Fundamentals and applications of canonical transformations in the extended phase space

Transcription

1 Fundamentals and applications of canonical transformations in the extended phase space Jürgen Struckmeier and Claus Riedel Gesellschaft für Schwerionenforschung GSI, Planckstrasse, 6429 Darmstadt, Germany Received 4 November 23 Realizing the principle of least action on the basis of the system s time t as the evolution parameter, the subsequent theory of canonical transformations implies the time t to be the common independent variable of both the original and the destination system. This description is not adequate if the nature of the given physical problem requires also a mapping of the respective time scales. In a more general realization of the principle of least action, the system s time t = ts is treated as an ordinary canonical variable that depen like all other canonical variables on a superordinated evolution parameter s. In this extended phase-space description, more general canonical transformations can be defined that allow to relate the original and the destination system at different instants of their respective time scales. We shall demonstrate that the Lorentz transformation can be formulated as a particular canonical transformation in this extended phase space. Furthermore, the generalized canonical transformation approach allows a direct mapping of explicitly time-dependent Hamiltonians into time-independent ones. An extended generating function will be presented that applies to a general class of time-dependent Hamiltonians. The subsequent mapping will turn out to linearly relate the system s global characteristics energy Ht and second moment q 2 t with the respective initial values. The properties of this mapping will be shown to yield information with regard to the regularity of the system s time evolution. In the appendix, we re-formulate the Lagrangian description of dynamics as it emerges from the generalized understanding of the principle of least action. As an application, we outline that the symmetry mapping associated with Noether s theorem is equivalent to a particular canonical transformation in the extended phase space. PACS numbers: d, a, 45.5.Jf I. INTRODUCTION The principle of least action constitutes the basis for both Hamilton s as well as Lagrange s formulation of dynamics, 2. Conventionally, this principle is worked out as a variational integral with the system s evolution parameter time t as the independent variable. The variation is then performed between fixed instants of time. This natural choice of the independent variable lea to the well-known canonical equations that completely and uniquely determine the system s time evolution for given initial conditions. From a mathematical viewpoint, this conventional formulation of the variational integral suffers from a lack of generality for not comprising the variation of time t. We may, however, easily reformulate the principle more generally treating the system s time t = ts as an additional canonical variable that depen on its part like the all other dependent variables qs and ps on a superordinated system evolution parameter, s. The variable conjugate to ts is given by the negative value Hs of the system s Hamiltonian Hq, p, t. Treating the time ts as an additional canonical variable is commonly referred to as the concept of the extended phase space 3 5. A discussion of the formulation of Hamiltonian mechanics in the extended phase space as it follows from a generalized understanding of the variational integral will be the starting point of our analysis. Despite the fact that the idea of a parameterization of time is all but new, regarding the existing address: j.struckmeier@gsi.de literature in the realm of classical dynamics it does not seem to be well understood. In various derivations 4, 6, we face a confusion of the Hamiltonian H with its value H, which ultimately lea the meaningless result of a variational integral without any Hamiltonian. A discussion of this problem which cannot be found in literature is given in Sec. II A, and, more detailed, in Appendix B. This will provide the basis for Sec. II B, in which generalized canonical transformations in the extended phase space are considered. We will see that the actual benefit of the generalized variational integral is to abolish the restriction of the conventional canonical transformation theory that presupposes the time t to be the common independent variable of both the original and the destination system. In other wor, the generalized formulation of the variational integral establishes the basis for canonical transformations that allow to relate both systems at different instants of their respective time scales. In Sec. III A we will observe that this property enables us to formulate the Lorentz transformation as a particular canonical transformation in the extended phase space. We thereby encounter an elegant way to convert non-lorentz-invariant Hamiltonians into Lorentz-invariant ones. We will furthermore see in Sec. III B that a canonical transformation of an explicitly time-dependent Hamiltonian into a time-independent Hamiltonian necessarily involves a transformation of the respective time-scales. The pertaining generating function for this extended canonical transformation will be shown to depend on a characteristic function ξt that emerges as a solution of a linear third-order equation. For a time-dependent harmonic oscillator system, a particular solution of this third-order equation yiel the wellknown Lewis invariant 7. For the general class of non-linear

2 2 and explicitly time-dependent Hamiltonian systems treated in Sec. III C, we will show that the complete set of fundamental solutions of the linear system for ξt describes the dependence of system s global quantities energy Ht and second moments qpt, q 2 t on their corresponding initial values. We hereby obtain additional information on the system s time evolution in terms of the sensitivity of the actual global quantities on variations of the initial conditions. In this respect, the fundamental solutions of the third-order equation reveal information on the system s regularity. Depending on the growth rates of an eventual instability of the solutions ξt, we demonstrate that these functions may serve as an indicator for the system s degree of chaos. In Sec. IV C, these new results will be verified by means of a numerical example, namely the well-studied Hénon-Heiles oscillator. We will observe that the time evolution of the ξt functions indeed provides a measure for the system s nonregularity. In Appendix A, the concept of the extended phase space will be applied to Lagrange s formulation of dynamics. We then encounter a formal similarity of the extended Lagrangian with that of a conventional time-independent Lagrangian. Making use of this similarity, we will outline a particular elegant derivation of Noether s theorem for general time-dependent Lagrangian systems. It will turn out that the symmetry transformation leading to Noether s theorem correspon to the extended canonical transformation of Sec. III C. II. HAMILTONIAN MECHANICS IN THE EXTENDED PHASE SPACE A. Principle of least action We consider an n-degree-of-freedom system of classical particles whose dynamics may be described in a 2ndimensional Cartesian phase space by an in general explicitly time-dependent Hamiltonian Hq, p, t. Herein, q denotes the n-dimensional vector of the configuration space variables, and p the vector of conjugate momenta. The principle of least action states that the actual system phase-space trajectory C minimizes the integral over the Poincaré-Cartan -form p dq Hdt. This means that the variation of the action integral due to a variation of C must vanish δ p dq H dt =. C Usually, the time t is used to parameterize the line integral. The system s time evolution is then represented by the phasespace path qt, pt. If the system s state is known at two distinct instants of time t and t 2, the principle of least action then writes t2 δ t n p i t dq it dt H qt, pt, t dt =. 2 This variational integral is well-known to vanish globally if the phase-space path qt, pt satisfies the canonical equations dq i dt = H p i, dp i dt = H q i, i =,..., n. 3 In the particular parameterization 2 of the line integral, the time t plays a twofold role, namely that of an argument of the Hamiltonian Hq, p, t and that of the integration variable, as well. Accordingly, the t-dependence of H is not varied performing the variation of the action integral in Eq. 2. This restriction is lifted by detaching the t-dependence of the Hamiltonian from the formal integration variable. With s the new system evolution parameter, we may parameterize the line integral as δ s2 s n p i s dq is H qs, ps, ts dts =. 4 The symmetric form of the integran in Eqs. and 4 suggests to conceive the time in conjunction with the negative value of the Hamiltonian as an additional pair of canonically conjugate coordinates. We thus introduce the 2n + 2- dimensional extended phase space by defining q n+ = t, p n+ = H as additional phase-space dimensions. The notation H is used to discriminate the value Hs R of the Hamiltonian as a new s-dependent canonical variable from the Hamilton function Hq, p, t itself Hs = H qs, ps, ts. 5 Provided that the Hamiltonian represents the sum of the system s kinetic and potential energies, then Hs quantifies the system s instantaneous energy content. The negative system energy content H thus embodies the canonical variable that is conjugate to the canonical variable time t. As well-known, Hs is not constant if the Hamiltonian H is explicitly timedependent. In order to cast the variational integral 4 into a form similar to Eq. 2, we define the zero Hamiltonian H as an implicit function of the extended phase-space variables H q, p, t, H dt Hq, p, t H =. 6 Inserting Hq, p, t dt/ from Eq. 6 into Eq. 4, we obtain the variational integral in terms of the extended phasespace Hamiltonian H with the upper summation index now ranging to n + s2 n+ δ s p i s dq is H qs, ps, ts, Hs =. 7 This expression provides the general parameterization of the line integral. We observe that this representation of the principle of least action formally agrees with Eq. 2. Similar to the case of Eqs. 3, the variational integral 7

3 3 thus vanishes globally on the extended phase-space path qs, ps, ts, Hs, given as the solution of the extended set of canonical equations dq i = H p i, dp i = H q i, i =,..., n +. 8 The additional pair of canonical equations that is induced by the extended phase-space description given in Eq. 8 for the index n+ writes in terms of the extended phase-space variables t and H dt = H H, dh = H. t In contrast to the time derivative of the original Hamiltonian H, the total s-derivative of H always vanishes identically by virtue of Eqs. 8 n+ dh H q i dq i + H dp i =. p i Thus, the extended Hamiltonian H = of Eq. 6 formally converts any given non-autonomous system Hq, p, t into an autonomous system in the 2n + 2-dimensional extended phase space. Regarding the form of the extended phase-space Hamiltonian 6, it is obvious that we must carefully distinguish the Hamilton function H : R 2n R R from its actual value H R as a canonical variable. Failing to do so, the authors of several textbooks cf, for example, Lanczos 6 mistakenly state that the extended phase-space Hamiltonian H q, p, t, H vanishes identically, and hence conclude that H can be omitted in the integrand of Eq. 7. As a result of this flaw, a variational integral emerges that does not contain any Hamiltonian! This is, of course, a senseless result as the Hamiltonian the source of all information on the given dynamical system can never drop out of a realization of the principle of least action. Because its partial derivatives do not vanish, the implicit function H = that constitutes the extended Hamiltonian of Eq. 6 must not be omitted in the integrand of the variational integral 7. A detailed discussion of this issue will be presented in Appendix B. Replacing H according to Eq. 6, we may express the extended set of canonical equations 8 in terms of the conventional Hamiltonian Hq, p, t dq i = dt H, p i dt = dt, dp i = dt H, q i dh = dt H t. 9a 9b The equations 9a are simply the conventional canonical equation 3 with s the independent variable instead of t. The last equation of Eqs. 9b states that the partial time derivative of H now constitutes a regular canonical equation the equation of motion for Hs. Yet, the conjugate equation of motion for ts merely constitutes an identity. The parameterization of the time t = ts remains thus undetermined. This means that the extended phase-space approach does not provide a substantial pair of additional canonical equations. As a consequence, the extended set of canonical equations is consistent with the principle of least action of Eq. for all differentiable parameterizations of time t = ts. Defining a priori ts = s as usually done in literature 3, 4, 6 is therefore a valid but useless parameterization that simply reduces the extended description to the conventional one. In contrast, we will see in the following section that it is exactly the freedom to appropriately adapt the parameterization of time t = ts in the extended phase space that allows us to generalize the conventional theory of canonical transformations. B. Canonical transformations The actual benefit of the extended phase-space formulation of the principle of least action is to lay the basis for more general canonical transformations, namely those that allow to relate two Hamiltonian systems H and H at different instants of their respective time scales, ts and t s, Hq, p, t canon. transf. H q, p, t. This contrasts with the conventional canonical transformation theory that is based on the specific variational integral of Eq. 2, where the time t plays by definition the role of the common independent variable of both the original and the destination system. Similar to the conventional theory, the transformation is referred to as canonical if and only if the variational principle 7 is maintained in the set of new primed canonical variables. The postulation thus states, explicitly s2 δ s s2 = δ s n n dq i p i H dt H q, p, t, H p dq i dt i H H q, p, t, H. This means that the integran of the variational integrals may differ at most by a total differential df of the extended phasespace variables q i, t, q i, t. We demand the canonical transformation to preserve the zero Hamiltonian H of Eq. 6 H q, p, t, H H q, p, t, H. Then, the condition for the variational principle to be maintained becomes n n p i dq i Hdt = p idq i H dt +df q, q, t, t. 2 The function F q, q, t, t is commonly called the generating function of the canonical transformation. Its total differential df writes df = n F dq i + F q i q i dq i + F t dt + F t dt. 3

4 4 Comparing Eq. 3 with Eq. 2, we obtain the transformation rules III. EXAMPLES OF CANONICAL TRANSFORMATIONS IN THE EXTENDED PHASE SPACE p i = F, p i = F q i q i, H = F t, H = F t. With the help of the Legendre transformation F 2 q, p, t, H = F q, q, t, t + n q ip i t H, 4 the generating function may be expressed equivalently in terms of the original configuration space and the new momentum coordinates. If we compare the coefficients pertaining to the respective differentials dq i, dp i, dt, and dh, we find the following coordinate transformation rules to apply for each index i =,..., n: p i = F 2, q i = F 2 q i p, H = F 2 i t, t = F 2 H. 5 An important feature of the extended transformation approach is that any conventional generating function f 2 q, p, t can always be reformulated as a particular generating function F 2 q, p, t, H within the extended phase space via F 2 q, p, t, H = f 2 q, p, t th. 6 The related transformation rules follow from Eqs. 5 as p i = f 2, q i = f 2 q i p, H = H + f 2 i t, t = t. The transformation rule for the Hamiltonians has been obtained by eliminating the extended phase-space variables H and H according to Eq. 5, respectively, hence by virtue of the projection into the conventional phase space. We note that this substitution is admissible only after having performed the partial differentiation of F 2 with respect to t. The conventional transformations generated by f 2 thus constitute the particular subset of general transformations generated by F 2, for which t = t. In other wor, the conventional canonical transformations distinguish themselves by the fact that the time t equally constitutes the independent variable of both the original and the destination system. This is the expected result recalling that the conventional canonical transformation theory is based on the specific variational integral of Eq. 2. According to the fourth rule of Eqs. 5, the extended functions F 2 generate non-trivial time transformations t t in general. As will become clear in the following example section, it is this freedom to relate a given system to a destination system at different instants of their respective time scales that enables us to formulate the Lorentz transformation as a particular canonical transformation in the extended phase space. We furthermore show that exclusively the generalized canonical transformation approach allows to directly transform an explicitly time-dependent Hamiltonian system into a time-independent one. A. Lorentz transformation We consider two Cartesian frames of reference x, y, z and x, y, z that move with respect to each other at constant velocity v. For simplicity, we first assume the coordinate axes to be aligned in the way that the relative motion occurs along the x-direction. Under these circumstances, the y- and z-coordinates are not affected by the Lorentz transformation y = y, z = z. As this transformation necessarily involves a non-trivial mapping of the respective time scales t t, it cannot be described in terms of a canonical transformation in the conventional phase space. Yet, in the extended phase space, a generating function F 2 exists that exactly produces the Lorentz transformation rules, F 2 x, p x, t, H = γ p xx βct H ct βx. 7 c With c denoting the speed of light, the common notation is used to abbreviate the scaled relative velocity by β = v/c. As usual, γ stan for the relativistic factor, defined by γ 2 = β 2. For the particular generating function 7 the general rules for a canonical transformation in the extended phase space of Eqs. 5 specialize to p x = F 2 x = γ x = F 2 p x H = F 2 t t = F 2 H = γ p x + β H, c = γ x βc t, = γ H + βc p x, t βc x. In complex notation, these transformation rules take on the familiar form of an orthogonal linear mapping x ict p x ih /c = γ iβγ iβγ γ γ iβγ iβγ γ x ict p x ih/c. 8 We observe that the generating function of Eq. 7 provides both the transformation rules for the x, ct coordinates which are commonly meant referring to the Lorentz transformation and at the same time the related rules for the conjugate coordinates momentum and energy p x, H/c. This is not astonishing as a canonical transformation always maintains the symplectic structure of the Hamiltonian in question which requires the transformation rules for all canonical variables to be uniquely defined. For the general case that both frames of reference are not aligned, their relative scaled velocity is expressed by the 3- component vector β = β i. With q = x, y, z = q i and

5 5 p = p x, p y, p z = p i, the general form of the generating function F 2 for the Lorentz transformation is then given by 3 F 2 q, p, t, H = γ H β i q i ct + c p i δ ik + γ β iβ k β 2 q k γc tβ i, k= which simplifies to Eq. 7 for the aligned case β = β x = β, β 2 = β y =, and β 3 = β z =. The particular class of covariant Hamiltonians, denoted by H L, is thus given by those Hamiltonians that are forminvariant under the canonical transformation generated by F 2 of Eq. 9. As a simple example how to cast a given non-lorentz-invariant Hamiltonian H NL into a Lorentz-invariant form, we consider the Hamiltonian of a free particle with constant mass m, H NL p = p2 2m. 2 As solely expressions of the form x 2 c 2 t 2 and p 2 H 2 /c 2 are invariant under the orthogonal transformation given by Eq. 8, the Hamiltonian 2 is obviously not Lorentzinvariant. In the extended phase space, however, a Lorentzinvariant form of Eq. 2 can easily be constructed by adding the required H-terms H L p, H = 2m p 2 H mc2 2 c 2 + mc 2. 2 Hereby, we chose the usual normalization to define H L = H = mc 2 for the particular case p =. The addition of the mc 2 terms merely describes a Lorentz invariant shift of the origin. According to Eq. 5, the Hamiltonian 2 is expressed in the conventional phase space replacing the Hamiltonian s value H by the Hamiltonian H L itself, H L p = 2m p 2 H L mc 2 2 c 2 + mc Solving Eq. 22 for H L, we find the well-known result H L p = p 2 c 2 + m 2 c 4 of the Lorentz-invariant form of the Hamiltonian of a free particle. B. Time-dependent harmonic oscillator As a simple non-trivial example, we shall demonstrate in the following that a system of n particles that is confined within a time-dependent harmonic oscillator potential can be mapped into a system with a time-independent harmonic potential by means of a single canonical transformation in the extended phase space. The Hamiltonian of original system is given by Hq, p, t = 2 p2 + 2 ω2 t q We demand a destination system H with t its independent variable of the same form, but with a potential that is independent of time t explicitly H q, p = 2 p ω2 q The function F 2 q, p, t, H that generates the mapping of the Hamiltonian 23 into the Hamiltonian 24 has been found to be F 2 q, p, t, H = q p ξt + ξt t 4ξt q2 H dτ ξτ, 25 with ξt a yet undetermined differentiable function of time. Because of the quadratic dependence on the canonical coordinates, the corresponding transformation rules 5 evaluate to the linear mapping q / ξ p = ξ/ 2 ξ ξ q. 26 p The transformations of time t and energy H between both systems emerge from the generating function 25 as t = t dτ ξτ, H = ξh 2 ξq p + 4 ξq We observe that the time shift transformation between both systems is determined by the yet unknown function ξt. The transformation of the Hamiltonians follows from the transformation rule for H of Eq. 27. Replacing H and H according to Eq. 5 by H and H, respectively, inserting the particular Hamiltonian of Eq. 23, and eliminating the unprimed variables according to Eq. 26, we find H q, p, t = 2 p q 2 2 ξ ξ 4 ξ 2 + ω 2 t ξ 2. Hence, the requested Hamiltonian 24 indeed arises, provided that we identify ω 2 = 2 ξ ξ 4 ξ 2 + ω 2 t ξ 2. The primed system s potential is not time-dependent exactly if dω 2 /dt =. This means, explicitly,... ξ t + 4 ξω 2 t + 4ξω ω =. 28 As a result of this requirement, the function ξt is now determined and hence the time correlation t t of both systems. With ξt a solution of Eq. 28, the value H of H is an integral of motion. Under this premise, the related transformation rule 27 depicts an integral of motion I of the original system 23 H = I = ξhq, p, t 2 ξ q p + 4 ξ q 2. 29

6 6 Defining ρ 2 t = ξt, the invariant 29 can be cast into the form of the Lewis invariant 7. Also, the explicit solution qt of the time-dependent harmonic oscillator 8 is directly obtained on the basis of the transformation rules 26 and 27. To this end, we must merely express the explicit solution q t of the ordinary harmonic oscillator 24 in terms of the coordinates q, p, and time t of the time-dependent system 23. For this linear system, we easily verify that a particular solution ξt of Eq. 28 is given by ξt = q 2 t, provided that qt represents a solution vector of the equation of motion for the time-dependent harmonic oscillator, q + ω 2 t q =. The function ξt is thereby attributed a direct physical meaning. Inserting ξt = q 2 t and its first and second time derivative into Eq. 29, the invariant I is rendered a function of q and p only I = q 2 p 2 q p 2 = 2 n p i q j q i p j 2. 3 i,j= We hereby observe that the invariant of the time-dependent harmonic oscillator has the form of a conservation law of the angular momentum in central force fiel. In the realm of ion optics, this invariant I for a time-dependent harmonic oscillator system of n particles is referred to as the root-meansquare-rms-emittance. The physical meaning of the two other fundamental solutions of the linear differential Eq. 28 will be discussed in Sec. IV. C. General time-dependent potential. Integrals of motion As a more demanding example, we now transform an n- degree-of-freedom Hamiltonian system with a general nonlinear time-dependent potential into a time-independent one. The Hamiltonian of the original system be given by Hq, p, t = 2 p2 + V q, t. 3 Again, we require a destination system H of the same form, but with a potential V that does not explicitly depend on the system s independent variable, t, H q, p = 2 p 2 + V q. 32 The most general extended generating function F 2 that retains both the quadratic momentum dependence of H, and a momentum-independent potential V turns out to be exactly the generating function of Eq. 25 of the previous example. Following Eq., an equivalent way to obtain the transformed Hamiltonian H is to simply express the extended Hamiltonian H q, p, t, H of Eq. 6 in terms of the primed variables. With the transformation rules of Eqs. 26 and 27, we find for the particular Hamiltonian of Eq. 3, considering that dt/ = ξ dt /, H = 2 p2 + V q, t H dt = 2 p q 2 ξ ξ 2 ξ 2 + ξ V ξ q, t H dt = 2 p 2 + V q, t H dt = H. Thus, a Hamiltonian H of the form of Eq. 32 comes out if we identify the transformed potential V with V q, t = 4 q 2 ξ ξ 2 ξ 2 + ξ V ξ q, t. 33 We can now make use of the freedom to modify the time correlation t t between the original system 3 and the destination system 32 in order to demand the new potential V to be independent of its time t explicitly V t! =. 34 By this requirement, we determine ξt, which was initially defined in the generating function 25 as a free differentiable function of time only. For the potential of Eq. 33, the condition 34 evaluates to... ξ q ξ V q, t + 2 q V q + 4ξ V t =. 35 The linear and homogeneous third-order equation 35 is equivalent to the linear system ξ d ξ ξ = ξ, 36 dt ξ g qt, t g 2 qt, t ξ with the coefficients g and g 2 defined by g qt, t = 4 V q 2 t, g 2 qt, t = 4 q 2 V q, t + 2 q V. q As by definition ξ = ξt embodies a function of t only, the coefficients g and g 2 must also be functions of time only in order for the system to be solvable. Thus, we can integrate the system 36 only if we henceforth conceive the vector of spatial coordinates q equally as a time function q = qt. This simply means that we must already know the trajectory q = qt as the solution of the equations of motion. We may therefore integrate Eq. 36 only in conjunction with the full set of canonical equations. With the set of canonical equations and Eq. 36 forming a closed coupled system, the functional dependence of Eq. 36 is well defined. Regarding the system matrix of Eq. 36, we observe that its trace is always zero. Hence independently of the particular form of the system s potential V q, t the Wronski determinant of any 3 3 solution matrix Ξt of Eq. 36 is always

7 7 constant. With the particular initial condition Ξ = E, E denoting the 3 3 unit matrix, we thus have ξ ξ 2 ξ 3 Ξt = ξ ξ ξ2 ξ2 ξ3, ξ3 det Ξt. 37 The transformation rule of Eq. 27 now provides an integral of motion I for the original system 3 exactly if ξt and its time derivatives represent a linear combination of the three linearly independent vectors of the solution matrix Ξt, H = I = ξt H 2 ξt q p + 4 ξt q 2 = const. 38 With the normalization Ξ = E, the three invariants, i.e. the three integration constants of the third order system 36 can be written in matrix form in terms of the transpose solution matrix Ξ T t, H ξ ξ ξ 2 q p = ξ 2 ξ2 ξ2 H 2 q p. 39 ξ 3 ξ3 ξ3 4 q2 4 q2 Due to the particular normalization Ξ = E, the invariants just represent the initial values of the Hamiltonian, H, and of the scalar products q p and q 2, as indicated by the subscript zero. For the general class of non-linear Hamiltonian systems 3, Eq. 39 thus expresses the remarkable result that the particular vector H, 2 q p, 4 q2 depen linearly on its initial state, and that this mapping is associated with a unit determinant. If the given system 3 is autonomous V/ t, then the linear equation 35 obviously possesses the particular solution ξ t. With regard to Eq. 39, this solution just represents the fact that the value of the Hamiltonian is a constant of motion H = H if H does not depend on time explicitly. This well-known feature of autonomous Hamiltonian systems thus appears in our analysis in a more global context. Particularly, we observe that two other invariants always exist for autonomous systems that are associated with the nonconstant solutions ξ 2 t and ξ 3 t. Following from the fact that the determinant of the solution matrix Ξt of the linear system 36 is always unity, we may cast Eq. 39 into its inverse form H 2 q p = Ξ T t H 2 q p 4 q2 t 4 q2 t= 4 with Ξ T t ξ 2 ξ3 ξ 2 ξ3 ξ ξ3 ξ ξ3 ξ ξ2 ξ ξ2 = ξ 2 ξ 3 ξ 2 ξ3 ξ ξ3 ξ ξ 3 ξ ξ 2 ξ ξ2. ξ 2 ξ3 ξ 2 ξ 3 ξ ξ 3 ξ ξ3 ξ ξ2 ξ ξ 2 Explicitly, this means that the system s actual energy Ht as well as its actual second moment q 2 t can be expressed as linear functions of the initial conditions H, q p, and q 2 Ht = ξ2 ξ3 ξ 2 ξ3 H + ξ ξ3 ξ ξ3 2 q p + ξ ξ2 ξ ξ2 4 q2, 4a 4 q2 t = ξ 2 ξ3 ξ 2 ξ 3 H + ξ ξ3 ξ ξ 3 2 q p + ξ ξ2 ξ ξ 2 4 q2. 4b As required, the equation for 2qpt turns out to be the time derivative of Eq. 4b. The complexity of the generally nonlinear mapping q, p qt, pt is thus transferred into a complexity of the time evolution of Ξt that defines the linear mapping of Eq. 4. An unstable time evolution of the ξ i -terms in the parentheses of Eqs. 4a and 4b can thus be related to an increasingly sensitive dependence of the actual state of the macroscopic quantities on their initial states. As the sensitivity to initial conditions is a characteristic feature of a chaotic motion, we may therefore deduce from the time evolution of Ξt whether the system as a whole evolves regularly or chaoticly. In the following section, we will discuss the physical meaning of the solutions of the linear third-order equation 35 on the basis of the previously addressed examples, the time-dependent harmonic oscillator and the general time-dependent potential. IV. DISCUSSION: PHYSICAL MEANING OF Ξt A. Time-dependent harmonic oscillator For the particular case of the time-dependent harmonic oscillator as treated in Sec. III B the third-order equation 28 for ξt possesses the particular property to be parametrically independent of qt. Then, the solution matrix Ξt does not depend on the specific choice of the initial condition q, p but solely on the external force function ω 2 t and its time derivative. With two arbitrary solutions ξ a t and ξ b t of Eq. 28, one fin under this precondition that the expression ξt = ξ a ξb ξ a ξ b is again solution of equation 28. As any solution ξt of the third-order Eq. 28 is uniquely determined by its three initial conditions ξ, ξ, and ξ, we easily verify the following relations for the fundamental solutions ξ t, ξ 2 t, and ξ 3 t that define the solution matrix 37 with Ξ = E ξ t = ξ ξ2 ξ ξ 2 + 4ω 2 ξ 3 t ξ 2 t = ξ ξ3 ξ ξ 3 ξ 3 t = ξ 2 ξ3 ξ 2 ξ 3. We conclude from Eq. 4b that the representation of the time evolution of the second moment q 2 t may be expressed in terms of the initial conditions q 2, q p and ω 2 = ω 2 as q 2 t = ξ t q 2 +2ξ 2 t q p +2ξ 3 t p 2 ω 2 q 2. 42

8 8 The second moment p 2 can be eliminated by means of the harmonic oscillator s invariant I of Eq. 3 q 2 t = ξ t q 2 + 2ξ 2 t q p + I + q 2ξ 3 t p 2 ω 2 q 2. q 2 42 We may now calculate the partial derivatives of q 2 t with respect to the initial conditions q 2 and 2q p, q 2 t q 2 = ξ t 4H q p q 2 ξ 3 t, q 2 t 2q p = ξ 2 t + 2q p ξ 3 t. q 2 For Hamiltonian systems 23, the sensitivity δq 2 t of the second moment q 2 t on a variation of the initial conditions q 2 and 2q p is thus given by δq 2 t = ξ t 4H q 2 q 2 ξ 3 t δq 2 + ξ 2 t + 2q p q 2 ξ 3 t δ2q p. 43 According to Eq. 43, an unstable time evolution of the ξ i t reflects an increasingly sensitive dependence of q 2 t on a variation of the initial conditions q p and q 2. The divergence of the terms in brackets thus directly indicates a chaotic evolution of the system as a whole. B. General potential For the general non-quadratic potential of Sec. III C, the function ξ t always establishes a particular solution of Eq. 35 if the potential does not explicitly depend on time, V = V q. Thus, for autonomous systems the expression det Ξt = simplifies to ξ 2 ξ3 ξ 2 ξ3 =. The partial derivatives of Eq. 4a then yield, as expected, Ht H q p,q 2 =, Ht q 2 =, H,q p Ht q p H,q 2 =. Since Eq. 35 is coupled to the system of canonical equations 3, at least the solutions ξ 2 t and ξ 3 t always depend on the particular trajectory qt which means that the solution matrix Ξt will be different for distinct initial conditions q, p and q, p. In order to calculate from Eq. 4b the variation of the actual macroscopic quantity q 2 t, it is then necessary to take into account additionally the variation of the ξ i t due to a variation of qt, δq 2 t = ξ 2 ξ3 ξ 2 ξ 3 4δH + 4H δ ξ 2 ξ3 ξ 2 ξ 3 + ξ ξ3 ξ ξ 3 ξ ξ3 ξ ξ 3 δ2qp +2qp δ + ξ ξ2 ξ ξ 2. ξ ξ2 ξ ξ 2 δq 2 + q 2 δ The δξ i t are given as solutions of the variational equation of Eq. 35 which on its part is coupled with the solution δqt of the variational equations of the canonical equations. In a simplified treatment, we may, however, set up the expression for the time evolution of the difference q 2 t q 2 t following from two adjacent initial conditions of identical macroscopic values H, q p, and q 2, q 2 t q 2 t = ξ2 ξ3 ξ2 ξ3 ξ 2 ξ3 + ξ 2 ξ 3 4H + ξ ξ3 ξ ξ3 ξ ξ3 + ξ ξ 3 2q p + ξ ξ2 ξ ξ2 ξ ξ2 + ξ ξ 2 q Thus, the time evolution of the separation of two initially identical macroscopic states of the systems can be expressed in terms of the separation of the respective solutions of Eq. 35. Following the usual understanding, we label a trajectory qt of a dynamical system chaotic if for arbitrary small initial distances q q < δ and appropriately small ε = εδ > we can always find an instant of time t > t such that q 2 t q 2 t ε. 45 This means that no matter how small we define a deviation in the initial conditions of the respective system, we always find a finite deviation of the system s state at some later instant of time. For a non-linear motion, the oscillations of the respective ξt-terms get rapidly out of phase. Consequently, the criterion of Eq. 45 is always satisfied if the solutions ξt of Eq. 35 are oscillatory unstable. The larger the related growth rates, the shorter are the time spans t t in order to satisfy Eq. 45. The growth rates of the ξ-terms may thus serve as an indicator for the system s degree of chaos. A numerical example will be presented in the following Sec. IV C. C. Numerical example: Hénon-Heiles oscillator In this section, we illustrate the physical meaning of the ξ,2,3 t-functions as the fundamental solutions of the linear system 35 by means of the well-studied Hénon-Heiles oscillator 4, 9. This oscillator models the motion within a twodimensional parabolic potential that is perturbed by a cubic potential term. With the perturbation being proportional to λ, its Hamiltonian can be written in normalized form as Hq, p = 2 p2 x + 2 p2 y + 2 x2 + 2 y2 +λ x 2 y 3 y3. 46 The subsequent canonical equations, together with the particular form of the third-order equation for ξt of Eq. 35 are given by ẍ + x + 2λxy =, ÿ + y + λx 2 y 2 = 47a... ξ + 4kt ξ =, kt = λ y 3x2 y 2 x 2 t + y 2 t. 47b As the system does not explicitly depend on t, a particular solution of Eq. 47b is given by ξ t =. According to

9 9.5 ξ2t Eq. 39, this solution represents the energy conservation law Ht = H for this autonomous dynamical system. Setting λ = and fixing the system s dimensionless energy to the limiting value of H = /6 for a bounded motion, we obtain the Poincare surface-of-section of Fig. for the initial condition x, px,, y, py, =,.5367,.2,. The points x =, px > py y FIG. : y, py -Poincare surface-of-section representation of an irregular trajectory in the He non-heiles oscillator 46 with the limiting energy H = /6 for the initial condition x, px,, y, py, =,.5367,.2, and λ =. display the y, py -coordinates of the particle whenever its x, px -coordinates satisfy the conditions x = and px > along the time axis. Figure shows that almost the entire available phase-space area-of-section is covered in the course of the trajectory s time evolution which means that this particular trajectory is chaotic. As each of these points can be considered on its part as an initial condition, we conclude that almost all initial conditions give rise to chaotic orbits. In to the disjunct volume of phase space where the motion is regular. This regular motion is displayed in Fig. 2 in the form of a real space trajectory. It is obtained by choosing the particle s initial conditions to lie within the blank islan of Fig.. We observe in Fig. 2 that the particle crosses the vertical line x = exactly four times in the course of one oscillation period. These four locations correspond to the four blank islan occurring in Fig.. According to Eq. 44, the variation of the non-constant solutions ξ2 t and ξ3 t of Eq. 47b determine the time evolution of the sensitivity of q 2 t x2 t + y 2 t on the system s initial conditions. As an example, the solution ξ 2 t ξ2t.4.2 y 8 FIG. 3: ξ 2 t as the solution of Eq. 47b for the regular trajectory of Fig. 2 as obtained for the initial values ξ2, ξ 2, ξ 2 =,, and λ = t t x FIG. 2: Real space projection of a regular trajectory in the He nonheiles oscillator 46 with the limiting energy H = /6 as obtained for the initial condition x, px,, y, py, =,.3765,.55, and λ =. contrast, the blank areas within the dotted region correspon FIG. 4: ξ 2 t as the solution of Eq. 47b for the irregular trajectory of Fig. as obtained for the initial values ξ2, ξ 2, ξ 2 =,, and λ =. of Eq. 47b with the initial condition ξ2, ξ 2, ξ 2 =,, for the regular particle motion xt, yt is displayed in Fig. 3. For this particular case, the solution functions ξ2 t and ξ3 t turn out to be strictly periodic. In other

10 wor, the periodic motion of the particle is reflected by a stable oscillatory behavior of ξ 2,3 t. This contrasts with the solutions ξ 2 t and ξ 3 t that emerge for the irregular particle motion arising from the initial condition x, p x,, y, p y, =,.5367,.2,. We observe in Fig. 4 that the irregular motion expresses itself as an unstable oscillation of ξ 2 t. A similar behavior is observed for ξ 3 t. Figure 5 displays in the form of a Lyapunov function Lt, defined by Lt = ln ξ2 t ξ 2 t t, 48 the time evolution of the difference ξ 2 t = ξ2 t ξ 2 t that is associated with two distinct trajectories xt, yt and xt, ȳt. An oscillatory unstable behavior of the fundamental solutions ξ 2,3 t thus reflects the fact that the system s actual state depen increasingly sensitive on its initial state. This suggests to use the growth rates of δξ 2,3 t as a global measure for the irregularity of dynamical systems. Lt FIG. 5: Lyapunov function Lt as defined by Eq. 48 for two solutions of Eq. 47b for two initially adjacent irregular trajectories with the limiting energy H = /6 and λ =. V. CONCLUSIONS On the basis of the fundamental Poincaré-Cartan -form p dq Hdt, we have worked out the variational principle in the form of Eq. 7 that generalizes its conventional formulation of Eq. 2. The idea underlying this derivation is to conceive the system s time t together with the negative value H of the Hamiltonian H as an additional pair of canonically conjugate variables. The distinction of the Hamilton function H from its value H is hereby crucial. Only then, the extended Hamiltonian H of Eq. 6 does not vanish identically but is correctly obtained as an implicit function H = of the extended phase-space variables. We hereby avoid the flaw of various textbooks in which the extended phase-space approach is asserted to yield a variational principle without any Hamiltonian. t The generalized variational integral of Eq. 7 lays the groundwork for both the extended set of canonical Eqs. 9, and the extended set of canonical transformation rules 5. A formal advantage of the extended phase-space formulation is that the partial time derivative of the Hamiltonian H now also establishes a canonical equation. On the other hand, the conjugate equation merely yiel an identity. This implies that the additional pair of canonical equations emerging from the generalized variational integral does not provide additional information on the system s dynamics. In contrast, the extended canonical transformation rules indeed establish a generalization of the conventional ones the latter being restricted to the identical time transformation t t. By lifting this restriction, the generalized transformation approach allows canonical mappings that relate two Hamiltonian systems at different instants of their respective time scales. It is this feature that constitutes the main benefit of the extended phase-space approach. As the first example, we have demonstrated that the Lorentz transformation embodies as a particular canonical transformation in the extended phase space. In the form of a canonical mapping, the conditions for a Hamiltonian to be Lorentzinvariant are clearly exposed. As shown for a free particle, we thus obtain a guideline how to convert non-lorentz-invariant Hamiltonians into Lorentz-invariant ones. Furthermore, the generalized understanding of canonical transformations allows a direct mapping of Hamiltonian systems with explicitly time-dependent potentials into timeindependent ones. For both the time-dependent harmonic oscillator and for a general time-dependent potential, an extended generating function that defines such a canonical mapping has been presented. This generating function turned out to depend on a characteristic time function ξt that is given as a solution of a linear third-order equation. For the case of an autonomous system, a particular solution was found to be given by ξ t =. This solution could be related to the energy conservation law Ht = H that applies for this class of systems. We thus observe that the canonical transformation approach generalizes this well-known relationship. The set of fundamental solutions of this third-order equation for ξt indeed provides additional information on the system s dynamics. We have shown that the fundamental solutions ξ,2,3 t describe the sensitivity of the macroscopic system quantities energy Ht and second moments qpt, q 2 t on their respective initial states regardless of the system s number of degrees of freedom. An oscillatory unstable time evolution of the difference of the ξt-functions for two initially adjacent phase-space trajectories could be related to a chaotic evolution of the system as a whole. This way, the test for regularity of dynamical systems can be reduced to the stability analysis of a linear third-order differential equation with time-dependent coefficients. Eventual growth rates of the δξt may then serve as a measure for the non-regularity of the given dynamical system. Especially the investigation of nonlinear and explicitly time-dependent many-body systems where Poincaré surfaces-of-section make no sense anymore will benefit from this global description. This opens a new access to the problem of analyzing the dynamical properties

11 of such systems. APPENDIX A: LAGRANGIAN MECHANICS IN THE EXTENDED PHASE SPACE. Extended Euler-Lagrange equations By means of a Legendre transformation, the extended Hamiltonian of Eq. 6 can be translated into the corresponding extended Lagrangian L, L q, dq dt, t, = p dq H dt H q, p, t, H. Inserting H from Eq. 6, we find that L is related to the conventional Lagrangian Lq, q, t by L q, dq dt, t, = L q, dq/ dt dt/, t. A This result agrees with earlier ones 6,, which confirms that the extended phase-space Hamiltonian H = must exactly possess the form of Eq. 6. As expected, L is consistent with the integrand of the variational integral of Eq. 4. We may write this integral more concisely in terms of the extended configuration space vector q, defined by q q =. A2 t The principle of least action in the extended phase-space formulation of Eq. 7 now takes on the short form s δ L q s, dq s =. s A3 Similar to the conventional variational problem with a Lagrangian Lq, q, we find that Eq. A3 is globally fulfilled if the extended set of Euler-Lagrange equations is satisfied, L d L =. A4 q dq / In order to rewrite Eq. A4 in terms of the conventional Lagrangian L, the following identities are readily derived from Eq. A L q = dt L q, L = dt L t t, L dq/ = L q, L L = L q dt/ q. A5a A5b The L / q-related part of the extended set of Euler-Lagrange equations A4 thus follows as dts L q d dt L =. A6 q The L / t- related part of Eq. A4 yiel dqs L q d L =. A7 dt q We observe that both Eq. A6 as well as Eq. A7 are globally fulfilled if and only if the equations in brackets the conventional Euler-Lagrange equations are satisfied. Thus, the extended set Euler-Lagrange equations A4 is equivalent to the conventional set and does not provide an additional equation of motion for t = ts. This result correspon to the observation of Eq. 9 that the extended set of canonical equations does not furnish a substantial canonical equation for dt/. Nevertheless, we may take advantage of the extended formulation of Lagrangian mechanics. Statements that have been derived on the basis of a time-independent Lagrangian Lq, q can straightforwardly be reformulated for the extended Lagrangian L q, dq /. A subsequent translation into a conventional time-dependent Lagrangian Lq, q, t then provides the generalization of the respective statement for timedependent Lagrangian systems. We will demonstrate this technique in the following section on the basis of Noether s theorem. 2. Noether s theorem Noether s theorem relates the symmetries of a Lagrangian system to its conserved quantities. It is easily proved for time-independent Lagrangians Lq, q. Rewriting the theorem in terms of the extended phase-space notation, we easily obtain its generalized formulation that applies for explicitly time-dependent Lagrangian systems. Let h ɛ be an ɛ-dependent transformation of the configuration space vector q into a new vector q, h ɛ : qt ɛ q t. A8 A simple example of such a transformation would be a rotation of the configuration space by an angle ɛ. The transformed velocity vector q t immediately follows as q t = d dt hɛ qt. As the Lagrangian L depen on q and its time derivative, q, the transformation of L by virtue of the prescription of Eq. A8 is fully determined. For this reason, Noether s theorem is most painlessly formulated in the context of Lagrange s formulation of mechanics. Noether s theorem now states that the quantity Iq, q = L q dh ɛ q dɛ + fq ɛ= A9 constitutes an integral of motion, provided that h ɛ maintains the Lagrangian Lq, q up to the total time derivative of a function fq. For the particular case of a time-independent Lagrangian, the proof is simple. The precondition that the ɛ-dependent

12 2 transformation A8 is supposed to maintain L up to an expression df/dt just means dlq, q dɛ + dfq dt =. Writing dl/dɛ explicitly, we get inserting the Euler-Lagrange equations dlq, q = d L dh ɛ q dɛ dt q. dɛ Together with df/dt, we immediately end up with the total time derivative of Eq. A9, performing the transition ɛ. In order to generalize the result of Eq. A9 for explicitly time-dependent Lagrangians, we formally rewrite it in terms of the extended phase-space notation. With L the extended Lagrangian of Eq. A, q the extended configuration space vector A2, and h ɛ an extended mapping similar to Eq. A8, h ɛ h ɛ = h ɛ t, h ɛ : q s ɛ q s, A the generalized form of Eq. A9 is obtained as I q, dq L dh ɛ = q dq / dɛ + fq = const. ɛ= A By virtue of the identities A5b and L q = p, Lq, q, t q p = Hq, p, t, the Noether invariant of Eq. A can be expressed in terms of the conventional time-dependent Hamiltonian Hq, p, t as Iq, p, t = p dhɛ q, t dɛ H dhɛ tt ɛ= dɛ + fq, t. ɛ= A2 This is the desired generalization of Eq. A9 in the sense that Eq. A2 now covers explicitly time-dependent Hamilton- Lagrange systems. The quantity Iq, p, t is thus an integral of motion provided that the mapping of Eq. A maintains the Euler-Lagrange equations in the primed coordinates L q d L L =, dt q q d L dt dq /dt =, which means that L q, dq /dt, t dt = L q, q, t dt + dfq, t. A3 For the particular case of an infinitesimal mapping A, we may restrict ourselves to linear functions in ɛ q = h ɛ q, t = q ɛ η q, t, t = h ɛ tt = t ɛξt. A4 Then Noether s theorem in the form of Eq. A2 states that the function Iq, p, t = ξt Hq, p, t ηq, t p + fq, t A5 is an integral of motion if the functions ξt and ηq, t that define the transformation A4 satisfy the condition A3. For a given Lagrangian Lq, q, t or, equivalently, for a given Hamiltonian Hq, p, t the functions ξt and ηq, t that preserve the Euler-Lagrange equations, and thus furnish the invariant I, are commonly not known. Therefore, the scheme must be carried out reversely: we require di/dt = and subsequently determine the coefficients ξt and ηq, t of the symmetry transformation A4. This means to construct the functions Iqt, pt, t that are invariant on the system path qt, pt for any given Hamilton-Lagrange system. We shall demonstrate the scheme in the following section on the basis of the Hamiltonian system 3 of Sec. III C. 3. General time-dependent potential From a general viewpoint, we refer to a quantity Iq, p, t as an integral of motion if its total time derivative di/dt vanishes along the system trajectory. For the Noether function of Eq. A5, the explicit representation of the requirement di/dt = yiel a hierarchy of three partial differential equations. With the Hamiltonian 3, Hq, p, t = 2 p2 + V q, t, the hierarchy writes, ordered by the respective powers in the canonical momenta 2, p 2 : p i p ξδ j 2 ij η i =, q i j j f p : p i η i =, q i i t p V : η i + q ξv q, t + ξ V i t + f t =. i On the basis of the first and the second equation, we find the solutions ηq, t = 2 ξt q, fq, t = 4 ξt q 2. A6 Hence, the invariant A5 can be expressed in terms of a single yet unknown function, ξt Iq, p, t = ξt Hq, p, t 2 ξt q p + 4 ξt q We observe that this invariant coincides with H of Eq. 38, which has been obtained as a result of a canonical transformation generated by Eq. 25. The conditional equation for ξt follows from the third equation of the above hierarchy, inserting the functions ηq, t and fq, t of Eq. A6. We thus find the following linear equation for ξt... ξ q ξ V q, t + 2 q V q + 4ξ V t =, A7 which agrees with Eq. 35 of the canonical transformation approach. We conclude that the canonical transformation in