Hilbert space description of classical dynamical systems I

Physica 145A (1987) 408-424

North-Holland, Amsterdam

HILBERT SPACE DESCRIPTION OF CLASSICAL DYNAMICAL SYSTEMS I

Krzysztof KOWALSKI

Department of Biophysics, Institute of Physiology and Biochemistry, Medical School of tddi. 3 Lindley St., 90-131 Addi, Poland

Received 20 August 1986

Revised manuscript received 9 March 1987

It is shown that classical dynamical systems can be described by a Hilbert space, Schrodinger-like

equation with the Hamiltonian (a non-Hermitian one) expressed in terms of Bose creation and

annihilation operators. The presented formalism includes the Carleman embedding as the special

case.

1. Introduction

An amazingly large number of natural laws and models of natural

phenomena are described by dynamical systems of the form X: = F(x, t). Let us

recall only a few examples of classical and of current interest, such as: the

many body problem of classical mechanics’), the Lorenz model showing

“chaotic behaviour”2), the nonconservative oscillators like that of Van der

Pol”) exhibiting limit cycles, the nonlinear rate equations of chemical kinetics

describing the behaviour of systems far from thermodynamical equilibrium”)

and models of ecological dynamics like that of Lotka-Volterra’).

The question naturally arises as to whether the systems X = F(x, t) having a

great diversity of nonlinearity of the F, admit any linear description. In 1932

Carleman6) showed that an autonomous finite dimensional system of differen-

tial equations X: = P(x), where P, are polynomials in X, can be embedded into

an infinite linear system of differential equations. In 1976 the Carleman’s

observations were introduced anew by Montroll and Helleman7) for working in

the field of nonlinear models. Recently, several nonlinear problems have been

succesfully investigated within this approach *-IS). For example, in ref. 13 the

Carleman embedding was used for calculating Lyapunov exponent characteriz-

ing chaos in nonlinear systems.

HILBERT SPACE DESCRIPTION OF DYNAMICAL SYSTEMS I 409

The appearance of infinite dimension in the Carleman technique suggests

that it can describe linearization of finite dimensional systems in a Hilbert space. In fact, recently Steeb16) found the expression of the embedding matrix in terms of Bose creation and annihilation operators. Nevertheless, he only treated this bosonic picture as a formal property of the Carleman embedding matrix.

The object of the present paper is to give the Hilbert space description of nonlinear finite systems It = F(x, t), where F is analytic in x (these systems shall be called almost holomorphic systems - AHS for abbreviation - throughout this work). The following paperl’) is devoted to the Hilbert space description of classical dynamical systems with infinite degree of freedom.

In section 2 we demonstrate that solutions of AHS are the time-dependent eigenvalues of Bose annihilation operators, where the time evolution is described by a non-Hermitian boson operator determined by the vector field F.

The linear abstract Schriidinger-like equation which obeys the time-dependent eigenvectors of Bose annihilation operators (coherent states) is the Hilbert space counterpart of AHS. We then show that the introduced canonical formalism generalizes the Carleman embedding technique as well as alternative methods of linearization of classical dynamical systems in infinite dimensional spaces. Finally, we discuss the problem of a priori possible probabilistic interpretation of the Schrijdinger-like evolution equation. It is demonstrated that whenever the solution of AHS remains in a ball with radius determined by the initial conditions then such probabilistic interpretation exists.

In section 3 we study the “Heisenberg equations of motion” which obey the time-dependent Bose annihilation operators. By integration of these equations the new “quantum-mechanical” form of the Lie series is obtained.

Section 4 is devoted to applications. As a nontrivial application of the presented formalism we introduce a method for finding the first integrals for dynamical systems.

2. Evolution equation

In this section the evolution equation in Hilbert space is introduced corresponding to the classical dynamical system with analytic nonlinearity (AHS). Consider the following classical dynamical system - AHS (complex or real):

i = F(z, t), z(0) = zo, (2.1)

where F: Ck + Ck, F is analytic in z and overdot denotes differentiation with respect to the time.

410 K. KOWALSKI

Let lz) be a normalized coherent state (see appendix B), where z satisfies

(2.1). We define the vectors 12, t) as follows:

I*, t) = etM2-lz,,l*) 14 (2.2)

Suppose now we are given a boson operator of the form

M(t) = u+ - F(a, t) ) (2.3)

where a+, a are the standard Bose creation and annihilation operators, respec-

tively.

By differentiating both sides of (2.2) one can easily find that the vectors

Iz, t) satisfy the following evolution equation in Hilbert space:

12, t> = W)lz, t> > IGO> = Izd (2.4)

Let Iz,, t) designate the solution of eq. (2.4) and let z(z,,, t) be the solution of

system (2.1). Taking into account (2.2) we find that the vectors Iz,,, t) are the

eigenvectors of the Bose annihilation operators (coherent states) corresponding

to the eigenvalues z(zo, t) i.e. the following relation holds:

al&), t) = z(zo, t)lzw l) . (2.5)

It thus appears that the integration of the nonlinear classical dynamical system

(2.1) is equivalent to the solution of the linear, abstract evolution equation

(2.4). Notice, that the eigenvalue equation (2.5) suggests the “quantization”

scheme of the form z-+ a, where z satisfies (2.1).

We now demonstrate that the presented canonical formalism generalizes

existing methods of linearization of classical dynamical systems in infinite

dimensional spaces. Besides the Carleman embedding technique the only

alternative approaches known to the author are two similarly structured ones

decribed in refs. 18 and 19. The first of themis) specializes on the discussion of

the Hamiltonian systems. Using the symplectic structure of the phase space

such systems are brought down to the linear problem in the space L’. The aim

of this treatment is to reduce the Hamiltonian systems to the Schrodinger

equation, unfortunately, such reduction meets serious difficulties. The second

approach”) takes into consideration the general dynamical systems of the form

(2.1), nevertheless, the linear operator equations derived within it are not built

into any Hilbert space structure. We now show that the presented formalism

includes both discussed methods as a special case. Consider the evolution

equation (2.4). This equation is equivalent to the operator evolution equation

HILBERIT; SPACE DESCRIPTION OF DYNAMICAL SYSTEMS I 411

of the form

v= M(t)V ) V(0) = z ) (2.6)

where V(t) is the evolution operator such that

(2.7)

Taking the conjugation of (2.6) the following equation is obtained:

V’=v+M+(t), v+(o)=z. (2.8)

Putting V+ = U and writing (2.8) in the Bargmann representation (see appendix B) we get

ir= ux, U(O)=Z, (2.9)

where X = F(z*, t) - d/dz* is the vector field corresponding to the system (2.1) and we assume that F * = F.

The operator equations of the form (2.9) are fundamental in both algorithms discussed above. We have thus shown that these methods correspond to the particular Bargmann representation in the presented canonical Hilbert space

formalism. We now demonstrate that the Carleman embedding technique is also in-

cluded by the introduced Hilbert space approach. Consider the evolution equation (2.4). This equation written in the occupation number representation (see appendix A) is the differential-difference equation

where z, = (~tlz, t) and M,,.(t) = (nl~(t)ln’) .

Taking into account (2.2) and (B.5) we find that z, is given by

z,(t) = (4z, t> = ( k (z.(t))“’ lcl -) exp(- 1 lz,12) ,

(2.10)

(2.11)

where z(t) fulfils (2.1). NOW, since one can introduce an order in the set Zk+, it follows from (2.11)

412 K. KOWALSKI

that we in fact embedded the nonlinear system (2.1) into the infinite linear system (2.10). The Carleman’s linearization ansatz in the form suggested by Steebr6) coincides up to the constant with (2.11). It thus appears that the presented canonical formalism is a far-reaching generalization of the Carleman embedding. On the one hand we linearize the general nonautonomous systems (2.1) including the autonomous systems with polynomial nonlinearity consi- dered by Carleman as the special case. On the other hand the Carleman embedding technique corresponds to the particular occupation number representation in the presented Hilbert space approach.

Example. Consider the following AHS:

i = (u(t) - z)z ) z(0) = Z” (2.12)

The corresponding evolution equation can be written as

(2, t> = Wt)lz, t> > Iz, 0) = Id > (2.13)

where the “Hamiltonian” is given by

M(t) = Nu(t) - a , (2.14)

where N= a+ - a is the total number operator. The “Hamiltonian” (2.14) satisfies the following relation:

[M(t), M(t’)] = 0 , t, t’ E R ; (2.15)

therefore, the solution Iz,,, t) of (2.12) is of the form

Hence, with the use of (2.11) the solution z(z,), t) of the system (2.12) can be obtained:


zi(Zf)> t> = (eilZo> t> expGlz,l*)

f

> I

= zgi z. - U(T) dr

‘Oi = ) i=l,..., k,

(2.17a)

(2.17b)

where ei, i = 1,. . . , k are the unit vectors e, = (0, . . . ,O, li, 0, . . . , 0). The relation (2.17b) holds for 1 JL z. - U(T) dTl< 1 only, nevertheless (2.17b)

is the solution of (2.12) for arbitrary zo, t such that the denominator of (2.17b) does not vanish (it can be analytically continued).

We now discuss the problem of a priori possible probabilistic interpretation of the evolution equation (2.4). Taking into account (2.2) we find that the squared norm of the solution Izo, t) of eq. (2.4) is given by

(z,, Go, t> = exp(lz(z,, t)l* - M*) . (2.18)

Thus, it turns out that the necessary condition for the evolution equation to possess a probabilistic interpretation can be written in the form

Iz(zo, t)12 s lzol* for arbitrary t . (2.19)

This means that each trajectory remains in a k-dimensional ball with radius lzol (if (2.19) holds for arbitrary initial conditions z. then the system (2.1) is Lagrange stable (each trajectory remains in a finite region)).

An example of AHS satisfying (2.19) are nonlinear rate equations of chemical kinetics describing reactions without autocatalytic steps. The inequality (2.19) is then a consequence of the conservation of the total particle number of reactants in the closed system. It should be noted that the squared norm (2.18) does not depend on time for arbitrary initial data z,, in the case of skew-Hermitian “Hamiltonian” M(t) only, when the evolution equation (2.4) is the Schrodinger one (the evolution operator V(t) generated by such “Hamil- tonian” is then unitary). The only system with skew-Hermitian “Hamiltonian” is a linear homogeneous one with the skew-Hermitian matrix. However, it must be borne in mind, that if one specializes to the real domain then the nonlinear systems can be easily introduced such that the squared norm (2.18) is conserved during evolution for arbitrary real initial data x0.

Example. An example satisfying relation (2.19) for arbitrary initial data is the

414 K. KOWALSKI

following three-dimensional autonomous system which is the special case of the

Lorenz system:

x = F(x) ) x(0) = x,, , (2.20)

where the vector field F: R’ -+ R” is given by

with U, b > 0.

In fact, we have

Hence,

(2.21)

(2.22)

where x(x0, t) designates the solution of the system (2.20).

The inequality (2.19) follows immediately from (2.22).

3. Operator evolution equations

We now demonstrate that solutions of classical dynamical systems are the

covariant symbols’“) of the time-dependent anihilation operators. Consider the

“Heisenberg equations of motion” satisfied by the time-dependent Bose

annihilation operators

b(t) = [a(r), M,,(r)1 > 40) = a > (3.1)

where Mu(t) = V(t)-‘M(t)V(t) is the “Hamiltonian” in “Heisenberg picture”;

V(t) is the evolution operator given by (2.6).

The solution of (3.1) is

u(r) = V(t)_‘aV(t) . (3.2)

Hence, taking into account the following formula2’):

HILBERT SPACE DESCRIPTION OF DYNAMICAL SYSTEMS I 41.5

V(t) = exp[ @(t>l , (3.3a)

where the “phase operator” Q(t) is given by

G’(t) = j dr M(T) + ; j dr, j dr, ~(7~ - ~r)[M(r~), M(r,)] + . . . , (3.3b) 0 . 0 0

where E(T) is the sign function, and using (2.5) we obtain the formal solution of the system (2.1)

z(zo7 4 = (zol4)lzo)

= z 0

+ i (ylY i=l

yj- (ZOlPW~ ’ . . 3 l@(t), aI. . .llzo> . (3.4)

Notice that whenever the “quantization” scheme z+ a is assumed, where z satisfies (2.1), then the first equation of (3.4) forms the “Ehrenfest theorem” within the presented formalism. On the other hand it appears that solutions of AHS (2.1) are the covariant symbols of the time-dependent Bose annihilation operators.

Example. Suppose that the “Hamiltonian” . 1s independent of time. Certainly this is the case with the autonomous AHS (holomorphic system)

i = F(z) ) z(0) = z. ) (3.5)

where F is analytic in z. The phase operator is then given by G(t) = M, where M = u+ * F(a) and the

solution of (3.5) can be written as the formal power series in t

;” (-t)’ z(z0, t> = zo + c -y- (z,l[K . . . , [M, al. . .llz,) *

i=l (3.6)

It can be easily shown that this solution coincides with the formal solution of the holomorphic system (3.5) given by Lie series. However, the solution (3.6) is purely algebraic. The differential operations are hidden in the commutators like [a+, f(u)] = -df/da.

It should be noted that the solution of the nonautonomous holomorphic system (2.1) with F analytic in z and t can also be written in the form (3.6). Indeed, it is well-known that k-dimensional system (2.1) can be treated (k + 1)-dimensional,

416 K. KOWALSKI

i’ = F’(d) ) z’(0) = z;, ) (3.7)

where z’ = (z, t) and ZJ’(z’) = (F(z), 1).

This observation is consistent with the celebrated Cauchy theorem on the

existence and the uniqueness of the solution of the holomorphic system (2.1).

Observe, that the formal similarity between the presented approach and

quantum mechanics allowed us to derive the new, algebraic form of the Lie

series. In fact, the introduced formalism makes it possible to translate the

notions and the methods familiar from quantum mechanics into the language of

the theory of classical dynamical systems. For example, the interaction picture

within the presented treatment corresponds to the classical method of variation

of constants. We now illustrate this observation on the following example.

Example. Consider the following AHS:

x = Lx + (c(c) - x)x ) x(0) = X” ) (3.8)

where L: Rk -+ Rk is a linear operator.

The evolution equation corresponding to (3.8) is of the following form:

where M,, = a+ * La and M,(t) = NC(~) - a; N = a+ - a is the total number

operator.

Passing to the interaction picture

127) = exp(-%)Iz, t) , (3.10)

we get

(3.11)

where k,(t) = exp(-tM,,)M,(t) exp(tM,) = Nv”(c(t), t) - a and i?(x,, t) desig-

nates the solution of the linear part of the system (3.8) with the operator L replaced by its transpose.

The solution of AHS corresponding to (3.11) has been already obtained (see

formula (2.17b)). Hence, with the use of (2.5) and (3.10) we get the solution

x(x0, t) of the system (3.8),


(3.12)

where u(x,,, t) is the solution of the linear part of the system (3.8). One can easily check that the above procedure of integration of the system

(3.8) is equivalent to the method of variation of constants.

4. An application

As a nontrivial application of the presented formalism we now introduce a method for determining the first integrals for classical dynamical systems. Consider the real autonomous system

i = F(x) . (4.1)

We seek the first integrals for the system (4.1) with exponential time de-

pendence22)

Z(x, t) = ?F(x) eeAr . (4.2)

Suppose that $ is analytic in X, then taking into account (B.6) we can write (4.2) in the form

Z(x, t) = etx2( IPlx) epAt , (4.3)

where Ix) is a normalized coherent state. It follows immediately from (4.3) and (2.2) that the vector I?P) should

satisfy the relation

M+(F) = hpty ) (4.4)

where M = a+ -F(a) is the “Hamiltonian” for the system (4.1). Thus, it turns out that the problem of finding the first integrals of the form

(4.2) with analytic x-dependent part is equivalent to solving Hilbert space eigenvalue equation (4.4).

Let us now specialize to the case with @ polynomial in x. Certainly, the vector IYP) is then of the form (see eq. (B.5))

P) = z_ a,ln) ) (4.5)

418 K. KOWALSKI

where the vectors In), n E Z: span the occupation number representation and

S is a bounded subset of Zk+.

Motivated by (4.5) we put IT’> to be common eigenvector of the operator M + and the linear combination c - N of the number operators N,, i = 1, . . , k

[M+, c*N](!P) =O. (4.6)

It appears that there exist nontrivial cases when the relations (4.6) and (4.4) allow to determine in a simple way the first integrals. We now illustrate this observation on the example of the Lorenz system.

Example. Consider the Lorenz system

i, = u(x* - x,) )

x2 = -x7 _ - x,x3 + TX1 ,

i3 = x,x2 - bx,

(4.7)

The conjugation M+ of the “Hamiltonian” corresponding to (4.7) is

M’ = -aN, - N2 - bN, + cala, + raTa - a:a:a, + a:a,‘a, .

We have

(43)

[M+, C-N] = (T(c, - c,)a:a, + r(c* - c,>a:a, + (c, + c3 - c?)a:a:az

+(c3 - c, - c2)a:a:a3. (4.9)

Putting c, = c2, cj = 2c, we get from (4.9) the following commutation relation:

[M’, c-N] =2c,a:ala,. (4.10)

Taking into account (4.10) and (4.6) we make the ansatz

Inserting (4.11) into (4.4) we obtain easily

A=-2a, b=2a,

IW = %$w + a,,PW >

(4.12a)

(4.12b)

where ~?a~,, = -(Y”, .


Hence, putting a*,, - - fi and using (4.3) we finally get the following first

integral:

Z,(x, t) = (XT - ~ux,) ezu’ . (4.13)

Analogously, for cg = c2, ci = 0, r = 0 we have

Pf+, c -iv] = - (+c*u;u, . (4.14)

The relation (4.14) implies the ansatz

IW = c %2n310~2~,) . (4.15) n2”3

Substituting (4.15) into (4.4) we find

A=-2, b=l, (4.16a)

(4.16b)

where a*0 = ao2. Hence, with the use of (4.3) the following first integral is

obtained:

Z2(x, t) = (xt + x:) e*’ . (4.17)

Let us finally consider the case with ci = c2 = c3. The relation (4.9) reduces then to

pf+, c-N] = c1u~(u~Lz2 - u;u,).

The condition (4.6) leads to the ansatz of the form

(4.18)

IF) = c 4In20) + ln02)) + c P,I~OO) * n m

Inserting (4.19) into (4.4) yields

(4.19)

A=-2, (+=l, b=l,

IW = 4w + low) + P,ww 7

(4.2Oa)

(4.20b)

where q,~ = -p2. Hence, taking into account (4.3) we get the following first integral:

420 K. KOWALSKI

13(x, t) = (-my + xi + 2~:) e” (4.21)

The integrals I,, 1, have been originally found by means of the singular-point

analysis23). The integral 1, has been recently reported by KuS14) (he used the

approach described in ref. 22 together with the Carleman linearization).

5. Conclusions

In the present work it is shown that very general nonlinear classical

dynamical systems (embracing everything that arises in practice) may be

brought down to the abstract, linear Schrodinger-like equation in Hilbert

space. The introduced approach amounts to a far-reaching generalization of

existing methods for linearization of classical dynamical systems in infinite

dimensional spaces. For example, the Carleman embedding corresponds to the

particular occupation number representation in the presented canonical Hilbert

space formalism as well as it is generalized to the case of nonautonomous

systems with analytic nonlinearities. On the other hand the introduced ap-

proach enables us to treat from the unique point of view the classical methods

which are frequently employed in the study of nonlinear ordinary differential

systems. For example, the technique of Lie series corresponds to the particular

“Heisenberg picture” and the method of variation of constants refers to the

particular “interaction picture” within the presented “quantum-mechanical”

Hilbert space formalism. Moreover, such correspondences indicate that the

treatment allows us to translate the methods and the notions of quantum

mechanics into the language of the theory of classical dynamical systems. As a

nontrivial application of the presented formalism a new technique of determin-

ing first integrals for autonomous dynamical system is introduced. Applying the

algorithm to the Lorenz system we found in a remarkably straightforward

manner the first integrals.

The problem of the physical interpretation of the introduced canonical

formalism which is similar in structure to quantum mechanics remains open.

The properties of the coherent states which are the solution of the Schrodinger-

like evolution equation (2.4) suggest that whenever such interpretation exists

then it should take into consideration the quasi-classical approximations for the

systems. Indeed, it is well-known that coherent states minimalize the Heisen-

berg uncertainty relations and in this sense are nearest to the classical ones.

The interesting feature of the presented approach is that there exist classical

dynamical systems such that the corresponding Schrodinger-like equation

admits an a priori possible probabilistic interpretation. As we have mentioned

above, an example is a large class of nonlinear rate equations describing


chemical reactions without autocatalytic steps. The attractive idea arises to connect the appearing probability with the chaotic (stochastic) behaviour of classical dynamical systems. This idea is supported by the exceptional role of dissipativity of systems showing the chaotic behaviour (the systems satisfying (2.19) are certainly the dissipative ones). Nevertheless, it appears that chemical rate equations describing reactions without autocatalytic steps, which admit the probablistic interpretation of the corresponding Schrodinger-like equation are globally asymptotically stable24) and show regular behaviour. We have demonstrated that the evolution operators within the presented canonical formalism are not unitary ones and the squared norm of the state vectors is not conserved (there is one exception when the evolution equation (2.4) is the Schrodinger equation). Thus, whenever the introduced formalism describes some “quantization” scheme, it should be completely different from this one established in orthodox quantum mechanics. Notice however, that the quantum theories with nonunitary evolution operators are not exceptional in physics. Let us only recall the Prigogine theory of irreversible processess”). Having in mind the introduced approach it is also worthwhile to recall the important remark of Einstein26) referring to the old quantum theory (the Born-Sommerfeld theory) that the quantization holds only for quasi-periodic motion described in classical mechanics by the integrable systems. In spite of the progress made in quantum mechanics this theorem still remains true27).

Acknowledgements

It is a pleasure to thank Dr. J. Rembielinski and Dr. P. Kosinski for their competent comments. I am also grateful to a referee for helpful remarks.

Appendix A

Occupation number representation

We recall the basic properties of the occupation number representation. Consider the Bose creation and annihilation operators a+, a. These operators obey the standard commutation relations

[a,,aJ]=6ij, [a,,aj]=[a+,ay]=O, i, j=l,. . . ,k. (A.11

Let us assume now that there exists in the Hilbert space of states Z’, a unique normed vector 10 ) (vacuum vector) satisfying

422 K. KOWALSKI

alo) = 0. VW

We also assume that there is no nontrivial closed subspace of 27 which is invariant under the action of operators a, a+. The state vectors In), n E Z:,

where Z, designates the set of non-negative integers defined as follows

G4.3)

are the common eigenvectors of the number operators IV, = u: a,, i = 1, . . . , k

i.e.

N(n) = nln) . G4.4)

These vectors satisfy the following relations:

orthogonality,

(A.9 2 In) (nl = I , completeness.

?lEZk,

The action of the Bose operators on the vectors In) has the following form:

ailn> =filn-e,), a’In)=~/~ln+e,), i=l,..., k. G4.6)

Appendix B

Coherent states representation

We recall now the basic properties of coherent states. Consider the coherent states lz), where z E Ck, . I.e. the eigenvectors of the annihilation operators a,

alz) = zlz) . (B.1)

The normalized coherent states can be defined as

lz) = e-flZI* eZ..+Iq , 03.2)

where US u = CFZI uiui and lz(’ = Ef,, jzi12, these vectors fulfil the following relations:


(B.3)

where the asterisk denotes the complex conjugation,

J dp(z) ]z) (z] = I, completeness, (B.4) tP

where

k 1 d&z) = n - d(Re zi) d(Im zi) .

;=I 7r

The passage from the occupation number representation to the coherent states representation is given by the kernel

bid = (fi $) ev(-M2). (B.5)

Suppose now that we are given an arbitrary state IF). It is easily shown that that function ?P(z*) = ( Z]F) has the following form:

F(z*) = +(z*) exp(- 1 lzl*) , 03.6)

where $(z*) is an analytic function (entire function). Hence, we get

(91~) = 1 dp(z) exp(-]z]2)G*(z*)6(z*).

R2k

03.7)

The representation (B.7) is called the Bargmann representation2*). The Bose operators a, a+ act in this representation as follows:

r&z*> = + @(z*) ) a++(z*) = z*@z*> , 033)

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424 K. KOWALSKI

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Hilbert space description of classical dynamical systems I