Construction of exact complex dynamical invariant of a two-dimensional classical system

P

—journal of December 2006

physics pp. 999–1009

Construction of exact complex dynamical invariant of a two-dimensional classical system

FAKIR CHAND and S C MISHRA

Department of Physics, Kurukshetra University, Kurukshetra 136 119, India E-mail: fchand kuk@yahoomail.com; subash kuk@rediffmail.com

MS received 5 May 2006; revised 16 August 2006; accepted 22 August 2006

Abstract. We present the construction of exact complex dynamical invariant of a two- dimensional classical dynamical system on an extended complex space utilizing Lie alge- braic approach. These invariants are expected to play a vital role in understanding the complex trajectories of both classical and quantum systems.

Keywords. Complex Hamiltonian; exact complex invariant.

PACS No. 02.30.Ik

1. Introduction

In recent years, considerable progress has been made to develop the theory of dy- namical invariants for both time-dependent and time-independent classical systems [1–3]. The existence and subsequently the construction, if possible, of these addi- tional invariants for a dynamical system help a lot in understanding the detailed properties of the corresponding system. Therefore, the importance of invariants in the domains of a variety of fields [4,5] like plasma physics, laser physics, accelerator physics etc. clearly offers a motivation to look for system(s) that can generate such structures in an unambiguous manner. To this effect several methods have been developed in the literature (see [2]) for such construction with varying degrees of applicability.

Although real Hamiltonians and their associated invariants have been studied extensively, same efforts have not been made for the study of complex Hamiltonian systems. It may be mentioned that not only the complex Hamiltonians but also the real Hamiltonians are found to admit complex invariants, e.g. harmonic oscillator system possesses a complex invariant, namelyu= ln(p+imωx)−iωt [6].

The complex Hamiltonians have been in use for a long time to study many physical systems such as the optical model of nucleus. Non-Hermitian PT- symmetric complex Hamiltonians are also used to study delocalization of transi- tions in condensed matter systems such as vortex flux line depinning in Type-II

superconductors [7], to study population biology [8], to study the complex tra- jectories particularly in laser physics [9] etc. The complex Hamiltonians are now gaining importance for explaining several phenomena [10] such as the resonance scattering in atomic, molecular and nuclear physics and also in some chemical re- actions. Therefore, the study of the associated complex and/or real dynamical invariants become desirable and if they exist and are available, they can provide insight into the analysis of complex trajectories. In fact the quantum mechanics of such systems has been studied extensively by many authors [11–14] in recent years, and the classical mechanics of such systems needs to be investigated. Very recently, Kaushal and Sweta Singh [15,16] have investigated the construction of complex invariants of one-dimensional complex Hamiltonian systems.

Although there are several schemes [9,15,17] for the complexification of Hamil- tonianH(x, p), in the present work we follow the approach used by Xavier and de Aguiar [9] to develop an algorithm for the computation of semiclassical coherent- state propagator for a particle going through a simple barrier potential, which is given by the expression

x=x1+ip2; p=p1+ix2, (1.1)

for canonical variables xand pin one dimension. In fact the transformation, eq.

(1.1), makes both xand p separately complex by extending each of them to the corresponding complex planes, i.e. inserting an imaginary component in each.

An important aspect of such a characterization is the manifestation of (x1, p1) and (x2, p2) as canonical pairs which in turn provides a link between the com- plex Hamiltonian and a pair of real Hamiltonians and can be helpful in estab- lishing the integrability ofH(x, p) [18]. It is worth mentioning that the transfor- mations (1.1) have been successfully used for solving Schr¨odinger equation for a large number of one-dimensional complex potentials [13,14]. Also the PT symme- try of non-Hermitian H(x, p) appears to be a special case of the general trans- formation (1.1) under certain limits (in the sense that under PT-symmetry this transformation reduces to a restriction on the variables (x1, p1, x2, p2), such that (x1, p1, x2, p2)→(−x1, p1,−x2, p2;i→ −i)).

Further, from the physics point of view, the imaginary partsx2, p2 respectively are some components of momentum and coordinate and their presence in (1.1) give some sort of momentum-coordinate coupling in the dynamical system and hence for dimensional consistency, eq. (1.1) needs modification, i.e.

x=x1+idp2; p=p1+id⁻¹x2,

and in the present study for simplicity we considerd= 1.

Most of the studies on this front are carried out for one-dimensional systems.

Here, in the present work we carried out the extended phase-plane approach in two dimensions with a view to obtain exact complex invariants of classical dynamical systems. Lie algebraic method [19] is explored for such constructions as this has been widely used in literature for the construction of exact and real invariants for a variety of classical dynamical systems and can easily be extended for quantum systems also.

The organization of the paper is as follows: in§2, the method of complexification and construction of invariants of dynamical systems is described. In§3, we apply

the results of§2 to obtain a complex invariant of a dynamical system and finally concluding remarks are given in§4.

2. Construction of complex invariants

Consider a two-dimensional real phase-space (x, y, px, py), which can be transformed into a complex space (x1, x2, x3, x4, p1, p2, p3, p4), by extending eq. (1.1) as

x=x1+ip3; y=x2+ip4,

px=p1+ix3; py=p2+ix4, (2.1)

in two dimensions. Therefore, the HamiltonianH(x, y, px, py) of a two-dimensional system in complex space can be expressed, using eq. (2.1), as

H=H1(x1, x2, x3, x4, p1, p2, p3, p4) +iH2(x1, x2, x3, x4, p1, p2, p3, p4).

(2.2) Clearly, from eq. (2.1) we get

∂

∂x = ∂

∂x1 −i ∂

∂p3; ∂

∂y = ∂

∂x2 −i ∂

∂p4,

∂

∂p_x = ∂

∂p₁−i ∂

∂x₃; ∂

∂p_y = ∂

∂p₂ −i ∂

∂x₄. (2.3)

The Hamilton’s equations of motion for complexH in eq. (2.2) can be written as

˙

x1= ∂H1

∂p1 +∂H2

∂x3; p˙3= ∂H2

∂p1 −∂H1

∂x3,

˙

x2= ∂H1

∂p2 +∂H2

∂x4; p˙4= ∂H2

∂p2 −∂H1

∂x4,

˙ p₁=−

µ∂H1

∂x1

+∂H2

∂p3

¶

; x˙₃=− µ∂H2

∂x1

−∂H1

∂p3

¶ ,

˙ p2=−

µ∂H1

∂x2

+∂H2

∂p4

¶

; x˙4=− µ∂H2

∂x2

−∂H1

∂p4

¶

. (2.4)

If the HamiltonianH in eq. (2.2) is to be analytic function of complex variables, thenH1 andH2 satisfy the Cauchy–Riemann conditions [18] and after employing such analyticity conditions, eq. (2.4) becomes

˙

x1= 2∂H1

∂p₁; p˙1=−2∂H1

∂x₁; x˙2= 2∂H1

∂p₂; p˙2=−2∂H1

∂x₂,

˙

x3= 2∂H1

∂p3; p˙3=−2∂H1

∂x3; x˙4= 2∂H1

∂p4; p˙4=−2∂H1

∂x4. (2.5) Note that (x1, p1), (x2, p2), (x3, p3) and (x4, p4) constitute canonical pairs.

Now consider a complex phase-space functionI(x, y, px, py, t) as

I=I1(x1, x2, x3, x4, p1, p2, p3, p4, t) +iI2(x1, x2, x3, x4, p1, p2, p3, p4, t).

(2.6) Thus for the functionIto be the time-dependent dynamical invariant of the system in complex phase-space, we write the invariance condition as

dI dt = ∂I

∂t + [I, H] = 0, (2.7)

where [·,·] is the Poisson bracket, which in view of the definition eq. (2.1) turns out as

[A, B](x,p)= [A, B](x1,p1)−i[A, B](x1,x3)−i[A, B](p3,p1)

−[A, B]_(p3,x3)+ [A, B]_(x2,p2)−i[A, B]_(x2,x4)

−i[A, B](p4,p2)−[A, B](p4,x4), (2.8) which indicates that the computation of Poisson bracket in the case of complex Hamiltonian systems becomes a bit tedious.

With a view to demonstrate the underlying elegance of the Lie algebraic approach at the classical level, we briefly describe this in order to construct complex invariants of the dynamical systems. In the Lie algebraic approach, one can express the complex HamiltonianH(x, y, px, py, t) of the system as

H=X

n

hn(t)Γn(x, y, px, py, t), (2.9) where the set of functions {Γ1, ...,Γn} is not explicitly time-dependent and hn(t) are complex coefficient functions of time. The Γn’s in eq. (2.9) generate a closed dynamical algebra, which implies

[Γn,Γm] =X

l

C_nm^l Γl, (2.10)

where C_nm^l are the complex structure constants of the algebra. If the Γn’s in eq.

(2.9) are not sufficient to close the algebra then the set of Γn must be extended by adding new Γl’s, such that Γl = [Γn,Γm], until the closure is obtained along with additionalhl(t)’s which are taken to be zero.

Since the complex dynamical invariantI is also a part of Lie algebra, then one can express this as

I(t) =X

k

λk(t)Γk(x, y, px, py), (2.11)

whereλk(t)’s are time-dependent complex coefficients. Thus by using eq. (2.9) and (2.11) forH and I respectively in eq. (2.7), we get a system of linear, first-order differential equations, namely

λ˙r+X

n

"

X

m

C_nm^r hm(t)

#

λn = 0, (2.12)

in λn’s. Therefore, the solutions of these differential equations in turn provide classical complex invariant of a given system from eq. (2.11).

In the next section we will use the prescription given in the present section to obtain complex invariant of a classical complex Hamiltonian system.

3. Illustrative example

Consider the case of a simple harmonic oscillator in two dimensions, for which the Hamiltonian is written as

H= 1

2(p²_x+p²_y) +1

2ω²(t)(x²+y²). (3.1)

Using the complexification eq. (2.1), the above Hamiltonian can be expressed as H = 1

2p²₁−1 2x²₃+1

2p²₂−1

2x²₄+ip2x4+ip1x3+iω²(t)p3x1

+iω²(t)p4x2+1

2ω²(t)x²₁−1

2ω²(t)p²₃+1

2ω²(t)x²₂−1 2ω²(t)p²₄

= X12

m=1

hm(t)Γm(x1, x2, x3, x4, p1, p2, p3, p4), (3.2)

and the various Γ’s andh(t)’s for the above complexH are expressed as Γ1=1

2p²₁; Γ2= 1

2x²₃; Γ3= 1

2p²₂; Γ4= 1 2x²₄, Γ5=p1x3; Γ6=p2x4; Γ7=p3x1; Γ8=p4x2; Γ9=1

2x²₁; Γ10= 1

2p²₃; Γ11= 1

2x²₂; Γ12= 1 2p²₄,

h1=h3= 1; h2=h4=−1; h5=h6=i; h7=h8=iω²(t), h9=h11=ω²(t); h10=h12=−ω²(t). (3.3) The dynamical algebra in this case is not closed unless one adds eight more phase- space functions (Γl)’s. The additional (Γl)’s and their correspondinghl’s are given as

Γ13=p1p3; Γ14=p1x1; Γ15=x1x3; Γ16=p3x3, Γ17=p2p4; Γ18=p2x2; Γ19=p4x4; Γ20=x2x4,

h13=h14=h15=h16=h17=h18=h19=h20= 0. (3.4) Now in the light of modified definition of Poisson bracket for complex systems, eq.

(2.8), we get large number of nonvanishing Poisson brackets, namely

[Γ1,Γ7] =−Γ13+iΓ14; [Γ1,Γ9] =−Γ14; [Γ1,Γ10] =iΓ13; [Γ1,Γ13] = 2iΓ1, [Γ1,Γ14] =−2Γ1; [Γ1,Γ15] =−Γ5; [Γ1,Γ16] =iΓ5; [Γ2,Γ7] = Γ15+iΓ16, [Γ₂,Γ₉] =iΓ₁₅; [Γ₂,Γ₁₀] = Γ₁₆; [Γ₂,Γ₁₃] = Γ₅; [Γ₂,Γ₁₄] =iΓ₅,

[Γ2,Γ15] = 2iΓ2; [Γ2,Γ16] = 2Γ2; [Γ3,Γ8] =−Γ17+iΓ16; [Γ3,Γ11] =−Γ18, [Γ3,Γ12] =iΓ17; [Γ3,Γ17] = 2iΓ3; [Γ3,Γ18] =−2Γ3; [Γ3,Γ19] =iΓ6, [Γ₃,Γ₂₀] =−Γ₆; [Γ₄,Γ₈] =iΓ₁₉+ Γ₂₀; [Γ₄,Γ₁₁] =iΓ₂₀; [Γ₄,Γ₁₂] = Γ₁₉; [Γ4,Γ17] = Γ6; [Γ4,Γ18] =iΓ6; [Γ4,Γ19] = 2Γ4; [Γ4,Γ20] = 2iΓ4,

[Γ5,Γ7] =iΓ13+ Γ14+iΓ15−Γ16; [Γ5,Γ9] =iΓ14−Γ15; [Γ5,Γ10] = Γ13+iΓ16, [Γ₅,Γ₁₃] = 2Γ₁+iΓ₁; [Γ₅,Γ₁₄] = 2iΓ₁−Γ₅; [Γ₅,Γ₁₅] =−2Γ₁₄+iΓ₅,

[Γ5,Γ16] = 2iΓ2+ Γ5; [Γ6,Γ8] =iΓ17+ Γ18−Γ19+iΓ20; [Γ6,Γ11] =iΓ18−Γ20, [Γ6,Γ12] = Γ17+iΓ19; [Γ6,Γ17] = 2Γ3+iΓ6; [Γ6,Γ18] = 2iΓ3−Γ6,

[Γ6,Γ19] = 2iΓ4+ Γ6; [Γ6,Γ20] =−2Γ4+iΓ6; [Γ7,Γ13] =−iΓ7+ 2Γ10, [Γ7,Γ14] = Γ7−2iΓ9; [Γ7,Γ15] =−iΓ7−2Γ9; [Γ7,Γ16] =−Γ7−2iΓ10, [Γ8,Γ17] =−iΓ8+ 2Γ12; [Γ8,Γ18] = Γ8−2iΓ11; [Γ8,Γ19] =−Γ8−2iΓ12, [Γ8,Γ20] =−iΓ8−2Γ11; [Γ9,Γ13] = Γ7; [Γ9,Γ14] = 2Γ9; [Γ9,Γ15] =−2iΓ9, [Γ9,Γ16] =−iΓ7; [Γ10,Γ13] =−2iΓ10; [Γ10,Γ14] =−iΓ7; [Γ10,Γ15] =−Γ7, [Γ10,Γ16] =−2Γ10; [Γ11,Γ17] = Γ8; [Γ11,Γ18] = 2Γ11; [Γ11,Γ19] =−iΓ8, [Γ11,Γ20] =−2iΓ11; [Γ12,Γ17] =−2iΓ12; [Γ12,Γ18] =−iΓ8; [Γ12,Γ19] =−2Γ12, [Γ12,Γ20] =−Γ8; [Γ13,Γ14] =−Γ13−iΓ14; [Γ13,Γ15] =−Γ14−Γ16,

[Γ13,Γ16] =−Γ13+iΓ16; [Γ14,Γ15] =−iΓ14−Γ15; [Γ14,Γ16] =−iΓ13−iΓ15; [Γ15,Γ16] = Γ15−iΓ16; [Γ17,Γ18] =−Γ17−iΓ18; [Γ17,Γ19] =−Γ17+iΓ19; [Γ17,Γ20] =−Γ18−Γ19; [Γ18,Γ19] =−iΓ17+iΓ20; [Γ18,Γ20] =−iΓ18−Γ20;

[Γ19,Γ20] =iΓ19−Γ20. (3.5)

Thus we obtained a set of partial differential equations using eq. (3.5) in eq. (2.12), which are

λ˙1=−4(iλ13−λ14), (3.6)

λ˙2= 4(iλ15+λ16), (3.7)

λ˙3=−4(iλ17−λ18), (3.8)

λ˙4= 4(λ19+iλ20), (3.9)

λ˙₅= 2(λ₁₃+iλ₁₄+λ₁₅−iλ₁₆), (3.10) λ˙6= 2(λ17+iλ18−iλ19+λ20), (3.11)

λ˙7=−2ω²(λ13+iλ14+λ15−iλ16), (3.12) λ˙8=−2ω²(λ17+iλ18−iλ19+λ20), (3.13)

λ˙₉=−4ω²(λ₁₄−iλ₁₅), (3.14)

λ˙10=−4ω²(iλ13+λ16), (3.15)

λ˙11=−4ω²(λ18−iλ20), (3.16)

λ˙12=−4ω²(iλ17+iλ19), (3.17)

λ˙13=−2ω²(iλ1+λ5−λ7+iλ10), (3.18) λ˙14=−2ω²(λ1−iλ5+iλ7−λ9), (3.19) λ˙15= 2ω²(iλ2−λ5+λ7+iλ9), (3.20) λ˙16=−2ω²(λ2+iλ5−iλ7−λ10), (3.21) λ˙17=−2ω²(iλ3+λ6−λ8+iλ12), (3.22) λ˙18=−2ω²(λ3−iλ6+iλ8−λ11), (3.23) λ˙19=−2ω²(λ4+iλ6−iλ8−λ12), (3.24) λ˙20= 2ω²(iλ4−λ6+λ8+iλ11). (3.25) In fact, it is difficult to solve these 20 coupled partial differential equations for complexλ’s. Thus, here we make certain choices forλ’s which facilitate one to find solutions of the above equations.

From eqs (3.10) and (3.12), we get ω²λ˙5+ ˙λ7 = 0, and consider ˙λ5 = 0, which immediately gives

λ5=c5 and λ7=c7, (3.26)

where c5 and c7 are complex integration constants. Again from eqs (3.11) and (3.13), we obtain

λ6=c6 and λ8=c8. (3.27) Hence both eqs (3.10) and (3.12) become

iλ13−λ14+iλ15+λ16= 0,

and the above equation, after using eqs (3.6) and (3.7), reduces to ˙λ1= ˙λ2, which on integration gives

λ1=ρ(t) +c1 and λ2=ρ(t) +c2, (3.28) whereρ(t) is some arbitrary complex function of time andc1 and c2 are arbitrary complex constants.

Similarly, from eqs (3.11) and (3.13), after using eqs (3.8) and (3.9), we can find λ3=ξ(t) +c3 and λ4=ξ(t) +c4. (3.29) Again,ξ(t) is some arbitrary complex function of time andc3andc4are arbitrary complex constants.

Now, in order to find solutions forλ9andλ10, subtract eqs (3.14) and (3.15) and then using eqs (3.10) or (3.12), we arrive at ˙λ9= ˙λ10, which gives

λ9=η(t) +c9 and λ10=η(t) +c10. (3.30) Hereη(t) is another arbitrary complex function of time andc9andc10are complex constants. In the same spirit, subtracting eq. (3.16) from eq. (3.17) and then with the help of eq. (3.11) or (3.12), we get

λ11=φ(t) +c11 and λ12=φ(t) +c12, (3.31) whereφ(t) is one more arbitrary function of time andc11andc12are again complex constants.

Now for finding the solutions ofλ13andλ14, subtractitimes eq. (3.19) from eq.

(3.18) and after using eq. (3.30), we get

iλ˙13+ ˙λ14= 2(2η+c9+c10). (3.32) On the other hand, time derivative of eq. (3.6) is written as

¨λ₁= 4(−iλ₁₃+λ₁₄) = ¨ρ. (3.33)

Hence using eqs (3.32) and (3.33), one immediately get λ13= i

8( ˙ρ−8σ) +c13, (3.34)

and

λ14= 1

8( ˙ρ+ 8σ) +c14, (3.35)

whereσ=R

(2η(t) +c9+c10)dt.

Similarly, from eqs (3.20) and (3.21) with eq. (3.7), from eqs (3.22) and (3.23) with eq. (3.8), and from eqs (3.24) and (3.25) with eq. (3.9), we obtain solutions for (λ15, λ16), (λ17, λ18) and (λ19, λ20) respectively as

λ15=−i

8( ˙ρ−8σ) +c15, (3.36)

λ16= 1

8( ˙ρ+ 8σ) +c16, (3.37)

λ17= i

8( ˙ξ−8θ) +c17, (3.38)

λ18= 1

8( ˙ξ+ 8θ) +c18, (3.39)

λ19= 1

8( ˙ξ+ 8θ) +c19, (3.40)

and

λ20=−i

8( ˙ξ−8θ) +c20, (3.41)

whereθ=R

(2ξ(t) +c11+c12)dt.

Thus, we have solved eqs (3.6) to (3.25) in terms of arbitrary functions ρ, ξ, η andφand complex constants,ci’s (i= 1, ...,20).

If one put back these solutions forλi(i= 1, ...,20) in eqs (3.6)–(3.25), we obtain a number of constraint relations amongci’s, andρ, ξ, η, φ, which limit the choices of these arbitrary complex quantities. These relations are given as

ic13=c14; ic15=−c16; ic17=c18, ic20=−c19; ic15=c14; ic13=−c16, ic20=c18; ic17=−c19; c1−c2= 2ic5,

c3−c4= 2ic6; c9−c10= 2ic7; c11−c12= 2ic8, (3.42) and the equations determining arbitrary functionsρ, ξ, ηandφare written as

¨

ρ+ 16ω²(ρ+c1−ic5) = 0, ξ¨+ 16ω²(ξ+c3−ic6) = 0,

¨

η+ 8ω²(2η+c9+c10) = 0,

φ¨+ 8ω²(2φ+c11+c12) = 0. (3.43)

Therefore, after substituting the solutions ofλi’s in eq. (2.11), the final form of the invariant for a two-dimensional complex oscillator becomes

I= 1

2ρ(p²₁+x²₃) +1

2(c1p²₁+c2x²₃) +1

2ξ(p²₂+x²₄) +1

2(c3p²₂+c4x²₄) +1

2η(x²₁+p²₃) +1

2(c9x²₁+c10p²₃) +1

2φ(x²₂+p²₄) +1

2(c11x²₂+c12p²₄) + i

8( ˙ρ−8σ)(p1p3−x1x3) +1

8( ˙ρ+ 8σ)

×(x1p1+x3p3) + i

8( ˙ξ−8θ)(p2p4−x2x4) +1

8( ˙ξ+ 8θ)

×(p2x2+p4x4) +c5p1x3+c6p2x4+c7p3x1+c8p4x2+c13p1p3

+c14p1x1+c15x1x3+c16p3x3+c17p2p4

+c18p2x2+c19p4x4+c20x2x4. (3.44) which conforms to condition eq. (2.7) in view of the Poisson bracket eq. (2.8).

4. Conclusion

In this work, a modest attempt has been made to obtain exact complex second constant of motion of a two-dimensional complex simple harmonic oscillator on an extended complex phase-space characterized by eq. (2.1). The transformations (1.1) (or eq. (2.1)) had been a part of many studies [9,17] and can give PT sym- metric Hamiltonians under certain boundary conditions. Just as the invariants of real Hamiltonian systems have played a vital role in understanding the underly- ing dynamics of the systems, we hope that the complex invariants could also be helpful in exploring some deep insight into features of complex dynamical systems including the real one. Lie algebraic method is used for the construction of complex invariants, which has been used extensively for the construction of time-dependent invariants of both classical and quantum systems. The degrees of freedom become just double after complexification of a system which make the construction of com- plex invariants a bit tedious in two or higher dimensions and needs some alternative and systematic approach which could address such complexities.

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