canonical trasformation hamilton ian system

8/3/2019 Canonical Trasformation Hamilton Ian System

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Ann. Phys. (Leipzig) 11 (2002) 1, 15 38

Canonical transformations and exact invariantsfor time-dependent Hamiltonian systems

Jurgen Struckmeiera and Claus Riedel

Gesellschaft fur Schwerionenforschung (GSI), Planckstrasse 1, 64291 Darmstadt, Germany

Received 15 May 2001, revised 22 June 2001, accepted 22 June 2001 by G. Ropke

Abstract. An exact invariant is derived for n-degree-of-freedom non-relativistic Hamiltonian sys-tems with general time-dependent potentials. To work out the invariant, an infinitesimal canonical

transformation is performed in the framework of the extended phase-space. We apply this approachto derive the invariant for a specific class of Hamiltonian systems. For the considered class of Hamil-tonian systems, the invariant is obtained equivalently performing in the extended phase-space a finitecanonical transformation of the initially time-dependent Hamiltonian to a time-independent one. It isfurthermore shown that the invariant can be expressed as an integral of an energy balance equation.

The invariant itself contains a time-dependent auxiliary function (t) that represents a solution ofa linear third-order differential equation, referred to as the auxiliary equation. The coefficients of theauxiliary equation depend in general on the explicitly known configuration space trajectory defined bythe systems time evolution. This complexity of the auxiliary equation reflects the generally involvedphase-space symmetry associated with the conserved quantity of a time-dependent non-linear Hamil-tonian system. Our results are applied to three examples of time-dependent damped and undampedoscillators. The known invariants for time-dependent and time-independent harmonic oscillators areshown to follow directly from our generalized formulation.

Keywords: time-dependent Hamiltonian system, canonical transformation, invariant

PACS: 45.05.+x, 41.85.-p, 45.50.Jf

1 Introduction

The derivation of invariants also referred to as conserved quantities or constants of motion

is a key objective studying analytically a given Hamiltonian system. If we succeed to

isolate an invariant, we always learn about a fundamental system property. For instance, in

the case of an autonomous Hamiltonian system the Hamilton function itself is easily shownto represent an invariant which can be identified with the energy conservation law.

This direct way to an invariant no longer exists in the general case of explicitly time-

dependent systems, where the Hamiltonian does not provide directly a conserved quantity.

Various methods have been applied to attribute an invariant to time-dependent (non-autono-

mous) systems: time-dependent canonical transformations [1, 2, 3, 4, 5], general dynamical

symmetry studies in conjunction with Noethers theorem [6, 7, 8, 9, 10], and direct ad hoc ap-

proaches [11, 12, 13, 14, 15]. For the time-dependent quadratic Hamiltonian of the harmonic

oscillator type, Leach [2] showed that a two-step time-dependent linear canonical transforma-

tion can be applied to find an invariant. The invariant itself was found to consist of both the

acorresponding author: [email protected]


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16 Ann. Phys. (Leipzig) 11 (2002) 1

canonical coordinates, given by the systems phase-space path, and an auxiliary function sat-

isfying a non-linear second-order auxiliary differential equation. With regard to its physical

interpretation, the invariant could be identified as the Hamiltonian of a related autonomous

harmonic oscillator system.In this article, we present a generalized method to determine invariants of n-degree-of-

freedom non-relativistic Hamiltonian systems with non-linear and explicitly time-dependent

potentials. The systems may include damping with forces linear in the velocities.

We first review in Sections 2 and 3 the formalism of canonical transformations in the

framework of the extended phase-space where the time and the negative Hamiltonian are

taken as an additional pair of canonically conjugate coordinates. In Section 4, an infinites-

imal canonical transformation is carried out for a particular class of Hamiltonian systems

to directly isolate a conserved quantity. In a second approach, presented in Section 5, a fi-

nite canonical transformation in the frame of the extended phase-space is shown to map the

time-dependent Hamiltonian of Section 4 onto a time-independent one. Expressing this newHamiltonian in terms of the old coordinates, one immediately obtains an invariant in the

original system.

In either case, the same invariant is obtained. It consists of both the canonical coordinates

and an auxiliary function, which follows from a homogeneous, linear third-order auxiliary

equation. Apart from isotropic linear systems, the coefficients of the auxiliary equation de-

pend on all spatial particle coordinates. As the consequence, this differential equation can

only be integrated in conjunction with the systems equations of motion. This enhanced com-

plexity of the general auxiliary equation reflects little surprisingly the involved nature

of a conserved quantity for time-dependent non-linear Hamiltonian systems. From the energy

balance equation for time-dependent Hamiltonian systems, it is shown that the invariant can

be interpreted as the sum of the systems time-varying energy content and the energy fed into

or detracted from it.

As illustrative examples, we derive the invariant and the auxiliary differential equation for

three specific systems in Section 7: the time-dependent damped harmonic oscillator, the time-

dependent anharmonic undamped oscillator, and the n-dimensional anisotropic oscillator.For the harmonic oscillator, invariant and auxiliary equation of Sections 4 and 5 are shown

to specialize to the known results [2]. For the non-linear oscillator, interesting insight is

revealed from the detailed behavior of the auxiliary function. As expected, the phase-space

symmetries of this non-linear system differ significantly from those of a linear system.

We will show in Appendix A that our invariant can also be derived from Noethers theo-

rem in the context of the Lagrangian formalism. Our invariant thus corresponds to the Noethersymmetry group that leaves the action integral invariant. Very similarly, the invariant may as

well be worked out on the basis of a direct approach, using an Ansatz function with a

quadratic dependence on the canonical momenta. This will be sketched in Appendix B.

2 Canonical transformations in the extended phase-space

We consider an n-degree-of-freedom system of classical particles that may be described ina 2n-dimensional Cartesian phase-space by an in general explicitly time-dependent

Hamilton function H = H(q, p,t). Herein, q denotes the n-dimensional vector of the config-


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J. Struckmeier and C. Riedel, Exact invariants for Hamiltonian systems 17

uration space variables, and p the vector of conjugate momenta. The systems time evolution,also referred to as the phase-space path (q(t), p(t)), can be visualized as a unique curve in the2n-dimensional phase-space. If the systems state is known at two distinct instants of time

t1 and t2, the systems actual phase-space path between these fixed times is known to obeyHamiltons variation principle

t2t1

n

i=1

pi(t)dqi(t)

dtH

q(t), p(t), t

dt = 0 . (1)

This variation integral (1) can easily be shown to vanish exactly if the canonical equations

dqidt

=H

pi,

dpidt

= H

qi, i = 1, . . . , n (2)

are satisfied. We observe that the time t plays the distinguished role of an external parameterwhich is contained in both the system path and the Hamilton function H itself. As will beworked out in the following, this distinguished role of the time t may not be desirable inthe general case of explicitly time-dependent (non-autonomous) Hamiltonian systems. We

therefore introduce an evolution parameter s that parameterizes the systems time evolutiont = t(s). With s the new integration variable, we may rewrite Hamiltons principle (1) as

s2s1

n

i=1

pi(s)dqi(s)

dsH

q(s), p(s), t(s)

dt(s)ds

ds = 0 . (3)

With this symmetric form of the integrand, it looks reasonable to conceive the negative Hamil-

tonian in conjunction with the time as an additional pair of canonically conjugate coordinates.We thus introduce the (2n + 2)-dimensional extended phase-space by defining

qn+1 = t , pn+1 = H

as additional phase-space dimensions. In this notation, H = H(s) is understood as an s-dependent variable that may be identified with the actual energy content of the Hamiltonian

system H. We furthermore introduce the quantity H as an implicit function

H =

H(q1, . . . , qn+1;p1, . . . , pn+1) = 0 (4)

of the extended phase-space variables. Equivalently to the initial Hamiltonian H, the functionH must contain all information about the dynamics of the given system. With H as definedby Eq. (4), the variation integral (3) its summation index now ranging to (n + 1) maybe written in the equivalent form of

s2s1

n+1i=1

pi(s)dqi(s)

ds H ds = 0 . (5)

Similar to the case of Eq. (1), the integral (5) vanishes exactly if the canonical equations

dqi

ds =

Hpi , dpids = Hqi , i = 1, . . . , n + 1 (6)


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hold for the extended Hamiltonian H. Since H by its definition (4) does not depend on sexplicitly, its total s-derivative vanishes by virtue of Eq. (6):

d Hds

=n+1i=1

Hqi

dqids

+ Hpi

dpids = 0 .

The Hamiltonian H thus describes an autonomous system in the (2n + 2)-dimensional ex-tended phase-space. Defining H according to [16]

H(q, p, t,H) = H(q, p,t)H , (7)the canonical equations (6) that follow from

Hcan be related in a simple way to the canonical

equations (2) of the original Hamiltonian H(q, p,t)

H

pi=

Hpi

,H

qi=

Hqi

,H

t=

Ht

,HH

= 1 . (8)

With the Hamiltonian (7), we conclude from Eq. (8) that the canonical variables q and p sat-isfy the original equations of motion. Then, the extension of the phase-space has no effect on

the description of the systems dynamics. This means that the extended phase-space formal-

ism can be applied to convert the description of a time-dependent Hamiltonian system to that

of an autonomous system with one additional degree of freedom.

Moreover, the extended phase-space formulation has the benefit to allow for more general

canonical transformations that also include a transformation of time

(q, p,t,H)canon. transf. (q , p , t, H) . (9)

The transformation (9) is referred to as canonical if and only if Hamiltons variation principle

(3) is maintained in the new (primed) set of canonical variables. The condition for an ex-

tended phase-space transformation (9) to be canonical can therefore be derived directly from

Hamiltons principle in the form of Eq. (3)

s2s1

n

i=1

pidqidsH

dt

ds

ds =

s2s1

n

i=1

pidqidsH

dt

ds

ds = 0 .

This means that the integrands of the variation integrals may differ at most by a total differ-

ential in the extended phase-space

ni=1

pidqi Hdt =n

i=1

pidq

i Hdt + dF1(q, q

, t , t) . (10)

The function F1(q, q, t , t) is commonly referred to as the generating function of the canon-

ical transformation. With the help of the Legendre transformation

F2(q, p, t , H ) = F1(q, q

, t , t) +n

i=1 qipi tH ,


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the generating function may be expressed equivalently in terms of the old configuration space

and the new momentum coordinates. If we compare the coefficients pertaining to the respec-

tive differentials dqi, dp

i, dt, and dH, we find the following coordinate transformation rules

to apply for each index i = 1, . . . , n:

qi =F2pi

, pi =F2qi

, t = F2H

, H = F2t

. (11)

In the following section, we will investigate the general properties of infinitesimal canonical

transformations in the extended phase-space. In Section 5, the rules (11) will be applied to

carry out a finite canonical transformation for a particular class of Hamiltonian systems.

3 Time-dependent symmetry mapping and corresponding infinitesimal canonical

transformation

In the extended phase-space, the generating function F2 of an infinitesimal canonical trans-formation is given by

F2(q, p, t , H ) =

ni=1

qip

i tH G(q, p,t,H) , (12)

with an infinitesimal parameter, and G(q, p,t,H) the function that characterizes the devia-tion of the canonical transformation from the identity. Since the transformation generated by

Eq. (12) is infinitesimal, to first order in the old (unprimed) canonical variables can be usedcalculating the derivatives ofG(q, p,t,H). From the transformation rules (11), we thus find

qi = qi G

pi, t = t +

G

H, pi = pi +

G

qi, H = H

G

t. (13)

The variation of the function G(q, p,t,H) is given by

G =n

i=1

G

qiqi +

G

pipi

+

G

tt +

G

HH .

Inserting Eqs. (13), we get

G = n

i=1

Gqi Gpi + Gpi Gqi+ Gt GH GH Gt = 0 , (14)which means that G(q, t , p,H) remains invariant by virtue of the canonical transformation.In other words, the infinitesimal part of Eq. (12) itself provides the quantity that is conserved

performing the canonical transformation generated by (12).

We now define a general time-dependent infinitesimal symmetry mapping that is sup-

posed to be consistent with Eq. (13)

t = t + t = t + (t) (15)

qi(t) = qi(t) + qi = qi(t) + i(q, p,t) (16)

p

i(t

) = pi(t) + pi = pi(t) + i(q, p,t) . (17)


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Comparing the partial derivatives of the function G(q, p,t,H), as given by Eq. (13), with theAnsatz (15), (16), and (17) for the the time-dependent symmetry mapping, we find

G

qi = i ,

G

pi = i ,

G

H = (t) . (18)

From the yet unused last expression of Eq. (13), we derive directly the condition for the

partial time derivative ofG(q, p,t,H)

H H = H =H

tt +

ni=1

H

qiqi +

H

pipi

,

which reads with Eqs. (15), (16), and (17)

G

t =

H

t (t)

n

i=1

H

qi i

n

i=1

H

pi i . (19)

A necessary and sufficient condition for transformations to be canonical is to conserve the

Poisson bracket [u, v] of two arbitrary differentiable functions u, v. This requirement imposesconditions on the functions , i, and i that ensure the canonical properties of Eqs. (15),(16), and (17). For the Poisson bracket of the pair of canonical variables (t,H), we have[t,H] = 1, hence

[t,H] =

1 + (t)

1

2G

tH

!

= 1 .

To first order in , this means

d

dt=

t

G

H

or

G

H= (t) ,

in agreement with the last expression in Eq. (18). In the same way, the Poisson bracket

condition [qi, pi] = 1 = [q

i, p

i] leads to the relation

[qi, p

i] = 1 +

iqi

+ipi

+ 2 [i, i]

!= 1 ,

which is fulfilled to first order in if

iqi

= ipi

. (20)

The quantities qi and pi in Eqs. (16) and (17) stand for the variation of the canonical vari-ables as a result of the canonical transformation at different instants of time. In order to

separate the time shift from the coordinate transformation rules, we split Eqs. (16) and (17)

into the coordinate transformation part at fixed time t, and the time shift part according to

qi = i =

qi(t) qi(t

)

+

qi(t) qi(t)

p

i=

i= pi(t)pi(t)+ pi(t)pi(t) .


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Since we are dealing with an infinitesimal transformation, the right brackets may be identified

without loss of generality with the first order term of a Taylor series

qi(t) qi(t) = qi t , pi(t

)pi(t) = pi t .

Solving for the terms in the left brackets, we finally have in conjunction with Eq. (15)

qi(t) qi(t

) =

i (t) qi

(21)

pi(t)pi(t

) =

i (t) pi

. (22)

With the requirement that the functions i = i(q, p,t) and i = i(q, p,t) fulfill condition(20), the coupled set of Eqs. (15), (21), and (22) provides the general form of the trans-

formation rules for infinitesimal canonical transformations in the extended phase-space. In

the following section, we will specialize this general transformation for a specific class of

Hamiltonian systems. The invariant G can then be worked out in explicit form making use of

Eq. (18). The conditional equation for (t) is finally established by Eq. (19).

4 Invariant for a class of Hamiltonian systems

We now consider a specific class of Hamiltonian systems, namely, an n-degree-of-freedomsystem of particles moving in an explicitly time-dependent potential V(q, t) with time-de-pendent damping forces proportional to the velocity. A system of this class is described by

the Hamiltonian

H = 1

2

eF(t)n

i=1 p2i + eF(t)V(q, t) with F(t) = t

t0

f() d , (23)

that provides the canonical equations

qi =H

pi= pi e

F(t) , pi = H

qi=

V

qieF(t) , (24)

hence the equation of motion

qi + f(t) qi +V(q, t)

qi= 0 . (24a)

In the following, we work out the invariant G of the Hamiltonian system (23) that corresponds

to the symmetry mapping (15), (21), and (22) containing the more specific functions i =i(qi, t) and i = i(qi, pi, t). We choose the connection between Eqs. (21) and (22) to beestablished by the first canonical equation of (24)

pi(t)pi(t

) = eF(t)d

dt

qi qi

t

= eF(t)d

dt

i (t) qi

.

Hereby, we determine the invariant G to represent a conserved quantity along the systemsevolution in time. With Eq. (22) and the first canonical equation of (24), we then obtain for

i

i(qi, pi, t) = iqi (t) + (t)f(t)pi + it eF(t) . (25)


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The function i(qi, t) can now be determined from Eq. (25) with the help of Eq. (20)

i

qi

= 12

(t) 12

(t) f(t) , (26)

which can be integrated to give

i(qi, t) =12

qi

f

+ i(t) . (27)

Herein, i(t) denotes an arbitrary function of time only. Inserting Eq. (26) and the partialtime derivative of Eq. (27) into Eq. (25), we eliminate its dependence on i

i(qi, pi, t) = 12pi

f

+ 1

2qie

F(t)

f f

+ ieF(t) . (28)

Now that i and i are specified by Eqs. (27) and (28), respectively, the invariant G(q, p,t,H)can be deduced from its partial derivatives (18)

G(q, p,t,H) = H 12

f

ni=1

qi pi +14

eF(t)

f f n

i=1

q2i

+n

i=1

i qi e

F(t) i pi

. (29)

The functions (t) and i(t) are determined by the condition (19) for G(q, p,t,H) to yieldan invariant. Calculating the partial time derivative of Eq. (29) and making use of the explicit

form of the Hamiltonian (23), Eq. (19) leads to the following linear differential equations for(t) and i(t)

...

ni=1

q2i + 4

V(q, t) + 1

2

ni=1

qiV

qi 1

2

f + 1

2f2 n

i=1

q2i

+ 4

V

t+ f

V(q, t) 1

2

ni=1

qiV

qi

1

4

f + ff

ni=1

q2i

= 0 , (30)

i qi i qi +

i qi i qi

f = 0 , i = 1, 2, . . . , n . (31)

Since the i(t) are arbitrary functions that do not depend on (t), the respective expressionsmust vanish separately. We thus obtain distinct differential equations for (t) and i(t), asgiven by Eqs. (30) and (31). It turns out that the total time derivative of the i-terms inEq. (29) vanish because of Eq. (31). Consequently, the related sum in Eq. (29) provides a

separate invariant

ni=1

i(t) qi e

F(t) i(t)pi

= I = const. ,

which means that the invariant G can be written as a sum of two invariants

G(q, p,t,H) = I+ I .


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The terms associated with the function (t) thus form the invariant I

I = H 12 f

n

i=1 qi pi + 14eF(t) f fn

i=1 q2i . (32)A discussion of the physical meaning of the invariant (32) and the solution (t) of the third-order equation (30) will be the subject of the following two sections.

5 Canonical transformation to the equivalent autonomous system

We will show in the following that the invariant I of Eq. (32) can be regarded as the Hamil-tonian H of an autonomous system, which is related to the Hamiltonian H of the originalsystem by a finite canonical transformation in the extended phase-space. Since this transfor-

mation is unique, we may conceive the transformed system described by H as the equivalentautonomous system ofH.

The Hamiltonian H(q, p,t) of Eq. (23) will be transformed by means of a canonical trans-formation into the new Hamiltonian

H(q , p ) = 12

ni=1

p 2i + V(q ) , (33)

which is supposed to be independent of time explicitly. The canonical transformation in the

extended phase-space be generated by

F2(q, p, t , H ) = 2(q, p

, t)Htt0

d

(). (34)

The function 2(q, p, t) contained herein be defined as the following generating function in

the usual (non-extended) phase-space

2(q, p, t) =

eF(t)

(t)

ni=1

qi p

i +14

eF(t)

(t)

(t) f(t)

n

i=1

q2i . (35)

For the moment, (t) may be regarded as an arbitrary differentiable function of time only.

Working out the transformation rules (11) for the specific generating function F2, as definedby Eq. (34), we find the following linear transformation between the old {qi, pi} and the newset of coordinates {qi, p

i}qipi

=

(t)/eF(t) 0

12

f

eF(t)/(t)

eF(t)/(t)

qipi

. (36)Furthermore, the transformations of time t and Hamiltonian H are given by

t

=

F2

H = t

t0

d

() , H =

F2

t =

2

t +

H

(t) . (37)


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From Eq. (37), the new Hamiltonian H follows as

H = (t)H+2

t . (38)The transformed Hamiltonian H of Eq. (38) is obtained in the desired form of Eq. (33) if theold Hamiltonian H as well as the partial time derivative of Eq. (35) are expressed in termsof the new (primed) coordinates. Explicitly, the effective potential V(q ) of the transformedsystem evaluates to

V(q ) = 14

1

22 2

f + 1

2f2 n

i=1

q 2i + eF(t) V

eF q , t

. (39)

The new potential V consists of two components, namely, a term related to the original

potential V, and an additional quadratic potential that describes the linear forces of inertiaoccurring due to the time-dependent linear transformation (36) to a new frame of reference.

The required property of the new Hamiltonian (33) to describe an autonomous system is

met if and only if the new potential V(q ) does not depend on time t explicitly. This meansthat the initially arbitrary function (t) defined in the generating function (34) is nowtailored to eliminate an explicit time-dependence of V at the location q . In order to setup the appropriate conditional equation for (t), we calculate the partial time derivative ofEq. (39)

V (q )

t

= 14

eF... 2f f f2 ff e

F

n

i=1 q 2i

+4

V + 1

2

ni=1

qiV

q i

+ 4

V

t+ f V 1

2f

ni=1

qiV

q i

= 0 .

(40)

Inserting the transformation rules that follow from Eq. (36)

q 2i =eF

q2i , q

i

V

q i= qi

V

qi,

we observe that Eq. (40) agrees with the auxiliary equation (30) for (t), derived in Section 4from condition (19). With (t) a solution of Eq. (40), we thus have

V (q )

t= 0 ,

which means that the Hamiltonian

H(q , p ) = 12

ni=1

p 2i + V(q ) = I (41)

does not depend on time explicitly, hence constitutes indeed the invariant I in question. The

Hamiltonian H

of Eq. (41) may be expressed as well in terms of the old coordinates {qi, pi}.


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We then encounter the invariant I in the form of Eq. (32), which was derived in Section 4on the basis of an infinitesimal canonical transformation. The effect of the finite canonical

transformation generated by Eq. (34) can be summarized as a transfer of the time-dependence

from the original Hamiltonian H(q, p,t) into the time-dependence of the frame of reference inthe extended phase-space of the new Hamiltonian H(q , p ). In other words, the autonomoussystems Hamiltonian H is canonically equivalentto the initial time-dependent Hamiltoniansystem H.

For (t) > 0, the Hamiltonian H represents a real physical system. Because of theuniqueness of the transformation rules (36) and (37), the Hamiltonian system H may thenbe conceived as the autonomous system that is physically equivalent to the initial system

described by H.The instants of time t with (t) = 0 mark the singular points where this transformation

does not exist. For time intervals with (t) < 0, the elements of coordinate transformation

matrix (36) become imaginary, and according to Eq. (37) the flow of the transformedtime t with respect to t becomes negative. Under these circumstances, the transformedsystem does not possess a physical meaning anymore, which means that the equivalent au-

tonomous system ceases to exist as a physical system. In other words, the particle motion

within the time-dependent non-linear system can no longer be expressed as the linearly trans-

formed motion within an autonomous system.

On the other hand, the invariant I in the representation of Eq. (32) exists for all (t) thatare solutions of Eq. (30), including the regions with (t) 0. We may regard Eq. (32) as animplicit function I = I(q, p,t) of the phase-space coordinates, visualized as a time-varying(2n 1)-dimensional surface in the 2n-dimensional phase-space. In Section 7.2, we willshow for a one-dimensional example that the times t with (t) = 0 mark the transition pointsbetween different topologies of this phase-space surface.

To end this section, we finally note that a canonical transformation that maps an explicitly

time-dependent Hamiltonian H into a time-independent H can equivalently be formulated inthe conventional (non-extended) phase-space. Presenting a two-step linear canonical transfor-

mation, this has been demonstrated for the damped linear oscillator by Leach [2]. A similar

transformation that is followed by a rescaling of time [15] has been shown capable to work

out invariants for more general cases of time-dependent non-linear Hamiltonian systems.

6 Discussion

The invariant (32) is easily shown to represent a time integral of Eq. (30) by calculating the

total time derivative of Eq. (32), and inserting the canonical equations (24). Hence, Eq. (32)

provides a conserved quantity exactly along the phase-space trajectory that represents the

systems time evolution. This trajectory is given as a solution of the 2n first-order canonicalequations (24) or, equivalently, of the n second-order equations of motion (24a). With thesesolutions, Eq. (30) must not be conceived as a conditional equation for the potential V(q, t).Rather, all qi-dependent terms of Eq. (30) are in fact functions of the parameter t only givenalong the systems phase-space trajectory. Correspondingly, the potential V(q(t), t) and itsderivatives in Eq. (30) are time-dependent coefficients of an ordinary third-order differential

equation for (t). According to the existence and uniqueness theorem for linear ordinary


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differential equations, a unique solution function (t) of Eq. (30) exists and consequentlythe invariant I if V and its partial derivatives are continuous along the systems phase-space path.

Vice versa, Eq. (30) together with the side-condition I = const. from Eq. (32) may beconceived as a conditional equation for a potential V(q, t) that is consistent with a solutionof the equations of motion (24). In other words, the invariant I = const. in conjunction withthe third-order equation (30) must imply the canonical equations (24). This can be shown

inserting Eq. (30) into the total time derivative of Eq. (32). Since dI/dt 0 must hold forall solutions (t) of Eq. (30), the respective sums of terms proportional to (t), (t), and (t)must vanish separately. For the terms proportional to (t), this means

12

eF(t)n

i=1

qi

qi pi e

F

0 . (42)

The identity (42) must be fulfilled for all initial conditions (q(0), p(0)) and resulting phase-space trajectories (q(t), p(t)) of the underlying dynamical system. Consequently, the expres-sion in parentheses must vanish separately for each index i, thereby establishing the firstcanonical Eq. (24). The remaining terms impose the following condition for a vanishing total

time derivative ofI

eF(t)n

i=1

pi

12

eF

f

qi

pi +

V

qieF 0 .

Again, this sum must vanish for all (t) that are solution of Eq. (30). This can be accom-

plished only if the rightmost expression in parentheses, hence the second canonical Eq. (24),is fulfilled for each index i = 1, . . . , n. Summarizing, we may state that the triple made up bythe canonical Eq. (24), the auxiliary Eq. (30), and the invariant I of Eq. (32) forms a logicaltriangle: if two equations are given at a time, the third can be deduced.

The physical interpretation of the invariant (32) can be worked out considering the total

time derivative of the Hamiltonian (23). Making use of the canonical equations (24), we find

dH

dt+ 1

2f eF

ni=1

p2i eF

f V(q, t) +

V

t

= 0 , (43)

which represents just the explicit form of the general theorem dH/dt = H/t for theHamiltonian (23). Equation (43) can be interpreted as an energy balance relation, stating

that the systems total energy change dH/dt is quantified by the dissipation and the explicittime-dependence of the external potential. Inserting V/t from the auxiliary Eq. (30) intoEq. (43), the sum over all terms can be written as the total time derivative

d

dt

H 1

2

f

ni=1

qi pi +14

eF

f f n

i=1

q2i

= 0 .

The expression in brackets coincides with the invariant (32). As the function (t) is thesolution of a homogeneous linear differential Eq. (30), it is determined up to an arbitrary

factor. We are, therefore, free to conceive (t) as a dimensionless quantity.


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With the initial conditions (0) = 1, (0) = (0) = 0 for the auxiliary Eq. (30), theinvariant I can now be interpreted as the conserved initial energy H0 for a non-autonomoussystem described by the Hamiltonian (23), comprising both the systems time-varying energy

content H and the energy fed into or detracted from the system.The meaning of (t) follows directly from the invariant (32) if the Hamiltonian H is

treated formally as an independent variable I = I(q, p,t,H). A vanishing total time deriva-tive of the invariant I then writes

dI

dt=

I

t

q,p,H

+H

t

I

H

q,p,t

+i

qi

I

qi

p,t,H

+ piI

pi

q,t,H

= 0 .

Inserting qi and pi from the canonical Eq. (24), and making again use of the auxiliary Eq. (30)to eliminate the third-order derivative

... (t), we find the expected result

I

Hq,p,t = (t) . (44)(t) thus quantifies how much the total energy I varies with respect to a change of the actualsystem energy H at fixed phase-space location (q, p) and time t.

Some additional remarks are appropriate at this point. In the calculation of pi, we usedthe relation between pi and qi, given by the first canonical equation (24). A similar situationis encountered in the framework of the Lagrangian formalism, where the Lagrange function

L(q, q, t) depends on q, i.e. the time derivative of the coordinates q(t). As will be shownin Appendix A, the invariant (29) and the corresponding conditional Eqs. (30) and (31) can

indeed be derived likewise in the Lagrangian context using Noethers theorem.

For a vanishing damping coefficient (f(t) F(t) 0), Eqs. (30) and (32) are consistent

with corresponding expressions presented earlier [14], where a direct approach to constructinvariants for time-dependent Hamiltonian systems has been applied. As has been shown

in [15], the direct approach can also serve as the basis to derive invariants for dissipative

non-linear systems. We will review this approach briefly in Appendix B.

Equations (30) and (32) can be shown to agree with the corresponding results of Ref. [15]

if the functions c(t) and f2(t) used there are expressed in terms of our actual notation accord-ing to c(t) = eF(t) and f2(t) = (t) e

F(t).

We further note that for the special case f(t) V/t 0, hence for autonomous sys-tems, (t) 1 is always a solution of Eq. (30). For this (t), the invariant (32) reducesto I H, hence provides the systems total energy, which is a known invariant for Hamil-

tonian systems with no explicit time-dependence. Nevertheless, Eq. (30) also allows forsolutions (t) = const. for these systems. We thereby obtain other non-trivial invariants forautonomous systems that exist in addition to the invariant representing the energy conserva-

tion law. This will be demonstrated for the simple case of the harmonic oscillator at the end

of Section 7.1.

For general potentials V(q, t), the dependence of Eq. (30) on q(t) cannot be eliminated.Under these circumstances, the function (t) can only be obtained integrating Eq. (30) simul-taneously with the equations of motion (24a). For the isotropic quadratic potential with d(t)denoting an arbitrary continuous function of time

V(q, t) = d(t)n

i=1 q2i , (45)


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Eq. (30) may be divided by

i q2i and hereby strip its dependence on the single particle

trajectories qi(t). Only for the particular linear system associated with Eq. (45), the third-order differential Eq. (30) for (t) decouples from the equations of motion (24a). Then, the

solution functions (t), (t), and (t) apply to all trajectories that follow from the equationsof motion (24a) with Eq. (45). For the potential (45), Eq. (30) may be integrated to yield a

non-linear second-order equation for (t)

12

2 + 2

4d f 12

f2

= c , (46)

c = const. denoting the integration constant. With the help of Eq. (46), we may eliminate (t)in the expression (32) for the invariant I. After reordering, Eq. (32) may then be rewritten as

8 I = eF(t)n

i=1 2qi fqi2

+ 2c q2i . (47)

Obviously, for a positive integration constant c > 0, the quantities I and (t) must have thesame sign. Thus, for c > 0, an invariant I > 0, and for the initial condition (0) > 0, thefunction (t) remains non-negative at all times t > 0 for systems governed by the potential(45). On the other hand, (t) may change sign for general non-linear systems (23). Ashas been shown in Section 5, the condition (t) > 0 provides a necessary criterion for thephysical existence of an equivalent autonomous system of Eq. (23).

7 Examples

7.1 Time-dependent damped harmonic oscillator

As a simple example, we treat the well-known one-dimensional time-dependent harmonic

oscillator with damping forces linear in the velocity. Its Hamiltonian is given by

H = 12p2eF(t) + 1

22(t) q2eF(t) , (48)

which yields the equation of motion

q = p e

F(t) , q + f(t) q + 2(t) q = 0 . (49)

Because of the quadratic structure of the potential V(q, t) = 12

2(t) q2, the dependence of theauxiliary Eq. (30) on the particle coordinate q cancels. The differential equation for (t) thussimplifies to

... +

42 2f f2

+

4 f ff

= 0 . (50)

The invariant I is immediately found as the one-degree-of-freedom version of Eq. (32)

I = H 12 fq p + 14eF(t) f f q2 , (51)


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with H standing for the Hamiltonian (48). For this particular case, the linear third-orderequation (50) for (t) can be integrated once to yield a non-linear second-order equation

1

2 2

+ 22 2 12 f 14f2 = 2c , (52)

c = const. denoting the integration constant. Using Eq. (52) to replace (t) in Eq. (51), weobtain the invariant I for the system (48) in the alternative form

I =ceF

2q2 + 1

2

p

/eF 12

q

f

eF/2

. (51a)

As already concluded from Eq. (47), (t) cannot change sign if we define c > 0. The sub-stitution (t) = 2(t) then exists for real (t) at all times t, which means that Eqs. (52) and(51a) can be expressed equivalently in terms of(t). Setting c = 1, this leads to the auxiliary

equation and the invariant in the form derived by Leach [2], who applied the method of linearcanonical transformations with time-dependent coefficients.

The expression for the invariant (51) becomes particularly simple if expressed in terms of

the coordinates of the canonically transformed system. Applying the related transformation

rules (36), we find inserting Eq. (52)

I = 12p 2 + 1

2c q 2 . (53)

According to the canonical transformation theory of Section 5, the invariant I may regardedas the Hamiltonian H of an autonomous system that is equivalent to (48)

H = 1

2

p 2 + V(q) , (54)

with the effective potential (39) in the transformed system simplifying to

V(q) = 14

1

22 + 22

2 1

2f 1

4f2

q 2 .

Comparing the expression in brackets with Eq. (52), we find that the new Hamiltonian (54)

indeed agrees with the invariant (53). Figure 1 shows a special case of a numerical integration

of the equation of motion (49). The results of the simultaneous numerical integration of the

auxiliary Eq. (50) are included in this figure. As coefficients of Eq. (49) we chose

(t) = cos(t/2) , f(t) = 1.8 102 sin(t/) .

The initial conditions were set to q(0) = 1, q(0) = 0, (0) = 1, (0) = 0, and (0) = 0.According to Eq. (51), we hereby define an invariant of I = H(0) = 0.5 for the sampleparticle.

In agreement with Eq. (47), the function (t) remains positive for this linear system. Wefurthermore observe that (t) is related to the energy transfer into the system according toEq. (44): (t) becomes large for strong changes of the actual system energy H(t) and viceversa.

For the autonomous analog of Eq. (48), we have f f 0, hence 0. Theauxiliary Eq. (50) then further simplifies to

...

+ 420 = 0 . (55)


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-5

0

5

10

15

0 5 10 15 20 25 30

t

q(t)(t)

H(t)

Fig. 1 Example of a simultaneous numerical integration of the equation of motion (49) and the aux-

iliary Eq. (50). In addition, H(t) displays the actual system energy given by the Hamiltonian (48).

Obviously, Eq. (55) has the special solution (t) 1. The invariant (51) for this solutionof Eq. (55) then coincides with the systems Hamiltonian I H, which represents the con-served total energy of the autonomous system. A further non-trivial invariant is obtained for

the linear independent solution of Eq. (55) with (t) = const.

(t) = c1 cos20t + c2 sin20t .

Setting c1 = 1 and c2 = 0, this leads to the following representation of the second invariantfor the time-independent harmonic oscillator

I = 12

p2 20q

2

cos20t + qp 0 sin20t ,

in agreement with the result obtained earlier by Lutzky [17]. Likewise, the third linear inde-

pendent solution for (t) and the two solutions for (t) of Eq. (31) yield invariants. Thesefive conserved quantities of which only two are functionally independent are correlated

to the five-parameter Noether subgroup of the complete eight-parameter symmetry group

SL(3, R) for the harmonic oscillator [17].

7.2 Time-dependent anharmonic undamped one-dimensional oscillator

As a second example, we investigate the one-dimensional non-linear system of a time-depend-

ent anharmonic oscillator without damping, defined by the Hamiltonian

H = 12p2 + 1

22(t) q2 + a(t) q3 + b(t) q4 . (56)

The associated equation of motion is given by

q = p , q + 2(t) q + 3a(t) q2 + 4b(t) q3 = 0 . (57)


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-20

-15

-10

-5

0

5

10

15

20

0 5 10 15 20 25 30

t

q(t)(t)

H(t)

Fig. 2 Example of a simultaneous numerical integration of the equation of motion (57) and the cou-

pled set (60), (61) for (t). In addition, H(t) displays the time-dependent system energy given by theHamiltonian (56).

Again, the invariant is immediately obtained writing the general invariant (32) for one di-

mension. With vanishing damping functions (F(t) f(t) 0), the invariant simplifiesto

I = 12

(t)

q2 + 2(t) q2 + 2a(t) q3 + 2b(t) q4

1

2(t) qq + 1

4(t) q2 . (58)

For this particular case, the linear third-order equation for the auxiliary function (t) reads... + 4 2(t) + 4 + 2q(t)

2 a + 5 a

+ 4q2(t)

b + 3 b

= 0 , (59)

which follows from the general form of Eq. (30). We observe that in contrast to the

previous linear example the particle trajectory q = q(t) is explicitly contained in in therelated auxiliary Eq. (59). Consequently, the integral function (t) can only be determined ifEq. (59) is integrated simultaneously with the equation of motion (57).

We may directly convince ourselves that I is indeed a conserved quantity. Calculating thetotal time derivative of Eq. (58), and inserting the equation of motion (57), we end up with

Eq. (59), which is fulfilled by definition of (t) for the given trajectory q = q(t).The third-order differential Eq. (59) may be converted into a coupled set of first- and

second-order equations. It is easily shown that the non-linear second-order equation

12

2 + 22(t) 2 = g(t) (60)

is equivalent to Eq. (59), provided that the time derivative of the function g(t), introduced inEq. (60), is given by

g(t) = 2q(t)

2 a + 5 a 4q2(t)

b + 3 b

. (61)

With the help of the auxiliary equation in the form of Eq. (60), the invariant (58) may be

expressed equivalently as

I =1

2 q2 qq + 24 q2 + 2 a q3 + 2 b q4+ g(t)4 q2 . (62)


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-8

-6

-4

-2

0

2

4

6

8

-10 -5 0 5 10

q

q

t = 3.3

-8

-6

-4

-2

0

2

4

6

8

-10 -5 0 5 10

q

q

t = 7.7

-8

-6

-4

-2

0

2

4

6

8

-10 -5 0 5 10

q

q

t = 9.7

-8

-6

-4

-2

0

2

4

6

8

-10 -5 0 5 10

q

q

t = 13.5

Fig. 3 Curves of constant invariant I = 0.58 in the (q, q) phase-space plane and location of thesample particles at four instants of time with (t = 3.3) = 3.2, (t = 7.7) = 3.7, (t = 9.7) = 6.1,

and (t = 13.5) = 8.3.

In contrast to Eq. (59), the equivalent coupled set of equations (60) and (61) does not contain

anymore the time derivative of the external function 2(t).For the time-dependent harmonic oscillator (a(t) = a(t) = b(t) = b(t) 0), Eq. (61)

leads to g(t) = 0, which means that g(t) = g0 = const. For this particular case, g0, and henceEq. (60) no longer depends on the specific particle trajectory q = q(t). Consequently, thesolution function (t) applies to arbitrary trajectories.

Setting g0 = 2 and (t) = 2(t) we obtain the well-known Lewis invariant [2]

I =1

22q2 + (q q)2 .For this linear case, we can rewrite Eq. (62) as

8I(t) =

2(t)q q(t)2

+ 2q2g0 ,

providing a special case of the general Eq. (47). For the initial condition g0 > 0, and I > 0,we have (t) 0 for all t > 0. As a consequence, the linear canonical transformation (36)is regular for all times t, and the equivalent autonomous system represents a real physicalsystem. In contrast, the function g(t) is no longer a constant in the general non-linear case,and (t) may become negative. For (t) < 0, the canonical transformation (36) becomes

imaginary, which means that the equivalent system is no longer physical. Nevertheless, the


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invariant (58) exists as a real number for all solutions of the third-order differential equa-

tion (59), independently of the sign of (t).Figure 2 shows a special case of a numerical integration of the equation of motion (57).

Included in this figure, we see the result of the simultaneous numerical integration of Eqs. (60)and (61). The coefficients of Eq. (57) were chosen as

(t) = cos(t/2) , a(t) = 5 102 sin(t/3) , b(t) = 8 102 cos2(t/3) .

The initial conditions were set to q(0) = 1, q(0) = 0, (0) = 1, (0) = 0, and (0) = 0.According to Eq. (62), we hereby define an invariant of I = H(0) = 0.58 for the sampleparticle. Owing to the systems non-linear dynamics, (t) now becomes piecewise negative.Also, the relationship (44) between (t) and H(t) appears more complicated compared to thelinear case of Fig. 1. Interesting insight into the dynamical evolution of the sample particle

can be obtained if the invariant (62) is regarded as an implicit representation of a phase-

space curve I = I(q, q, t). Figure 3 displays snapshots of these curves at four differentinstants of time t. As expected, the particle lies exactly on these curves of constant I, therebyproviding a numerical verification of Eq. (62). The two upper pictures display situations

with (t) > 0. Then, the phase-space curves are of closed elliptic type, being more or lessdeformed because of the non-linear terms in the Hamiltonian (56). When the function (t)becomes negative, as given for the lower two pictures, topological changes of the phase-space

curves to more complex shapes are observed. For the special case of a conservative system,

we have (t) = 0 = const., a(t) = a0 = const., b(t) = b0 = const., and Eq. (59) reduces to

... (t) + (t)

420 + 10q(t)a0 + 12q

2(t) b0= 0 . (63)

Obviously, this equation has the special solution (t) 1. For this case, Eq. (58) simplifiesto

I = 12

q2 + 12

20q2 + a0q

3 + b0q4 = H ,

thus agrees with the systems Hamiltonian, which represents the conserved total energy.

Another non-trivial invariant is obtained inserting a solution of the auxiliary Eq. (63) with

(t) = const. into the expression for the invariant (58).

7.3 n-dimensional anisotropic oscillator with interaction

As the last example, we investigate the n-dimensional system of a time-dependent anisotropicoscillator with interaction between the particles. With an interaction potential V(q) that de-pends on all configuration space variables, the Hamiltonian be defined as

H =n

i=1

12p2i +

12

2i (t) q2i

+ V(q) . (64)

The associated canonical equations follow as

qi = pi , pi = 2

i (t) qi

V(q)

qi , i = 1, . . . , n .


20/24


With H the Hamiltonian (64), the invariant is obtained from the general invariant (32) settingthe damping functions to zero (F(t) f(t) 0)

I = (t) H 12 (t)

ni=1

qi pi + 14 (t)

ni=1

q2i .

For this example, the auxiliary Eq. (30) for (t) specializes to

ni=1

... + 4 2i (t) + 4 i i

q2i (t) + 4

V(q) + 1

2

ni=1

qiV

qi

= 0 . (65)

Defining 2(t) as the average force function

2(t) =n

i=1 2i (t) q2i n

i=1 q2i ,the linear third-order auxiliary equation (65) may again be expressed equivalently in terms of

a coupled set of a non-linear second-order equation for (t) and a first-order equation for g(t)

12

2 + 22(t) 2 = g(t)

g(t) = 4

2

ni=1

2i

2

qi pi

V(q) + 1

2

ni=1

qiV

qi

ni=1

q2i . (66)

We learn from Eq. (66) that g(t) is determined by two quantities of different physical nature:

the interaction potential V(q) and the systems anisotropy. In contrast to Eq. (52) of theone-dimensional example sketched in Section 7.1, the r.h.s. of the second-order equation

for (t), i.e. the function g(t), is generally not constant. Even in the linear case, whichis obtained here for a vanishing interaction potential (V(q) 0), the particle trajectoriesqi(t) are inevitably contained in the auxiliary equation (65). The dependence of Eq. (65) onthe particle trajectories vanishes exclusively for the isotropic linear oscillator, hence for the

particular case where V(q) 0, and the oscillator frequencies agree in all degrees of freedom(t) i(t). With respect to the auxiliary function (t) and hence the invariant I ananisotropy thus induces an effective coupling of all particles.

8 Conclusions

For a fairly general class of time-dependent Hamiltonian systems, we have derived an in-

variant I, hence a quantity that is conserved along the systems phase-space trajectory. Therepresentation of I was found to depend on a function (t), which in turn embodies a so-lution of a linear homogeneous third-order differential equation, referred to as the auxiliary

equation.

The invariant I for time-dependent Hamiltonian systems H was derived in the frameworkof the extended phase-space, where time and the negative Hamiltonian itself are regarded

as canonically conjugate coordinates. In this description, an invariant may be isolated di-

rectly from the generating function of an infinitesimal canonical transformation, hence from


21/24


symmetry mappings of the canonical variables and time that leave the canonical equations in-

variant. The auxiliary equation for (t) then emerges from the requirement that the symmetrymapping be canonical. The invariant could be identified with the conserved total energy for

non-autonomous systems, which is obtained if we add to the time-varying energy representedby the Hamiltonian H the energies fed into or detracted from the system.

We have furthermore shown that the invariant I is also obtained applying in the extendedphase-space a finite canonical transformation to the initial Hamiltonian H(q, p,t). Imposingthe condition on the new Hamiltonian H(q , p ) not to depend on time explicitly then coin-cides with the auxiliary equation for (t) derived beforehand in the context of infinitesimalcanonical symmetry mappings.

Expressed in terms of the old coordinates, the new Hamiltonian H was found to agreewith the invariant I for all (t) that follow as solutions of the auxiliary equation. On the otherhand, the new Hamiltonian H represents a real physical system only for time intervals with

(t) > 0. Then, H

may be interpreted as the Hamiltonian of the autonomous system that isequivalent to the non-autonomous system H.

In general, the auxiliary equation depends on the systems spatial coordinates. As the

consequence, the auxiliary equation can only be integrated in conjunction with the equations

of motion. From this viewpoint, the 2n first-order canonical equations that determineuniquely the time evolution of the n particle system form together with the three first-order equations of the auxiliary equation a closed coupled set of 2n + 3 first-order equationsthat uniquely determine the invariant I.

For the particular case of isotropic quadratic Hamiltonians, the dependence of the auxil-

iary equation on the spatial coordinates cancels. The third-order auxiliary equation may then

be analytically integrated to yield a well-known non-linear second-order equation that has

been derived earlier for the time-dependent harmonic oscillator.

In the special case of autonomous systems hence Hamiltonian systems with no explicit

time-dependence the function (t) 1 is always a solution of the auxiliary equation. Withthis solution, the invariant Icoincides with the invariant that is given by the systems Hamilto-nian H itself. In view of this result, the familiar invariant I = H just represents the particularcase where the auxiliary equation possesses the special solution (t) 1 which is exactlygiven for autonomous Hamiltonian systems. In addition to the invariant I = H, anothernon-trivial invariant for autonomous systems always exists that is associated with a solution

(t) = const. of the auxiliary equation. The dependence of the invariants of a Hamiltoniansystem on solutions (t) of an auxiliary equation thus constitutes a general feature which

disappears solely for the particular case of the invariant I = H of an autonomous system.For the case of explicitly time-dependent Hamiltonian systems, solutions of the auxil-

iary equation with (t) = const. do not exist. Therefore, the invariants for non-autonomoussystems always depend on solutions (t) of the auxiliary equation. The additional complex-ity that arises for the invariants of non-isotropic linear and general non-linear Hamiltonian

systems is that the auxiliary equation now depends on the systems spatial coordinates. The

authors believe that this generalized viewpoint of the concept of an invariant is the main result

of the present article.

It has been shown that the solution function (t) of the auxiliary equation remains non-negative for linear isotropic systems. In these cases, (t) may be interpreted as an amplitude

function of the particle motion [18]. For all other Hamiltonian systems, the auxiliary function


22/24


(t) may become negative. A connection between theses solutions of the auxiliary equationand the characteristics of the solutions of the equations of motion has not yet been established.

Furthermore, the physical implications that are associated with an unstable behavior of (t)

still await clarification.

The authors are indebted to H. Stephani (University of Jena, Germany) for valuable discussions.

A Invariants derived from Noethers theorem

We will show in this Appendix that the invariant (29) can straightforwardly be derived on the

basis of Noethers theorem [6]. This theorem relates the conserved quantities of a Lagrangian

system L(q, q, t) to the one-parameter groups that leave the action integral invariant. The firstextension of the infinitesimal generator of a one-parameter Lie group [19] of point transfor-

mations is given by

E = (t)

t+

ni=1

i(qi, t)

qi+

ni=1

i qi

qi

(67)

with

i(qi, t) =it

+iqi

qi , and = (t) .

Noethers theorem then states that the solutions of the Euler-Lagrange equations

L

qi d

dt Lqi = 0 , i = 1, 2, . . . , n (68)admit the conserved quantity [20, 21]

I =n

i=1

(qi i)L

qi L + f0(q, t) , (69)

provided that the action integral is invariant under the group generator (67) which is the

case if the total time derivative of f0(q, t) fulfills the condition

df0(q, t)

dt

= EL + L . (70)

The particular Lagrangian, whose Euler-Lagrange Eqs. (68) lead to the equations of motion

(24a) is

L(q, q, t) =

12

ni=1

q2i V(q, t)

eF(t) with F(t) =

tt0

f()d . (71)

Inserting the Lagrangian (71) into Eq. (70), and equating separately the coefficients of powers

of qi to zero, we obtain a set of equation for (t), i(qi, t), and f0(q, t) which can be solvedto find

i(qi, t) = 12 (t) (t)f(t) qi + i(t) (72)


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and

f0(q, t) = eF(t)

14 f f

n

i=1 q2i +

n

i=1 iqi . (73)The terms of Eq. (70) that do not depend on qi sum up to

f0t

+ eF(t)

n

i=1

iV

qi+

V

t+ f(t)V

+ V

= 0 . (74)

Inserting Eq. (72) and the partial time derivative of Eq. (73) into Eq. (74), we get the third-

order differential equation for (t) of Eq. (30), together with the n differential equations forthe arbitrary functions i(t), given by Eq. (31). Using the relation

pi =

L

qi = qieF(t)

,

we finally find the invariant (29) inserting Eqs. (71), (72), and (73) into Eq. (69).

B Invariants derived from a direct approach

Finally, we will show that the invariant (29) can as well be derived using a direct ansatz

function being quadratic in the canonical momentum. This approach has been used earlier

by Lewis and Leach [13] to work out a class of potentials for which an invariant exists. In

our actual case of an n-degree-of-freedom system with the canonical equations given by (24),this ansatz for the invariant may be defined as

I =n

i=1

12

(t)piqi n

i=1

(qi, t)pi + f1(q, t) . (75)

The quantity I embodies a constant of motion if its total time derivative vanishes

dI

dt!

= 0 .

Calculating the total time derivative of (75), and inserting the particular canonical equations

(24), we may separately equate to zero the sums proportional to q2

i , q1

i , and q0

i similarto the procedure pursued in the approach based on Noethers theorem. I thus constitutes aninvariant which is defined exactly on the phase-space path that represents the systems time

evolution.

For the function (qi, t) contained in (75), we find

i(qi, t) =12

(t) (t)f(t)

qi + i(t) , (76)

which obviously agrees with (72). The function f1(q, t) of (75) evaluates to

f1(q, t) = eF(t) V(q, t) + 14 f f

n

i=1 q2i +n

i=1 iqi . (77)


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Inserting (76) and (77) into (75), one directly obtains the invariant (29). The third-order

differential equation for (t) follows from the terms of dI/dt = 0 that do not depend on theqi

f1t

+ eF(t)n

i=1

iV

qi= 0 . (78)

The differential equations (30) and (31) for (t) and the i(t) are found inserting (76) andthe partial time derivative of (77) into (78). We hereby observe that the ansatz approach of

Eq. (75) to derive the invariant I is equivalent to the strategy based on Noethers theorem.

References

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canonical trasformation hamilton ian system

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