Hamiltonian mechanics in the “extended” phase spacestruck/hp/antritt/extphsp.pdf · 2005. 10....

Hamiltonian mechanics in the“extended” phase space

Jurgen Struckmeier

[email protected]

GSI Accelerator Seminar

Darmstadt, 03 July 2003

Extended phase space – p. 1

Motivation• Given task: investigate the dynamical properties of an

explicitly time-dependent Hamiltonian system.




• Simple example: time-dependent harmonic oscillator

H(q, p, t) = 1

2p2 + 1

2ω2(t) q2 .





H(q, p, t) = 1

2p2 + 1

2ω2(t) q2 .

• Solution strategy: canonical mapping into thetime-independent harmonic oscillator H ′(q′, p′),back transformation of its solution into the originaltime-dependent system. H ′(q, p, t) is an invariant!





H(q, p, t) = 1

2p2 + 1

2ω2(t) q2 .


• Problem: a conventional generating function f2(q, p′, t)

that defines such a mapping does not exist.





H(q, p, t) = 1

2p2 + 1

2ω2(t) q2 .


• Problem: a conventional generating function f2(q, p′, t)

that defines such a mapping does not exist.

• Necessary: (re-)formulation of the canonicaltransformation theory in the “extended” phase space.


Outline

• Principle of least action and its general formulation

• Extended form of the canonical equations

• Canonical transformations in the extended phasespace

• Example 1: time-dependent harmonic oscillator

• Example 2: general time-dependent potential

• Conclusions and outlook


Principle of least action

Given: dynamical system of n degrees of freedom with ~q and~p the n-dimensional vectors of generalized coordinates.




A path γ in the 2n dimensional phase space with the time t

the independent variable is defined by

γ :{(

~q, ~p)

∈ R2n∣

∣ ~q = ~q(t), ~p = ~p(t), t0 ≤ t ≤ t1}

.




A path γ in the 2n dimensional phase space with the time t

the independent variable is defined by

γ :{(

~q, ~p)

∈ R2n∣

∣ ~q = ~q(t), ~p = ~p(t), t0 ≤ t ≤ t1}

.

We formulate the principle of least action via a functional Φ,hence with a mapping of the set of paths γ into R

Φ(γ) =

∫

t1

t0

[

~p(t)d~q(t)

dt− H

(

~q(t), ~p(t), t)

]

dt .

The function H : R2n × R → R denotes the Hamiltonian.


��

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Water

vw

vsSand

Watch Bay

Example of a functional Φ : γ 7→ ∆t ∈ R.


Principle of least action (Leibnitz, Maupertuis, Euler,Lagrange):

Among all thinkable paths γ, a dynamical system “chooses”exactly that oneγext, whereΦ(γext) takes on a minimum.




In the picture of the rescuer, the system takes always theoptimum path.





Max Planck: The principle applies for all reversiblephenomena of physics!





Max Planck: The principle applies for all reversiblephenomena of physics!

Calculus of variations: the functional Φ(γ) takes on aminimum (δΦ(γ) = 0), exactly if the phase-space path(

~q(t), ~p(t))

satisfies the “canonical equations”

d~q

dt=

∂H

∂~p,

d~p

dt= −∂H

∂~q.


Let us look back to the variational problem δΦ(γ)!= 0

δΦ(γ) = δ

∫

t1

t0

[

~p(t)d~q(t)

dt− H

(

~q(t), ~p(t), t)

]

dt!= 0 .

We observe: the time t plays a twofold role, namely that ofthe formal integration variable and that of an externalparameter in the argument list of the Hamiltonian.



δΦ(γ) = δ

∫

t1

t0

[

~p(t)d~q(t)

dt− H

(

~q(t), ~p(t), t)

]

dt!= 0 .


Calculating the variation δΦ(γ), the time t is not varied.

; Not the most general formulation of the principle ofleast action!



δΦ(γ) = δ

∫

t1

t0

[

~p(t)d~q(t)

dt− H

(

~q(t), ~p(t), t)

]

dt!= 0 .


Calculating the variation δΦ(γ), the time t is not varied.

; Not the most general formulation of the principle ofleast action!

; We must separate the explicit t-dependence of theHamiltonian from the formal integration variable.


A more general form of the variational problem is obtainedsubstituting t = t(s), with s the new integration variable

Φ(γ) =

∫

s1

s0

[

~p(s)d~q(s)

ds− H

(

~q(s), ~p(s), t(s))dt(s)

ds

]

ds .



Φ(γ) =

∫

s1

s0

[

~p(s)d~q(s)

ds− H

(

~q(s), ~p(s), t(s))dt(s)

ds

]

ds .

The symmetric form of the integrand suggests to define the2n + 2 dimensional “extended” phase space, introducing

qn+1 = t , pn+1 = −H

as additional s-dependent phase-space variables.



Φ(γ) =

∫

s1

s0

[

~p(s)d~q(s)

ds− H

(

~q(s), ~p(s), t(s))dt(s)

ds

]

ds .

The symmetric form of the integrand suggests to define the2n + 2 dimensional “extended” phase space, introducing

qn+1 = t , pn+1 = −H

as additional s-dependent phase-space variables.

; H = H(s) ∈ R must be understood as the value of theHamiltonian H(~q, ~p, t), hence as the system’s“instantaneous energy”

H(s) = H(~q(s), ~p(s), t(s)) .


Defining the extended vectors ~q1 = (~q, t) and ~p1 = (~p,−H),the variational integral can be cast into the familiar form

δ

∫

s1

s0

[

~p1(s)d~q1(s)

ds− H1

(

~q1(s), ~p1(s))

]

ds!= 0 ,

with the extended Hamiltonian H1 = 0 as the implicit function

H1

(

~q1, ~p1

)

≡[

H(~q, ~p, t) −H] dt

ds= 0 .



δ

∫

s1

s0

[

~p1(s)d~q1(s)

ds− H1

(

~q1(s), ~p1(s))

]

ds!= 0 ,


H1

(

~q1, ~p1

)

≡[

H(~q, ~p, t) −H] dt

ds= 0 .

We observe: the extended functional has exactly the formof the conventional functional.



δ

∫

s1

s0

[

~p1(s)d~q1(s)

ds− H1

(

~q1(s), ~p1(s))

]

ds!= 0 ,


H1

(

~q1, ~p1

)

≡[

H(~q, ~p, t) −H] dt

ds= 0 .

We observe: the extended functional has exactly the formof the conventional functional.

; The variation of the functional vanishes again if theextended phase-space path

(

~q1(s), ~p1(s))

satisfies theextended set of canonical equations

d~q1

ds=

∂H1

∂~p1

,d~p1

ds= −∂H1

∂~q1

.


In terms of the conv. quantities ~q, ~p, t, H and H, this means

d~q

ds=

∂H1

∂~p=

dt

ds

∂H

∂~p,

d~p

ds= −∂H1

∂~q= − dt

ds

∂H

∂~q,

dt

ds= −∂H1

∂H =dt

ds,

dHds

=∂H1

∂t=

dt

ds

∂H

∂t.



d~q

ds=

∂H1

∂~p=

dt

ds

∂H

∂~p,

d~p

ds= −∂H1

∂~q= − dt

ds

∂H

∂~q,

dt

ds= −∂H1

∂H =dt

ds,

dHds

=∂H1

∂t=

dt

ds

∂H

∂t.

• The partial time derivative of H now induces an“ordinary” canonical equation.



d~q

ds=

∂H1

∂~p=

dt

ds

∂H

∂~p,

d~p

ds= −∂H1

∂~q= − dt

ds

∂H

∂~q,

dt

ds= −∂H1

∂H =dt

ds,

dHds

=∂H1

∂t=

dt

ds

∂H

∂t.


• The conjugate canonical equation constitutes only anidentity. ; The parameterization of time t = t(s)

remains undetermined.



d~q

ds=

∂H1

∂~p=

dt

ds

∂H

∂~p,

d~p

ds= −∂H1

∂~q= − dt

ds

∂H

∂~q,

dt

ds= −∂H1

∂H =dt

ds,

dHds

=∂H1

∂t=

dt

ds

∂H

∂t.




• The principle of least action is equally satisfied forall differentiable parameterizations of time t = t(s).



d~q

ds=

∂H1

∂~p=

dt

ds

∂H

∂~p,

d~p

ds= −∂H1

∂~q= − dt

ds

∂H

∂~q,

dt

ds= −∂H1

∂H =dt

ds,

dHds

=∂H1

∂t=

dt

ds

∂H

∂t.




• The principle of least action is equally satisfied forall differentiable parameterizations of time t = t(s).

• Exactly this freedom to appropriately adapt t = t(s)

allows to define more general canonical transformationsin the extended phase space.


Canonical transformationsGeneral condition for canonical transformations:

The variational principle must be maintained.

This means in the conventional description

δ

∫

t1

t0

[

~p ~q − H(

~q, ~p, t)

]

dt = δ

∫

t1

t0

[

~p ′~q ′ − H ′(

~q ′, ~p ′, t)

]

dt .





δ

∫

t1

t0

[

~p ~q − H(

~q, ~p, t)

]

dt = δ

∫

t1

t0

[

~p ′~q ′ − H ′(

~q ′, ~p ′, t)

]

dt .

; The time t is the common independent variable of boththe original system H and the destination system H ′.





δ

∫

t1

t0

[

~p ~q − H(

~q, ~p, t)

]

dt = δ

∫

t1

t0

[

~p ′~q ′ − H ′(

~q ′, ~p ′, t)

]

dt .


; Canonical transformations that correlate two systems onthe basis of their own time scales t, t′ are not possible.





δ

∫

t1

t0

[

~p ~q − H(

~q, ~p, t)

]

dt = δ

∫

t1

t0

[

~p ′~q ′ − H ′(

~q ′, ~p ′, t)

]

dt .


; Canonical transformations that correlate two systems onthe basis of their own time scales t, t′ are not possible.

; Only a CT in the extended phase space can do that job

H(~q, ~p, t)CT in the extended PhSp−−−−−−−−−−−−−−→ H ′(~q ′, ~p ′, t′) .


The general condition for transformations to be canonicalwrites analogously in the extended phase-space description

δ

∫

s2

s1

[

~p1

d~q1

ds− H1

(

~q1, ~p1

)

]

ds = δ

∫

s2

s1

[

~p ′

1

d~q ′

1

ds− H ′

1

(

~q ′

1, ~p′

1

)

]

ds .

; The integrands may differ at most by the totaldifferential dF1 of a function F1(~q1, ~q

′

1).


The general condition for transformations to be canonicalwrites analogously in the extended phase-space description

δ

∫

s2

s1

[

~p1

d~q1

ds− H1

(

~q1, ~p1

)

]

ds = δ

∫

s2

s1

[

~p ′

1

d~q ′

1

ds− H ′

1

(

~q ′

1, ~p′

1

)

]

ds .

; The integrands may differ at most by the totaldifferential dF1 of a function F1(~q1, ~q

′

1).

If we require the extended Hamiltonian H1 to be conserved

H1(~q1, ~p1) ≡ H ′

1(~q′

1, ~p′

1) ,

this yields the general condition

~p1 d~q1 = ~p ′

1 d~q ′

1 + dF1

(

~q1, ~q′

1

)

,

withdF1 =

∂F1

∂~q1

d~q1 +∂F1

∂~q ′

1

d~q ′

1 .


Comparing the coefficients, we find the transformation rules

~p1 =∂F1

∂~q1

, ~p ′

1 = −∂F1

∂~q ′

1

,



~p1 =∂F1

∂~q1

, ~p ′

1 = −∂F1

∂~q ′

1

,

or, equivalently, in terms of the quantities ~q, ~p, t and H

~p =∂F1

∂~q, ~p ′ = −∂F1

∂~q ′, H = −∂F1

∂t, H′ =

∂F1

∂t′.

Accordingly, we refer to F1(~q, t, ~q′, t′) as the generating

function of the extended canonical transformation.



~p1 =∂F1

∂~q1

, ~p ′

1 = −∂F1

∂~q ′

1

,

or, equivalently, in terms of the quantities ~q, ~p, t and H

~p =∂F1

∂~q, ~p ′ = −∂F1

∂~q ′, H = −∂F1

∂t, H′ =

∂F1

∂t′.

Accordingly, we refer to F1(~q, t, ~q′, t′) as the generating

function of the extended canonical transformation.

By means of a Legendre transformation

F2

(

~q1, ~p′

1

)

= F1

(

~q1, ~q′

1

)

+ ~q ′

1 ~p ′

1 ,

the generating function F1 can be converted into agenerating function of type F2.


The transformation rules associated with F2(~q, ~p′, t,H′) are

~p =∂F2

∂~q, ~q ′ =

∂F2

∂~p ′, H = − ∂F2

∂t, t′ = − ∂F2

∂H′.



~p =∂F2

∂~q, ~q ′ =

∂F2

∂~p ′, H = − ∂F2

∂t, t′ = − ∂F2

∂H′.

The “conventional” CTs, generated by f2(~q, ~p′, t), constitute

a subset of CTs in the extended phase space. Defining

F2(~q, ~p′, t,H′) = f2(~q, ~p

′, t) − tH′ ,

we find the well-known conventional transformation rules

~p =∂f2

∂~q, ~q ′ =

∂f2

∂~p ′, H ′ = H +

∂f2

∂t, t′ = t ,

substituting finally H = H and H′ = H ′.



~p =∂F2

∂~q, ~q ′ =

∂F2

∂~p ′, H = − ∂F2

∂t, t′ = − ∂F2

∂H′.

The “conventional” CTs, generated by f2(~q, ~p′, t), constitute

a subset of CTs in the extended phase space. Defining

F2(~q, ~p′, t,H′) = f2(~q, ~p

′, t) − tH′ ,

we find the well-known conventional transformation rules

~p =∂f2

∂~q, ~q ′ =

∂f2

∂~p ′, H ′ = H +

∂f2

∂t, t′ = t ,

substituting finally H = H and H′ = H ′.

; The extended transformation rules allow more generalrelations of H ↔ H ′ and t ↔ t′ than the conventional ones.


Example 1: harmonic oscillator

We consider the time-dependent 1-D Hamiltonian system

H(q, p, t) = 1

2p2 + 1

2ω2(t) q2 .




H(q, p, t) = 1

2p2 + 1

2ω2(t) q2 .

We want to transform this system into atime-independent system of the same form

H ′(q′, p′) = 1

2p′ 2 + 1

2ω2

0 q′ 2 .




H(q, p, t) = 1

2p2 + 1

2ω2(t) q2 .

We want to transform this system into atime-independent system of the same form

H ′(q′, p′) = 1

2p′ 2 + 1

2ω2

0 q′ 2 .

The generating function F2 that does the job has beenfound to be

F2

(

q, p′, t,H′)

=q p′√

ξ(t)+

ξ(t)

4ξ(t)q2 −H′

∫

t

0

dτ

ξ(τ).

ξ(t) denotes a yet undetermined differentiable function of time.Extended phase space – p. 15

For this particular F2, the transformation rules follow as(

q′

p′

)

=

(

1/√

ξ 0

−1

2ξ/√

ξ√

ξ

)(

q

p

)

,

t′ =

∫

t

0

dτ

ξ(τ), H′ = ξ H− 1

2ξ q p + 1

4ξ q 2 .



q′

p′

)

=

(

1/√

ξ 0

−1

2ξ/√

ξ√

ξ

)(

q

p

)

,

t′ =

∫

t

0

dτ

ξ(τ), H′ = ξ H− 1

2ξ q p + 1

4ξ q 2 .

Replacing H′ and H by H ′ and H, and eliminating theunprimed variables, the requested Hamiltonian H ′ emerges

H ′(q′, p′) = 1

2p′ 2 + ω2

0 q′ 2 ,

withω2

0 = 1

2ξξ − 1

4ξ2 + ω2(t) ξ2 .

Due to ω20

!= const., the function ξ(t) is now determined.



q′

p′

)

=

(

1/√

ξ 0

−1

2ξ/√

ξ√

ξ

)(

q

p

)

,

t′ =

∫

t

0

dτ

ξ(τ), H′ = ξ H− 1

2ξ q p + 1

4ξ q 2 .

Replacing H′ and H by H ′ and H, and eliminating theunprimed variables, the requested Hamiltonian H ′ emerges

H ′(q′, p′) = 1

2p′ 2 + ω2

0 q′ 2 ,

withω2

0 = 1

2ξξ − 1

4ξ2 + ω2(t) ξ2 .

Due to ω20

!= const., the function ξ(t) is now determined.

With this ξ(t), the value H′ of H ′ embodies an invariant I

H ′ ≡ I(q, p, t) = ξH − 1

2ξ q p + 1

4ξ q2 .


Question: what is the physical meaning of ξ(t)? We easilyverify that

ξ(t) = q 2(t)

satisfies 1

2ξξ − 1

4ξ2 + ω2(t) ξ2 = const., provided that q(t) is

a solution of the equation of motion of the time-dependentharmonic oscillator

q + ω2(t) q = 0 .



ξ(t) = q 2(t)

satisfies 1

2ξξ − 1



q + ω2(t) q = 0 .

With ξ(t) = q 2(t), q(t) denoting a second solution of theequation of motion, the invariant takes on the form

I = 1

2(p q − q p)2 .



ξ(t) = q 2(t)

satisfies 1

2ξξ − 1



q + ω2(t) q = 0 .

With ξ(t) = q 2(t), q(t) denoting a second solution of theequation of motion, the invariant takes on the form

I = 1

2(p q − q p)2 .

; The invariant of the time-dependent harmonic oscillatorhas the form of a conservation law of the angularmomentum in central force fields.

; Accelerator physics: I is referred to as “rms emittance”.Extended phase space – p. 17

Example 2: time-dependent potential

We now consider the general n-dimensional non-lineartime-dependent Hamiltonian system

H(~q, ~p, t) = 1

2~p 2 + V (~q, t) .

Again, we want to transform it into a time-independentHamiltonian system of the same form

H ′(~q ′, ~p ′) = 1

2~p ′ 2 + V ′(~q ′) = I .


Example 2: time-dependent potential

We now consider the general n-dimensional non-lineartime-dependent Hamiltonian system

H(~q, ~p, t) = 1

2~p 2 + V (~q, t) .

Again, we want to transform it into a time-independentHamiltonian system of the same form

H ′(~q ′, ~p ′) = 1

2~p ′ 2 + V ′(~q ′) = I .

The most general function F2 generating thetransformation that maintains the form of H is given by

F2

(

~q, ~p ′, t,H′)

=~q ~p ′

√

ξ(t)+

ξ(t)

4ξ(t)~q 2 −H′

∫

t

0

dτ

ξ(τ).


The subsequent transformation rules are(

~q ′

~p ′

)

=

(

1/√

ξ 0

−1

2ξ/√

ξ√

ξ

)(

~q

~p

)

,

t′ =

∫

t

0

dτ

ξ(τ), H′ = ξH− 1

2ξ~q ~p + 1

4ξ~q 2 .



~q ′

~p ′

)

=

(

1/√

ξ 0

−1

2ξ/√

ξ√

ξ

)(

~q

~p

)

,

t′ =

∫

t

0

dτ

ξ(τ), H′ = ξH− 1

2ξ~q ~p + 1

4ξ~q 2 .

Replacing H′ and H by H ′ and H, and eliminating theunprimed variables, the new potential V ′ evaluates to

V ′(

~q ′, t′)

= 1

4~q ′ 2

(

ξξ − 1

2ξ2

)

+ ξ V(√

ξ ~q ′, t)

.



~q ′

~p ′

)

=

(

1/√

ξ 0

−1

2ξ/√

ξ√

ξ

)(

~q

~p

)

,

t′ =

∫

t

0

dτ

ξ(τ), H′ = ξH− 1

2ξ~q ~p + 1

4ξ~q 2 .

Replacing H′ and H by H ′ and H, and eliminating theunprimed variables, the new potential V ′ evaluates to

V ′(

~q ′, t′)

= 1

4~q ′ 2

(

ξξ − 1

2ξ2

)

+ ξ V(√

ξ ~q ′, t)

.

We now make use of the freedom to choose ξ(t) byrequiring

∂V ′

∂t′!= 0 .

Hereby, we determine transformation of time t′(t).The value H′ of H ′ then constitutes a constant of motion.


This leads to a linear, homogeneous third-order system

d

dt

ξ

ξ

ξ

=

0 1 0

0 0 1

−f1(~q(t), t) −f2(~q(t), t) 0

ξ

ξ

ξ

For known ~q = ~q (t), the coefficients are functions of time only

f1(~q (t), t) =4

~q 2

∂V

∂t, f2(~q (t), t) =

4

~q 2

[

V (~q, t) + 1

2~q

∂V

∂~q

]

.



d

dt

ξ

ξ

ξ

=

0 1 0

0 0 1

−f1(~q(t), t) −f2(~q(t), t) 0

ξ

ξ

ξ


f1(~q (t), t) =4

~q 2

∂V

∂t, f2(~q (t), t) =

4

~q 2

[

V (~q, t) + 1

2~q

∂V

∂~q

]

.

• Because of the ~q-dependence, the system can only beintegrated in conjunction with the canonical equations.



d

dt

ξ

ξ

ξ

=

0 1 0

0 0 1

−f1(~q(t), t) −f2(~q(t), t) 0

ξ

ξ

ξ


f1(~q (t), t) =4

~q 2

∂V

∂t, f2(~q (t), t) =

4

~q 2

[

V (~q, t) + 1

2~q

∂V

∂~q

]

.


• Exception: for ∂V/∂t ≡ 0, a solution ξ(t) ≡ 1 exists.



d

dt

ξ

ξ

ξ

=

0 1 0

0 0 1

−f1(~q(t), t) −f2(~q(t), t) 0

ξ

ξ

ξ


f1(~q (t), t) =4

~q 2

∂V

∂t, f2(~q (t), t) =

4

~q 2

[

V (~q, t) + 1

2~q

∂V

∂~q

]

.


• Exception: for ∂V/∂t ≡ 0, a solution ξ(t) ≡ 1 exists.

• The trace of the system matrix is zero. ; The Wronskideterminant of any solution matrix Ξ(t) is constant.; With Ξ(0) = E, we get det Ξ(t) ≡ 1.


The transformed Hamiltonian

H ′(

~q ′, ~p ′)

= 1

2~p ′ 2 + V ′

(

~q ′)

= const.

can be expressed in terms of the original coordinates

H ′(~q, ~p, t) = ξ(t) H − 1

2ξ(t) ~q ~p + 1

4ξ(t) ~q 2 .



H ′(

~q ′, ~p ′)

= 1

2~p ′ 2 + V ′

(

~q ′)

= const.


H ′(~q, ~p, t) = ξ(t) H − 1

2ξ(t) ~q ~p + 1

4ξ(t) ~q 2 .

With the 3 × 3 solution matrix Ξ(t) of the third-order system(Ξ(0) = E), this writes in terms of the transpose matrix ΞT (t)

H0

−1

2~q0 ~p0

1

4~q 20

=

ξ1 ξ1 ξ1

ξ2 ξ2 ξ2

ξ3 ξ3 ξ3

H

−1

2~q ~p

1

4~q 2

, det Ξ = 1 .



H ′(

~q ′, ~p ′)

= 1

2~p ′ 2 + V ′

(

~q ′)

= const.


H ′(~q, ~p, t) = ξ(t) H − 1

2ξ(t) ~q ~p + 1

4ξ(t) ~q 2 .

With the 3 × 3 solution matrix Ξ(t) of the third-order system(Ξ(0) = E), this writes in terms of the transpose matrix ΞT (t)

H0

−1

2~q0 ~p0

1

4~q 20

=

ξ1 ξ1 ξ1

ξ2 ξ2 ξ2

ξ3 ξ3 ξ3

H

−1

2~q ~p

1

4~q 2

, det Ξ = 1 .

We find: the vector(

H,−1

2~q ~p, 1

4~q 2)

has always alinear correlation to its initial state.

For ∂V/∂t ≡ 0, a solution ξ1(t) ≡ 1 exists ; H = H0.Extended phase space – p. 21

Finally, we consider the transformation of the 3-formdH d(~q ~p ) d(~q 2):

dH0 d(~q0 ~p0 ) d(~q 20 ) =

∂ (H0, ~q0 ~p0, ~q2

0 )

∂ (H, ~q ~p, ~q 2)dH d(~q ~p ) d(~q 2) .



dH0 d(~q0 ~p0 ) d(~q 20 ) =

∂ (H0, ~q0 ~p0, ~q2

0 )

∂ (H, ~q ~p, ~q 2)dH d(~q ~p ) d(~q 2) .

According to the transformation rule for(

H,−1

2~q ~p, 1

4~q 2)

,the Jacobi determinant is given by det Ξ.



dH0 d(~q0 ~p0 ) d(~q 20 ) =

∂ (H0, ~q0 ~p0, ~q2

0 )

∂ (H, ~q ~p, ~q 2)dH d(~q ~p ) d(~q 2) .

According to the transformation rule for(

H,−1

2~q ~p, 1

4~q 2)

,the Jacobi determinant is given by det Ξ.

Because of det Ξ = 1 conclude that

J = dH d(~q ~p ) d(~q 2) = const.

; The 3-form J is invariant with regard to the timeevolution of the general Hamiltonian system

H(~q, ~p, t) = 1

2~p 2 + V (~q, t) .


Conclusions and outlook• In the extended phase space, we can define more

general canonical transformations that allow todirectly map explicitly time-dependent Hamiltoniansystems into time-independent ones.




• For general the system with V (~q, t), we found a linearcorrelation of the vector of macroscopic quantities(H, ~q 2, ~q ~p ) into its initial state. This correlation isgoverned by a 3 × 3 matrix Ξ(t) with det Ξ(t) = 1 thatis given as a solution of a linear third-order system.





• To investigate: what does a chaotic evolution of Ξ(t)

tell us about the evolution of the system as a whole?





• To investigate: what does a chaotic evolution of Ξ(t)

tell us about the evolution of the system as a whole?

• To investigate: physical meaning of the invariant 3-form

J = dH d(~q ~p ) d(~q 2) = const.Extended phase space – p. 23

Publications

• Phys. Rev. Lett. 85, 3830 (2000)

• Phys. Rev. E 64, 026503 (2001)

• Phys. Rev. E 66, 066605 (2002)

• Ann. Phys. (Leipzig) 11, 15 (2002)

• Habilitation thesis (GSI Report 2002-06)

• This talk may be downloaded from“http://www.gsi.de/ ˜struck”


Hamiltonian mechanics in the “extended” phase spacestruck/hp/antritt/extphsp.pdf · 2005. 10....

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