C:/Documents and Settings/Profesor/Escritorio/Cubo/Cubo/Cubo/Cubo 2011/Cubo2011-13-01/Vol13 n\2721/Art N\2603 .dvi


CUBO A Mathematical Journal
Vol.13, No¯ 01, (25–43). March 2011

Evolutionary method of construction of solutions of
polynomials and related generalized dynamics

Robert M. Yamaleev

Facultad de Estudios Superiores,

Universidad Nacional Autonoma de Mexico,

Cuautitlán Izcalli, Campo 1, C.P.54740, México.

Joint Institute for Nuclear Research, LIT, Dubna, Russia.

email: iamaleev@servidor.unam.mx

ABSTRACT

Invariant theory as a study of properties of polynomials under translational transfor-

mations is developed. Class of polynomials with congruent set of eigenvalues is intro-

duced. Evolution equations for eigenvalues and coefficients remaining the polynomial

within proper class of polynomials are formulated. The connection with equations for

hyper-elliptic Weierstrass and hyper-elliptic Jacobian functions is found. Algorithm of

calculation of eigenvalues of the polynomials based on the evolution process is elabo-

rated. Elements of the generalized dynamics with n-order characteristic polynomials

are built.

RESUMEN

La teoŕıa de invariantes es un estudio de las propiedades de los polinomios que se

desarrolla en las transformaciones de traslación. Se introduce una clase de polinomios

congruentes con un conjunto de valores propios. Se formulan ecuaciones de evolución de

los valores propios y los coeficientes del polinomio restante dentro de la clase adecuada



26 Robert M. Yamaleev CUBO
13, 1 (2011)

de los polinomios. Se encuentra la conexión con las ecuaciones de Weierstrass hiper-

eĺıpticas y funciones jacobiano hiper-eĺıptica. Son elaborados algoritmos de cálculo de

valores propios de los polinomios basado en el proceso de evolución.

Keywords: Nonspreading mapping, maximal monotone operator, inverse strongly-monotone map-

ping, fixed point, iteration procedure

Mathematics Subject Classification: 12Yxx

1 Introduction

The problem of construction of solutions of the polynomial equations as a certain functions of

the coefficients is one of the oldest mathematical problems. E.Galois and H.Abel had proved that

the polynomial higher than fourth order, in general, does not admit a presentation of solutions

by radicals [3]. This rigorous mathematical theory directed mathematicians to look for the other

ways of solutions. In particularly, the eigenvalues of the polynomials can be expressed in analytical

way as certain functions of the coefficients [2], [7]. Ch.Hermite was first who found an elegant

expression of eigenvalues of the quintic equation by modular functions [6]. The theory of elliptic

functions originally was related with the problem of finding of eigenvalues of the cubic polynomial.

In fact, the eigenvalues of the cubic equation admit presentation by Weierstrass elliptic functions

at the periods [12]. It is clear, however, that for a search of analytical solutions of the n > 5-

degree polynomials one needs of mathematical tools beyond the elliptic functions. In this context

as hopeful tools one may consider the theories of hyper-elliptic functions [9] and multi-complex

algebras [8], [13].

The main purpose of the present paper is to construct an Algorithm for calculation roots

of the polynomials. For that purpose, firstly, we study invariant properties of the polynomials

with respect to simultaneous translations of the roots. Secondly, we build the system of evolution

equations for the coefficients transforming the given polynomial into the polynomial with one trivial

solution. The evolution process is directed in a such way that remain the original polynomial within

proper class where the roots of the polynomials form congruent set of values. The coefficients of

the polynomial with respect to the parameter of evolution are solutions of the Cauchy problem

for ordinary differential equations. As soon as the Cauchy problem is resolved, the eigenvalues

of initial polynomial are found simply by simultaneous translations of the set of eigenvalues of

the final polynomial. Since the final polynomial possesses with one trivial solution, the degree of

the polynomial is reduced from n to (n − 1) which simplifies the process of solution. If solution
of the obtained polynomial still is a difficult problem, then the method can be applied again in

order to reduce the degree of the obtained polynomial. This process can be continued up till

linear equation. The inverse process is fulfilled simply by simultaneous translations of the roots

from the solution of the linear equation up till solutions of n-degree polynomial. It is shown,

the evolution equations for the coefficients can be identified with equations for Weierstrass-type



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Evolutionary method of construction of solutions of polynomials
and related generalized dynamics 27

hyper-elliptic functions, whereas the evolution equations for the roots are given by equations for

Jacobian-type hyper-elliptic functions. Furthermore, it is demonstrated a link between evolution

equations for the coefficients of the polynomials and the classical dynamic equations. In n-order

generalized mechanics the inner- and outer-momenta are inter-related as roots and coefficients of

the characteristic n-degree polynomial. In particular n = 2 case, we return to well-known equations

of the relativistic dynamics closely related with quadratic polynomial of the mass-shell equation.

Besides the Introduction the paper contains the following sections.

In Section 2, the equations of evolution for the coefficients of n-degree polynomial are formu-

lated. In Section 3, the Algorithm of finding of eigenvalues of the n-degree polynomial is built.

In Section 4, some peculiarities of the cubic equation is explored. In Section 5, the elements of

the relativistic dynamics based on quadratic characteristic polynomial is presented. In Section 6

elements of the generalized dynamics related with n-degree characteristic polynomial is outlined.

2 Evolution equations for eigenvalues and coefficients of n-

degree polynomial

If F is a field and q1, ..., qn are algebraically independent over F , the polynomial

p(X) =
n∏

i=1

(X − qi)

is referred to as generic polynomial over F of degree n. The polynomial equation of n-degree over

field F is written in the form

p(X) := Xn +

n−1∑

k=1

(−)k(n − k + 1)PkXn−k + (−)nP 2 = 0, (2.1)

where the coefficients Pk ∈ F (q1, ..., qn). In this paper we shall restrict our attention only to
polynomials with real coefficients and with simple roots. The signs at the coefficients in Eq.(2.1)

are changed from term to term which allows in Vièta formulae to keep only the positive signs.

The expressions at the coefficients are included for a convenience and have no real bearing on

the theory. The mapping from the set of eigenvalues onto the set of coefficients is given by Vièta

formulae

(a) nP1 =
n∑

i=1

qi, (b) P
2 =

n∏

i=1

qi, (c) Pk =
∑

1≤r1<...<rk≤n

k∏

j=1

qrj , k = 2, ..., n − 1, (2.2)

here P1, Pk, P
2 are defined via elementary symmetric k-degree polynomials of the eigenvalues. The

number of monomials inside k-th elementary polynomial is equal to binomial coefficient:

Ckn =

(
k

n

)
=

n!

k!(n − k)!
. (2.3)



28 Robert M. Yamaleev CUBO
13, 1 (2011)

Since the roots of the generic polynomial p(X) are algebraically independent, this polynomial is,

in some sense, the most general polynomial possible.

In p(X) replace X with X = Y + P1. This transformation will eliminate the (n − 1)-degree
term, resulting in a polynomial of the form

r(Y ) := Y n +
n−1∑

k=2

(−)kRkY n−k + (−)nR0 = 0. (2.4)

The polynomials p(X) and r(Y ) have the same splitting field and hence the same Galois group.

Let E be the splitting field of r(Y ), let yk, k = 1, .., n be its roots in E and G = GF (E) be its

Galois group.

Lemma 2.1

The coefficients Rk, k = 2, ..., n − 1, R0 of Eq.(2.4) remain unaltered with respect to simulta-
neous translations of the eigenvalues of Eq.(2.1)

The Proof of the statement it follows from formula Y = X − P1 which in terms of the
eigenvalues is expressed as follows

yk = qk −
1

n

n∑

i=1

qi =
1

n

n∑

i6=k

(qk − qi), k = 1, ..., n. (2.5)

It is seen that the eigenvalues of Eq.(2.4) are represented by differences between the eigenvalues

of Eq.(2.1), hence, they are invariants with respect to the simultaneous translations. Since the

coefficients R0, Rk, k = 2, ..., n−1, are sum of uniform monomials of yk, k = 1, ..., n they have same
feature, namely, they are invariants with respect to simultaneous translations of the eigenvalues of

Eq.(2.1).

End of Proof.

The polynomial r(Y ) we denominate as invariant polynomial (with respect to translations of

the roots of (2.1)).

The main task of this section is to introduce evolution equations for the coefficients of Eq.(2.1)

for which the coefficients of Eq.(2.4) Rk, k = 2, ..., n − 1, R0 serve as the first integrals. This result
is given by the following

Theorem 2.2

Let qk, k = 1, ..., n be set of eigenvalues of polynomial equation of n- degree (2.1). Let differ-

entials of all eigenvalues are equal to each other

dq1 = dq2 = . . . = dqk = . . . = dqn. (2.6)

Then differentials of the coefficients satisfy the following system of equations:

2Pn−1dP1 = dP
2, (2.7)



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Evolutionary method of construction of solutions of polynomials
and related generalized dynamics 29

dPn−k = (k + 2)Pn−k−1 dP1, k = 1, ..., n − 2; (2.8)

Proof.

From (2.2a) it follows

dqk = dP1, k = 1, ..., n.

Coefficients of the polynomial (2.1) are symmetric forms of the eigenvalues where k-th coefficient

(n − k + 1)Pk consists of Ckn monomials of k-degree. Thus, the derivative of this coefficient (n − k +
1)dPk contains kC

k
n monomials and, since the derivatives of all eigenvalues are equal to dP1, the

derivative of (n − k + 1)Pk is proportional to λdP1 where the coefficient of proportionality consists
of kCkn (k − 1)-degree symmetric monomials. On the other hand, the symmetric polynomial of
(k − 1)-th degree can be expressed only by Ck−1n symmetric monomials of (k − 1)-degree. This
means the expression for λ consists of kCkn/C

k−1
n = n − k + 1 same symmetric polynomials of

(k − 1)-degree which are equal to the (k − 1)-th coefficient (n − k + 2)Pk−1. The result is expressed
as follows

(n − k + 1)dPk = (n − k + 1)(n − k + 2)Pk−1 dP1, k = 2, 3, ..., n − 1.

Differentiation of the last coefficient is obtained by direct calculation

dP 2 = 2Pn−1dP1, (2.9)

which completes the system of differential equations for the coefficients of Eq.(2.1).

End of Proof.

The following Lemma demonstrates an important role of the invariant polynomial in the

evolution process.

Lemma 2.3

The first integrals of evolution equations (2.7)-(2.8) are given by coefficients of invariant poly-

nomial (2.4).

Proof

In fact, the roots of Eq.(2.4), yk, k = 1, ..., n, according to formulae (2.5), are the first integrals

of evolution equations for eigenvalues (2.6). Equations (2.7)-(2.8) are consequences of Eqs.(2.6),

hence coefficients Rk as algebraic functions of the solutions of evolution equations are first integrals

of Eqs.(2.7)-(2.8).

End of Proof

Inversely by substituting Y with X − P1 in Eq.(2.4) we come back from invariant equation
(2.4) to equation (2.1). Substitute X − P1 instead of Y and gather together powers of X, and
the coefficients of the obtained polynomial compare with coefficients of Eq.(2.1). We shall see

that the coefficients Pk, k = 1, ..., n − 1 now are expressed as k-degree polynomials of P1 with
coefficients consisting of Rk, k = 2, ..., n − 1. Especially notice, the invariant R0 is defined by



30 Robert M. Yamaleev CUBO
13, 1 (2011)

n-degree polynomial of P1 the coefficients of which are certain polynomials of Rk.k = 2, ..., n − 1.
The task is to define a explicit form of the polynomial of P1. The general form of the desired

polynomial can be presented as follows

P n1 +

n−1∑

k=2

P n−k1 fk(R1, ..., Rk) + R0 = P
2.

Now, the next task is to find an explicit form of the function fk. For that purpose explore firstly the

case P 2 = 0. In this case one of the solutions of Eq.(2.1) is equal to zero. Then the corresponding

solution of the invariant polynomial is

y1 = −P1(P 2 = 0) = −P1(0),

which satisfy Eq.(2.4). By replacing Y with (−P1(0)) in Eq.(2.4) we come to the following equation
for P1(0):

P n1 (0) +

n−1∑

k=2

RkP
n−k
1 (0) + R0 = 0. (2.10)

Notice, here the signs at all coefficients are positive. Since we use evolution equations (2.7)-(2.8)

in order to pass from Eq.(2.1) with coefficients { Pk, P 2 6= 0 } to the polynomial with coefficients
{ P̃k, P 2 = 0 }, these two polynomials possess with the same invariants as the original one. Hence,
when P 2 6= 0 equation (2.10) is transformed as follows

P n1 +
n−1∑

k=2

RkP
n−k
1 + R0 = P

2. (2.11)

Notice, the situation is somewhat similar the conventional Classical Invariant Theory of Poly-

nomials [10]. If classical invariant theory is a study of properties of a polynomial p(x) that are

unchanged under fractional linear transformations, within the framework of the present approach

we study properties of polynomials under translational transformations.

Theorem 2.4

If evolution of the coefficients of Eq.(2.1) obey equations (2.7)-(2.8), then

(k + 1)Pn−k =
1

k!

dk

dP1
k
P 2. (2.12)

Proof.

Differentiate equation (2.11) by taking into account that R0, Rk, k = 1, ..., n − 1 are constants.
We get

dP1 ( nP
n−1
1 +

n−2∑

k=2

(n − k)RkP n−k−11 + Rn−1 ) = dP
2. (2.13)



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Evolutionary method of construction of solutions of polynomials
and related generalized dynamics 31

Compare this equation with Eq.(2.9). It is seen, the expression inside brackets at dP1 is nothing

else than the coefficient 2Pn−1 expressed as a polynomial of P1 with invariant coefficients. Hence,

2Pn−1 = nP
n−1
1 +

n−2∑

k=2

(n − k)RkP n−k−11 + Rn−1 =
d

dP1
P 2. (2.14)

Next, differentiate (2.14) with respect to P1, this leads to the equation

dPn−1 = 3Pn−2 dP1, (2.15)

where 3Pn−2 is the following coefficient of Eq.(2.1):

3Pn−2 =
n(n − 1)

2
P n−21 +

n−3∑

k=2

(n − k)(n − k − 1)
2

RkP
n−k−2
1 + Rn−2. (2.16)

Consequently,

3Pn−2 =
d2

dP1
2
P 2.

The next step of this procedure leads to the equation

dPn−2 = 4Pn−3 dP1, (2.17)

where

4Pn−3 =
n(n − 1)(n − 2)

1 · 2 · 3
P n−31 +

n−4∑

k=2

(n − k)(n − k − 1)(n − k − 2)
1 · 2 · 3

RkP
n−k−3
1 + Rn−3. (2.18)

Hence,

4Pn−3 =
d3

dP1
3
P 2.

By induction we come to the general formula for l-th coefficient Pn−l:

(l + 1)Pn−l =

(
l

n

)
P n−l1 +

n−l−1∑

k=2

(
l

n − k

)
RkP

n−k−l
1 + Rn−l =

1

l!

dl

dP1
l
P 2. (2.18)

End of Proof.

This theorem has some interesting consequences.

Corollary 2.5

The following representation for polynomial p(X) holds true

p(X) = exp(−X
d

dP1
)P 2 = 0. (2.19)

Proof



32 Robert M. Yamaleev CUBO
13, 1 (2011)

The generator of translation defined by Euler operator is represented by the following expan-

sion

exp(−X
d

dP1
) = 1 − X

d

dP1
+ X2

1

2!

d2

dP 21
+ ... + Xn

1

n!

d

dP n1
+ ... . (2.20)

By differentiating Eq.(2.11) n-times with respect to P1, we get

d

dP n1
P 2 = n!. (2.21)

Hence the sum in (2.20) is completed by this term. Thus,

exp(−X
d

dP1
)P 2 = p(X). (2.22)

Here the variable X means one of the roots of Eq.(2.1) in quality of which let us take qn.

The derivative with respect to P1 due to (2.6) can be expressed by the following sum

d

dP1
=

n∑

k=1

∂qk
∂P1

∂

∂qk
=

n∑

k=1

∂

∂qk
. (2.23)

Then,

exp(−qn
d

dP1
)P 2 =

n−1∏

k=1

exp(−qn
∂

∂qk
) exp(−qn

∂

∂qn
). (2.24)

Now, take into account Vièta formula for P 2 given by (2.2). The Euler operator (2.24) will translate

each root by qn. The last operator in (2.24) acts only upon n-th root resulting qn − qn = 0. Hence,

exp(−X
d

dP1
)P 2 = p(X) = 0.

End of Proof.

3 Algorithm of finding the eigenvalues of n-degree polyno-

mials

In the previous section we have found that evolution governed by equations (2.7)-(2.8) remains

the polynomials within certain class of polynomials with congruent set of eigenvalues. This fact

prompts an idea to use the evolution equations in order to come to the simplified polynomial. The

main purpose of this section is to elaborate an algorithm of solution of n-degree polynomial equation

by reducing the degree of the polynomial equation. The evolution from original polynomial to

the simplified polynomial is fulfilled in such a way which remains unaltered coefficients of the

invariant polynomial. Hence, initial and final polynomials of this evolution process will possess

with congruent set of eigenvalues, so that the solutions of the former can be obtained from the

solutions of the latter by simultaneous transformations of translation.



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Evolutionary method of construction of solutions of polynomials
and related generalized dynamics 33

The algorithm is directed in a such way that at the final point of evolution we shall come to

the polynomial with P 2 = 0. The last coefficient P 2 is used as a parameter of the evolution. This

evolution will remain unaltered the coefficients of Eq.(2.4), hence from solutions of the polynomial

with P 2 = 0 we may come to the solutions of the original equation simply by translation of the set

of solutions of the simplified polynomial with new coefficients.

Denote x = P 2. Re-write Eq.(2.8) with respect to x. We get

2Pn−1dP1 = dx,

dPn−k
dx

= (k + 2)Pn−k−1
dP1
dx

, k = 1, ..., n − 3; (3.1)

dP2
dx

= nP1
dP1
dx

.

This is a Cauchy problem with initial data Pk(x = P
2), k = 1, 2, 3, ..., n − 1. The variable x runs

from x = P 2 up till x = 0. These equations usually are resolved by using the Cauchy-Lipschitz

method of calculation [4]. The procedure is fulfilled by dividing the interval (x0, 0) into N parts:

∆x0 = x1 − x0, ∆xi = xi+1 − xi, ∆xn−1 = x − xn−1, (3.2)

where xi < xi+1, xN = 0.

In this way the continuous evolution process is transformed into discrete process consisting of

N steps. At the last step of discrete evolution process we come to n-degree polynomial free of the

last coefficient:
(1)

Xn +

n−1∑

k=1

(−)k(n − k + 1)
(1)

Pk

(1)

Xk = 0. (3.4)

One of the solutions is trivial, excluding this solution we come to the polynomial of (n − 1)-degree:
(1)

Xn−1 +

n−1∑

k=1

(−)k(n − k + 1)
(1)

Pk

(1)

Xk + 2
(1)

Pn−1 = 0. (3.5)

Let us mention that Eq.(3.4) possesses with same invariants as original one, i.e. Eq.(2.1). Suppose

that the solutions of Eq.(3.5) are
(1)
q k, k = 1, ..., n − 1. Complete this set of solutions by

(1)
q n = 0.

Then the solutions of original equation one may find simply by using the following translations

qk =
(1)
qk + P1 −

(1)

P1, k = 1, ..., n. (3.6)

If solution of the (n−1)-degree equation still is difficult problem, then one may apply the algorithm
again in order to reduce the problem of solution of (n−1)-degree equation to the problem of solution
(n − 2)-degree equation. This process can be continued up till quadratic or linear equation. At
each step of the iteration one will find an information on coefficient

(r)

P 1. At the r-th iteration one

deals with (n − r)-degree equation in the form
(r)

Xn−r +

n−r−1∑

k=1

(−)k(n − k + 1)
(r)

Pk

(r)

Xk + (r + 1)
(r)

Pn−r = 0. (3.7)



34 Robert M. Yamaleev CUBO
13, 1 (2011)

At this stage the norm of the coefficient
(r)

P n−r will fulfil the role of evolution parameter of the next

evolution process. As soon as the solutions of the lowest polynomial is found, the inverse process

of iterations will consist of translations of the known set of eigenvalues with known parameter of

translation. Let the last step of iteration be a linear equation with unique solution:

(n−1)
q1 = n

(n−1)

P1 .

Then the solutions of the original equation (2.1) are found as a result of the following set of

translations

q1 = n
(n−1)

P1 + n

n∑

r=2

1

r
(

(n−r)

P1 −
(n−r+1)

P1 ), (3.8a)

qs =

n∑

r=s

1

r
(

(n−r)

P1 −
(n−r+1)

P1 ), s = 2, 3, ..., n; (3.8b)

where
(0)

P1 = P1.

4 Relativistic Lorentz-force equations and related quadratic

polynomial

There exist an certain inter-relation between the evolution of the polynomials and the dynamic

systems. The aim of this section is to demonstrate as evolution of the quadratic polynomial is

related with the relativistic Lorentz-force equations. In the sequel, we shall use this example as a

starting platform to pass to case of higher degree polynomials and related generalized dynamics.

Let us start with generic quadratic polynomial

X2 − 2P0 X + P 2 = 0, (4.1)

with real coefficients P 20 ≥ P 2, and real eigenvalues p21, p22. This polynomial closely related with the
relativistic dynamics [14], [5]. In order to demonstrate this fact let us start from the Lorentz-force

equations describing a motion of the charged particle inside external electromagnetic fields E, B 1

dP

dτ
= (E P0 + [P × B]),

dP0
dτ

= (E · P), (4.2)

dr

dτ
= P,

dt

dτ
= P0. (4.3)

Consider projection of equations (4.2) on direction of the motion defined by unit vector ~n = P/P :

dP

dτ
= (E · n) P0,

dP0
dτ

= (E · n) P, (4.4)

1Hereafter for the sake of simplicity we omit all parameters like charge, mass, light-velocity and other parameters

regularizing physical dimensions taking them equal to unit.



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Evolutionary method of construction of solutions of polynomials
and related generalized dynamics 35

Then one deals only with the lengths of the momenta P0, P . Here parameter of evolution τ means

a proper time, simplify Eqs.(4.4) by introducing a new evolution parameter by

dP

ds
= P0,

dP0
ds

= P, (4.5)

where
ds

dτ
= (E · n). (4.6)

The first constant of motion is easily found

P 20 − P 2 = M 2. (4.7)

Here M 2 is a constant of motion. Conventionally, this constant is interpreted as a square of inertial

mass [11]. Inside stationary potential field, when E = −∇V (r), dynamic equations imply the other
integral of motion, the energy,

E0 = P0 + V (r). (4.8)

At the rest state where P = 0 one obtains P0(P = 0) = M . Consider the following linear

combinations of two kinds of energies:

p21 = P0 − M, p22 = P0 + M. (4.9)

From formulae (4.7) and (4.9) it follows

P = p1p2, P0 =
1

2
(p22 + p

2
1), M =

1

2
(p22 − p21). (4.10)

Notice, the first two formulae of (4.10) mean Vièta formulae for quadratic polynomial equation

(4.1). Substitution X = Y + P0 in (4.1) leads to invariant equation

Y 2 = P 20 − P 2 = M 2, (4.11)

the invariant coefficient of which is equal to the invariant of physical motion. Quadratic equation

for P0 is given by

P 20 − M 2 = P 2. (4.12)

Differentiating this equation we derive evolution equations for the coefficients, which in the case

of quadratic polynomial, of course, is a simple task:

2P0 P =
d

ds
P 2,

d

ds
P0 = P. (4.13)

Solutions of these evolution equations are given by hyperbolic cosine-sine functions

P = M sinh(s), P0 = M cosh(s). (4.14)

Then the eigenvalues of Eq.(5.1) are expressed by hyperbolic cosine-sine functions of one-half

argument

p21 =
√

2M sinh(
s

2
), p22 =

√
2M cosh(

s

2
). (4.15)



36 Robert M. Yamaleev CUBO
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Now, consider so-called the effective potential representation. With this purpose let us come

back to the equations written with respect to proper-time τ . Then from the second equation of

(4.13) it follows

E0 = P0 + V (r).

Further, replace P0 by E0 − V (r) in the first equation of (4.13), this gives

d

ds
P = −∇V (r) (E0 − V (r) = −∇W (r, E0), (4.16)

where the effective potential is defined by

W (rE0) = E0V (r) −
1

2
V 2(r). (4.17)

Eq.(4.16) is the Newtonian equation of motion with effective potential W and with the energy

E =
1

2
P 2 + W (r, E0) =

1

2
(E20 − M 2). (4.18)

5 Cubic polynomial equation and related dynamics

In the previous section we have demonstrated as the relativistic dynamics is related with the

evolution of quadratic polynomial. This example provides us with appropriate tool in order to

construct generalized scheme based on the evolution of polynomials of higher order. In this section

we explore the case of cubic polynomial

p(X) = X3 − 3P1 X2 + 2P2 X − P 2 = 0, (5.1)

where coefficients and eigenvalues are connected by the following formulae

3P1 = q1 + q2 + q3, 2P2 = q1q2 + q2q3 + q3q1, P
2 = q1q2q3. (5.2)

Let us assume that p(X) is irreducible. By replacing X with X = Y + P1 we come to invariant

polynomial

Y 3 + R2 Y − R0 = 0, (5.3)

where

(a) R2 = 2P2 − 3P 21 , (b) P 31 + R2 P1 + R0 = P 2. (5.4)

The eigenvalues and the coefficients of this polynomial are invariants with respect to simultaneous

translations of the eigenvalues of Eq. (5.1). Obviously this statement is a consequence of the

formula Y = X − P1 from which it follows

3y1 = e2 − e3, 3y2 = e3 − e1, 3y3 = e1 − e2, (5.5)

where

e1 = q3 − q2, e2 = q1 − q3, e3 = q2 − q1.



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Evolutionary method of construction of solutions of polynomials
and related generalized dynamics 37

Evolution equations remaining unaltered coefficients R0, R2 are obtained directly by differentiating

Eqs.(5.4). We get

dP1 (3P
2
1 + R2) = 2P2 dP1 = dP

2, (5.6)

dP2 = 3dP1.

The following equations for the eigenvalues hold true

d

ds
qk =

d

ds
P1 = P, k = 1, 2, 3. (5.7)

The generalized dynamics of third order is an object of special interest because this dynamics is

closely related with the classical elliptic functions. The solutions of the evolution equations for

the eigenvalues and the coefficients of the cubic polynomial can be represented via elliptic Jacobi

and Weierstrass functions, correspondingly [1]. On making use (5.7) from (5.4b) we come to the

following differential equation

P 31 + R1 P1 + R0 = (
dP1
ds

)2. (5.8)

Write this equation in the following designations

(
d℘

dz
)2 = 4℘(z)3 − g2℘(z) − g3, (5.9)

where z = 4s, g2 = −4R1, g3 = −4R0 and ℘(2s) = P1(s). The integral formula for ℘(z) is given
by

z =

∫ ∞

℘

(4x3 − g2x − g3)−1/2dx. (5.10)

The functions ℘(z) and ℘′(z) are Weierstrass elliptic functions with periods 2ω1, 2ω2. Define

ω3 = −ω1 − ω2. Then the values
℘(ω1), ℘(ω2), ℘(ω3)

are the roots of cubic equation

4x3 − g2x − g3 = 0. (5.11)

Introduce new variables pk, k = 1, 2, 3 by p
2
k = qk. Then P = p1p2p3. For pk evolution

equations are derived from (5.7):

dp1
ds

= p2p3,
dp2
ds

= p1p3,
dp3
ds

= p2p1. (5.12)

Solutions of these equations are given by quotients of Jacobi elliptic functions. Let sn(u), cn(u), dn(u)

be a set of Jacobi elliptic functions. Define quotients of these functions (in Glaisher notations [1])

by

ns = −
1

sn
, cs = −

cn

sn
, ds = −

dn

sn
,

which obey the following differential equations

d

du
ns(u) = cs(u)ds(u),

d

du
cs(u) = ns(u)ds(u),

d

du
ds(u) = cs(u)ns(u), (5.13)



38 Robert M. Yamaleev CUBO
13, 1 (2011)

with

ns2 − cs2 = 1, ds2 − cs2 = 1 − k2.

From Eqs.(5.12) it follows that the values

e1 = p
2
3 − p22, e2 = p21 − p23,

are constants of motion. Hence, the solutions of Eqs.(5.12) are proportional to the Jacobi elliptic

functions

p3 =
√

e1ns(u, k), p2 =
√

e1cs(u, k), p1 =
√

e1ds(u, k), k
2 = 1 −

e2
e1

. (5.14)

The final result is formulated by the following

Statement:

If the evolution of squared roots of the eigenvalues of the cubic equation obey equations for

Jacobi elliptic functions, then the evolution of the coefficients are governed by equations for Weier-

strass elliptic functions.

For the cubic equation there exist also another possibility to express its eigenvalues, namely,

by using trigonometric functions. The eigenvalues of the invariant polynomial (5.3) are given by

formulae:

y1 = −
2

3

√
3R1 cos θ, y2 =

1

3

√
3R1(cos θ +

√
3 sin θ), y3 =

1

3

√
3R1(cos θ −

√
3 sin θ). (5.15)

It is easy to verify that

y1 + y2 + y3 = 0, y1y2 + y2y3 + y3y1 = −R1.

Equation for the last coefficient, R0, leads to trigonometric equation for cos(θ):

−R0 = y1y2y3

= cos θ(cos2 θ − 3 sin2 θ) = 4 cos3 θ − 3 cos θ =
27

2

R0

(
√

3R1)3
.

By using the trigonometric formula

cos(3θ) = 4 cos3 θ − 3 cos θ

we come to simple trigonometric equation:

cos(3θ) =
3
√

3

2

R0√
R31

. (5.16)

This equation is valid if only if the cubic equation possesses with real solutions which provided by

the following inequality

(
R0
2

)2 < (
R1
3

)3. (5.17)



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Evolutionary method of construction of solutions of polynomials
and related generalized dynamics 39

The algorithm elaborated in the previous section in the case of cubic equation (5.1) is simplified

as follows:

2P2dP1 = 2P2dx, 2P2dP2 = 3P1dx, (5.18)

with the initial data P1(x = P
2) = P1, P2(x = P

2) = P2. At the final stage when x = P
2 = 0

quantities P1(P
2 = 0) =

(1)

P1, P2(P
2 = 0) =

(1)

P2 satisfy the following relationships

R2 = 2
(1)

P2 − 3
(1)

P 21 ,
(1)

P 31 + R1
(1)

P1 + R0 = 0.

Thus the new equation possesses with same invariants as the original one,

X3(0) − 3
(1)

P1X
2(0) + 2

(1)

P2X(0) = 0. (5.19)

Equations (5.1) and (5.19) possess with congruent eigenvalues where, notice, Eq.(5.19) has one

trivial solution. Therefore the problem is reduced to the problem of solution of quadratic equation

X2(0) − 3
(1)

P1X
2(0) + 2

(1)

P2X(0) = 0. (5.20)

From solutions of Eq.(5.20) to the solutions of Eq.(5.1) one comes simply by the set of translations:

q1 = P1 −
(1)

P1, q2 = q2(0) + P1 −
(1)

P1, q3 = q3(0) + P1 −
(1)

P1.

The dynamics related with the cubic polynomial, evidently, is characterized with two first

constants of motion R2, R0 which do not depend of potential field. In quality of evolution equations

we have to use Eqs.(5.6)-(5.7) written with respect to parameter of evolution defined in (4.6):

P2 =
d

ds
P,

d

ds
P2 = 3dP1,

d

ds
P1 = P. (5.21)

Evolution equations for the eigenvalues, correspondingly, are given by

d

ds
p2k = P, k = 1, 2, 3. (5.22)

From these equations it follows that the constants of motion are given by formulae

M1 = p
2
2 − p23, M2 = p23 − p21, M3 = p21 − p22. (5.23)

In order to include into this scheme the potential field V (r) we have to use equation (4.6):

ds

dτ
= (U · n), U = ∇V (r), P n = P. (5.24)

Furthermore, the set of evolution equations has to be completed with the interrelation between

momentum and velocity with respect to time-like parameter:

P =
dr

dτ
. (5.25)



40 Robert M. Yamaleev CUBO
13, 1 (2011)

In these designations evolution equations (5.21) are transformed into the following dynamic equa-

tions
d

dτ
P = −∇V (r) P2,

d

dτ
P2 = −3(P · ∇)V (r)P1,

d

dτ
P1 = −(P · ∇)V (r). (5.26)

In the case of stationary potential, beside the first integrals R2, R0, the equations imply another

integral of motion, the energy

E1 = P1 + V. (5.27)

By substituting the expression for P1 from (5.27) into the second equation of Eqs.(5.26), we get

E2 = P2 + 3E1V −
3

2
V 2 =

1

2
(R2 + 3E21 ). (5.28)

This is second expression for the energy. Next, express from this formula P2 and substitute into

first equation of (5.26). This leads to Newtonian equation

d

dτ
P = −∇W (r, E1), (5.29)

with effective potential

2W (r, E1) = E2V (r) − 3E1V 2(r) + V 3(r). (5.30)

From Eq.(5.29) one may find formula for the total energy

E =
1

2
P 2 + W =

1

2
(R0 + R2E1 + E31 ). (5.31)

6 Generalized dynamics related with evolution of n-degree

polynomial

Now we have accumulated enough experience in order to be able to build a general form of the

dynamics related with n-degree polynomials. Let us start with evolution equations (2.8) written for

n-degree polynomial (2.1). Dynamic equations with respect to some time-like evolution parameter

τ describing a motion inside stationary potential field V (r) are formulated in the following form

[16]:
dP

dτ
= U Pn−1, (6.1a)

dPk
dτ

= (U · P)Pk−1(n − k + 2), k = 2, . . . , n − 1, (6.1b)

dP1
dτ

= (U · P). (6.1c)

The set of evolution equations has to be completed with the interrelation between momentum and

velocity given by Eq.(5.25). This system of dynamic equations remain invariant the coefficients

R0, Rk, k = 1, ..., n − 1. The motion of a physical system obeying to these equations possesses with



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Evolutionary method of construction of solutions of polynomials
and related generalized dynamics 41

the set of inner and outer momenta. Evolution of the set of outer-momenta {P 2, Pk, k = 1, ..., n−1 }
are given by Eqs.(6.1a,b,c), whereas the evolution of the inner-momenta {pk, k = 1, ..., n} obey

dp2k
dτ

= (U · P), k = 1, ..., n. (6.2)

The first integrals of this system are given by

Mik = p
2
i − p2k, i 6= k. (6.3).

From Eq.(6.1c) we find the first constant of integration, the energy E1 = P1 +V , or P1 = E1−V.
By substituting P1 into k = 2-th equation of (6.1b) we find the other constant of integration (the

second expression for the energy):

E2 = P2 + nE1 V −
n

2
V 2.

Next, by substituting P2 from the last expression into the following equation with k = 3 we shall

find the third constant of integration:

E3 = P3 − (n − 1)E2 V +
n(n − 1)

2
E1 V

2 −
n(n − 1)

2 · 3
V 3.

Continue this process up till (n−1)-th stage. Then, at (n−1)-th stage we shall find the expression
for Pn−1. By using this formula in Eq.(6.1a) we come to Newtonian equation

d

dτ
P = −∇W (r, E1), (6.4)

where the effective potential is given by the following series

W (r, E1) =
1

2

n∑

k=1

(k + 1)En−kV k(r)(−1)k+1, with E0 =
1

n + 1
. (6.5)

From Eq.(6.4) it follows formula for the total energy

E =
1

2
P 2 + W =

1

2
(R0 +

n−1∑

k=2

RkEk−11 + E
n
1 ). (6.6)

Concluding remarks

This is a prodigious fact that the evolution equations elaborated for polynomials serve as

dynamic equations for the generalized dynamics. Notice, the Algorithm of calculation of eigenvalues

of the polynomials elaborated in this paper is distinct of the numerical methods of calculations

which principally are based on iteration process with initial sampling, or tentative data of solutions,

so that effectiveness of these algorithms depends of the initial data. The iteration process based on

the present Algorithm uses in the quality of initial data the coefficients of the original polynomial.

The method elaborated here can be considered also as a theory of functional connection between



42 Robert M. Yamaleev CUBO
13, 1 (2011)

coefficients and eigenvalues of the polynomial expressed as one-valued Weierstrass-type and Jacobi-

type hyper-elliptic functions. Evidently, the present method without any principal difficulties can

be continued to the case of polynomials defined over the field of complex numbers.

We have restricted our attention only with the dynamic equations over one-dimensional coor-

dinate space. On the theory of the generalized dynamics in 4D space with physical units one may

consult in Refs.[15] and [16] and the references therein.

Received: February 2009. Revised: September 2009.

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