1／(εα) － g₁]g_－2 + [－1／(εα) － g_－1]g₀ + Σ_1≠s≠－1 (－g_s)g_－1－s = 1／(ε²α²).
[1／(εα)]g_－2 － [1／(εα)]g₀ － 1／(ε²α²) = Σ_s g_s g_－1－s.

Therefore, the complete set of conditions for determine g are as follows.
|r| ≧ 2 ⇒ [1／(εα)]g_r－1 － [1／(εα)]g_r+1 = Σ_s g_s g_r－s,
[1／(εα)]g_－1 － [1／(εα)]g₁ + 2／(ε²α²) = Σ_s g_s g_－s,
[1／(εα)]g₀ － [1／(εα)]g₂ － 1／(ε²α²) = Σ_s g_s g_1－s,
[1／(εα)]g_－2 － [1／(εα)]g₀ － 1／(ε²α²) = Σ_s g_s g_－1－s.

After determining g from these conditions, f is determined from the condition derived from the condition for F + G.

The number of equations for determining g is equal to the number of elements of g, and so perhaps the solution of the set of equations exists, but it is too difficult to be solved.

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Though I can not write the exact solution because of such a difficulty, perturbation method saves us from giving up knowing further.

In convenience, here I paste again the main equation to be solved.
(d／dt)² x^j = [1／(εα)](dx^j-1／dt － dx^j+1／dt) + [1／(ε²α²)](2x^j － x^j-1 － x^j+1).

To apply perturbation method to this equation, I replace the coefficient of the scalar potential part by a new constant k.
(d／dt)² x^j = [1／(εα)](dx^j-1／dt － dx^j+1／dt) + k(2x^j － x^j-1 － x^j+1).

Then I substitute a power expansion of x in terms of k:
x^j = Σ_n=0^∞ kⁿ x_n^j
into the equation, and I equate both sides order by order as usual.

Below I try calculating the functions corresponding to the first two orders.
(d／dt)² x₀^j = [1／(εα)](dx₀^j-1／dt － dx₀^j+1／dt).
(d／dt)² x₁^j = [1／(εα)](dx₁^j-1／dt － dx₁^j+1／dt) + (2x₀^j － x₀^j-1 － x₀^j+1).

x₀ has already been derived above when I omitted the scalar potential term.

From the second equation, it is derived that
(d／dt)² x₁ = [1／(εα)] A dx₁／dt + B x₀,
where
A_jk = δ_j-1,k － δ_j+1,k,
B_jk = 2δ_jk － δ_j-1,k － δ_j+1,k.
Solving this equation, we get the expression for dx₁／dt:
dx₁／dt = exp[tA／(εα)]∫_c^tdt' exp[－t'A／(εα)]Bx₀(t').
Substituting x₀ into this expression, we get the following expression.
dx₁／dt = exp[tA／(εα)]∫_c^tdt' exp[－t'A／(εα)]B{aεα exp[t'A／(εα)] u + w}.

To reduce this expression, here I calculate the commutation relation between A and B.
[A, B]_jk = Σ_s A_jsB_sk － Σ_s B_jsA_sk
= Σ_s(δ_j-1,s － δ_j+1,s)(2δ_sk － δ_s-1,k － δ_s+1,k) － Σ_s(2δ_js － δ_j-1,s － δ_j+1,s)(δ_s-1,k － δ_s+1,k)
= (2δ_j-1,k － δ_j-2,k － δ_j,k － 2δ_j+1,k + δ_j,k + δ_j+2,k) － (2δ_j-1,k － 2δ_j+1,k － δ_j-2,k + δ_j,k － δ_j,k + δ_j+2,k)
= 0.

Because of this vanishment,
exp[－t'A／(εα)] B exp[t'A／(εα)]= B,
so the expression for dx₁／dt is reduced to the following form.
dx₁／dt = exp[tA／(εα)]∫_c^tdt' {aεαBu + exp[－t'A／(εα)]Bw}
= exp[tA／(εα)]{aεα(t － c)Bu － εα exp[－tA／(εα)]w' + εα exp[－cA／(εα)]w'}
= εα{a(t － c)exp[tA／(εα)]Bu － w' + exp[(t － c)A／(εα)]w'}.
where w' and w are chosen such that Aw' = Bw.

Let's consider the special case such that w = w' = 0 to sketch roughly the effect of scalar potential term.
(d／dt)(x₀ + kx₁)
= a exp[tA／(εα)]v + kεαa(t － c)exp[tA／(εα)]Bu
= a exp[tA／(εα)][v + kεα(t － c)Bu].
This means that a gradually varying vector, which is a linear function of t, is transformed by rotation with some constant angular velocity.

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Last edited at 2014/06/09/17:20JST