Noethet Theorem Invariants

7/31/2019 Noethet Theorem Invariants

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Noethers Theorem and Invariants

for time-dependentHamilton-Lagrange Systems

Jurgen Struckmeier

[email protected]

Kolloquiums-Vortrag

im Institut fur Theoretische Physik

der Johann Wolfgang Goethe-Universitat

Frankfurt am Main, 10. Januar 2002

Noethers Theorem . 1/30


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Outline Review of Noethers theorem Noethers theorem in the Hamiltonian

formulation

Invariant derived from Noethers theorem Example 1: 1D harmonic oscillator,

time-independent and time-dependent Ex. 2: 1D time-dependent non-linear oscillator

Ex. 3: 2D time-dependent harmonic oscillator

Ex. 4: System of Coulomb-interacting particles Application: verification of computer simulations

Conclusions



3/70

Review of Noethers theorem

Noethers theorem: relates conserved quantities I

to infinitesimal point transformations that leave

the Lagrange action L(q, q, t)dt invariant.; transformation that does not change the physics.



4/70

Review of Noethers theorem

Noethers theorem: relates conserved quantities I

to infinitesimal point transformations that leave

the Lagrange action L(q, q, t)dt invariant.; transformation that does not change the physics.

A point transformation maps points into points

(q, t) (q , t )

; the point transformation uniquely determines themapping of the velocity vector

q q



5/70

Definition of an infinitesimal point transformation ofan n-degree-of-freedom Lagrangian system:

t = t +t + . . . = t +(t) + . . .

qi = qi+qi+ . . . = qi+i(qi, t) + . . .

qi = qi+qi+ . . .

(t) =t

=0

, i(qi, t) =qi

=0

.



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t = t +t + . . . = t +(t) + . . .

qi = qi+qi+ . . . = qi+i(qi, t) + . . .

qi = qi+qi+ . . .

(t) =t

=0

, i(qi, t) =qi

=0

.

The mapping q q is hereby uniquely determined

qi =

dqidt =

dqi + didt + d =

qi + i

1 + = qi + i qi + O(

2

) ,



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t = t +t + . . . = t +(t) + . . .

qi = qi+qi+ . . . = qi+i(qi, t) + . . .

qi = qi+qi+ . . .

(t) =t

=0

, i(qi, t) =qi

=0

.


qi =

dqidt =

dqi + didt + d =

qi + i

1 + = qi + i qi + O(

2

) ,

which means that to first order the variation qi is

qi = i qi

.



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t = t +t + . . . = t +(t) + . . .

qi = qi+qi+ . . . = qi+i(qi, t) + . . .

qi = qi+qi+ . . . = qi+iqi

+ . . .

(t) =t

=0

, i(qi, t) =qi

=0

.


qi =

dqidt =

dqi + didt + d =

qi + i

1 + = qi + i qi + O(

2

) ,

which means that to first order the variation qi is

qi = i qi

.



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We now consider the subset of infinitesimal pointtransformations that leave Ldt invariant

L

q, q, t

dt

!= L

q, q, t

dt .



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L

q, q, t

dt

!= L

q, q, t

dt .

The functional relation between L and L may beexpressed introducing an auxiliary function f0(q, t)

Lq, q, t

= L

q, q, t

f0(q, t) + O(

2

) .



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L

q, q, t

dt

!= L

q, q, t

dt .


Lq, q, t

= L

q, q, t

f0(q, t) + O(

2

) .

On the other hand, we may express Lq, q, t interms of a truncated Taylor expansion ofL

q, q, t

Lq, q, t = Lq, q, t +

L

tt +

n

i=1

L

qi

qi +L

qi

qi + . . .



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L

q, q, t

dt

!= L

q, q, t

dt .


Lq, q, t

= L

q, q, t

f0(q, t) + O(

2

) .

On the other hand, we may express Lq, q, t interms of a truncated Taylor expansion ofL

q, q, t

Lq, q, t = Lq, q, t +

L

tt +

n

i=1

L

qi

qi +L

qi

qi + . . .



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Inserting t, qi, and qi, we obtain an equation forf0(q, t) that is uniquely determined by the (t), i(q, t)

f0(q, t) L(q, q, t) Lt

n

i=1

i

L

qi+

i qi L

qi

= 0



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n

i=1

i

L

qi+

i qi L

qi

= 0

The terms can be split into a total time derivative anda sum containing the Euler-Lagrange equations

ddt

f0(q, t) L +

ni=1

(qi i) Lqi

+

ni=1

(qi i)

Lqi

ddt

Lqi

= 0



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n

i=1

i

L

qi+

i qi L

qi

= 0

The terms can be split into a total time derivative anda sum containing the Euler-Lagrange equations

ddt

f0(q, t) L +

ni=1

(qi i) Lqi

+

ni=1

(qi i)

Lqi

ddt

Lqi

= 0

;

[. . . ]constitutes a conserved quantity

Ialong the

solutionq(t), q(t)

of the Euler-Lagrange equations

I = f0(q, t) L+n

i=1(qi i)

L

qi L

qid

dt

L

qi = 0, i = 1, . . . , n



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Hamiltonian formulation

A Legendre transformation relates a given L(q, q, t)with the corresponding Hamiltonian H(q, p, t)

L(q, q, t) =n

i=1

pi qi H(q, p, t) , pi = Lqi

, pi = Lqi

, Lt

= dHdt



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Hamiltonian formulation

A Legendre transformation relates a given L(q, q, t)with the corresponding Hamiltonian H(q, p, t)

L(q, q, t) =n

i=1

pi qi H(q, p, t) , pi = Lqi

, pi = Lqi

, Lt

= dHdt

Applying these rules to the Noether invariant, we find

I = (t) H(q, p, t)n

i=1

i(qi, t)pi + f0(q, t)

The conditional equation for f0(q, t) translates into

d

dt

(t) H(q, p, t)

ni=1 i(qi, t)pi + f

0

(q, t)

= 0 dI

dt

!

= 0



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Invariant

We now work out the Noether invariant for a class ofexplicitly time-dependent Hamiltonians H(p, q, t)

H =1

2

ni=1

p2

i + V(q, t) , qi = pi , pi = V

qi



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Invariant

We now work out the Noether invariant for a class ofexplicitly time-dependent Hamiltonians H(p, q, t)

H =1

2

ni=1

p2

i + V(q, t) , qi = pi , pi = V

qi

We insert H(p, q, t) into the equation dI/dt = 0 and

equate to zero the terms proportional to p2i , p1i , and p0i

p2i :1

2(t) i(qi, t)

qi= 0

p1i :f0(q, t)

qi i(qi, t)

t= 0

p0i : (t)V(q, t) + V

t

+f0

t

+ i

iV

qi= 0



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Combining these equations, we easily eliminate the

functions i(qi, t) and f0(q, t) to obtain a third-orderauxiliary equation for (t) that does not depend on p

...

ni=1

q2i + 4

V(q, t) +

12

ni=1

qiV

qi

+ 4

V

t = 0

Noethers Theorem . /30


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Combining these equations, we easily eliminate the

functions i(qi, t) and f0(q, t) to obtain a third-orderauxiliary equation for (t) that does not depend on p

...

ni=1

q2i + 4

V(q, t) +

12

ni=1

qiV

qi

+ 4

V

t = 0

The Noether invariant I for the particular Hamiltoniansystem H(p, q, t) =

i p

2i/2 + V(q, t) writes

I= (t) H 1

2

(t) i

pi qi +1

4

(t) i

q2

i

Prior to presenting examples 1a, 1b, 2, 3, and 4, well

discuss the meaning ofIand the equation for (t).

Noethers Theorem . /30

We summarize:


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We summarize:

The auxiliary equation for (t) depends on q(t).


We summarize:


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We summarize:

The auxiliary equation for (t) depends on q(t). ; The auxiliary equation can only be integrated

in conjunction with the canonical equations.


We summarize:


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We summarize:



The 2n first-order canonical equations formtogether with the three first-order equations of theauxiliary equation a closed coupled set of2n + 3first-order equations that uniquely determine q(t),

p(t), and (t) and hence the invariant I.


We summarize:


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We summarize:




p(t), and (t) and hence the invariant I. The invariant cannot ease the problem of

integrating the systems equations of motion.


We summarize:


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We summarize:




p(t), and (t) and hence the invariant I. The invariant cannot ease the problem of

integrating the systems equations of motion.

In the special case of autonomous systems, thefunction (t) = 1 is always a solution of the

auxiliary equation. With this solution, theinvariant Icoincides with the Hamiltonian H.Noethers Theorem . 10/30


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1D harmonic oscillator

Hamiltonian:

H(q, p) = 12p2 + V(q) , V(q) = 1

220q

2

Invariant:I = (t) H 1

2(t)pq+ 1

4(t) q2

with (t) a solution of the 3rd-order auxiliary equation

...(t) + 420 (t) = 0



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1D harmonic oscillator

Hamiltonian:

H(q, p) = 12p2 + V(q) , V(q) = 1

220q

2


2(t)pq+ 1

4(t) q2

with (t) a solution of the 3rd-order auxiliary equation

...(t) + 420 (t) = 0

General solution with a, b, c the integration constants:

(t) = a + b cos 20t + c sin20t

(1) Case a = 1, b = 0, c = 0:

(t) = 1 = I1 = HNoethers Theorem . 11/30

(2) Case a = 0, b = 1, c = 0:


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(t) = cos 20t = I2 = 12 p2 20q2 cos20t + 0pqsin20t

(3) Case a = 0, b = 0, c = 1:

(t) = sin 20t = I3 = 12 p2 20q2 sin20t 0pqcos20t


(2) Case a = 0, b = 1, c = 0:


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(t) = cos 20t = I2 = 12 p2 20q2 cos20t + 0pqsin20t

(3) Case a = 0, b = 0, c = 1:

(t) = sin 20t = I3 = 12 p2 20q2 sin20t 0pqcos20tThe mutual Poisson brackets yield

[I2, I1] = 20I3 , [I1, I3] = 20I2 , [I2, I3] = 20I1

Defining E1 = T V, E2 = 2

T V, the invariants I2 and I3 follow as

I2I3

=cos20t sin20t

sin20t cos20tE1

E2

Only two invariants are functionally independent:

I21 = I2

2 + I2

3


We summarize:


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We summarize:

Autonomous system: (t) = 1 is a solution of theauxiliary equation. With this solution, theinvariant I1 coincides with the Hamiltonian H,hence provides the conserved total energy.


We summarize:


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We summarize:


The other invariants I2, I3 are associated with(t) = const.


We summarize:


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We summarize:



The invariants I2 and I3 are related to the timeevolution of both the kinetic energy T and thepotential energy V, starting from the particularinitial energies T

0and V

0.


We summarize:


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We summarize:



The invariants I2 and I3 are related to the timeevolution of both the kinetic energy T and thepotential energy V, starting from the particularinitial energies T

0and V

0.

1D linear system: the q(t)-dependence of theauxiliary equation cancels in contrast to the

general case.


Hamiltonian of the time-dependent linear oscillator:


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H(q,p,t) = 12p2 + V(q, t) , V(q, t) = 1

22(t) q2


2(t)pq+ 1

4(t) q2

with (t) a solution of the auxiliary equation...(t) + 42(t) (t) + 4 (t) = 0


Hamiltonian of the time-dependent linear oscillator:


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H(q,p,t) = 12p2 + V(q, t) , V(q, t) = 1

22(t) q2


2(t)pq+ 1

4(t) q2

with (t) a solution of the auxiliary equation...(t) + 42(t) (t) + 4 (t) = 0

We observe:

Non-autonomous system: all invariants areassociated with (t) = const.

1D linear system: the auxiliary equation is notcoupled to the canonical equations.

The auxiliary equation agrees with the equationfor q2(t) ; we may identify (t) =

i q2i (t) in

this 1D linear case.Noethers Theorem . 14/30

1D non linear oscillator


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1D non-linear oscillator

Hamiltonian:

H(q,p,t) = 12p2 + V(q, t) , V(q, t) = 1

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

Invariant: I = (t) H 12

(t)pq+ 14

(t) q2

(t) is now a solution of the auxiliary equation

... + 4 2 + 4 + 2q(t)

2a + 5 a

+ 4q2(t)

b + 3 b

= 0


1D non linear oscillator


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1D non-linear oscillator

Hamiltonian:

H(q,p,t) = 12p2 + V(q, t) , V(q, t) = 1

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

Invariant: I = (t) H 12

(t)pq+ 14

(t) q2

(t) is now a solution of the auxiliary equation

... + 4 2 + 4 + 2q(t)

2a + 5 a

+ 4q2(t)

b + 3 b

= 0

Non-autonomous system: a solution (t) = const

does not exist. Non-linear system: the canonical equations andthe auxiliary equation are now coupled.;

The auxiliary equation can only be integratedsimultaneously with the canonical equations.Noethers Theorem . 15/30

2D linear oscillator


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Hamiltonian:

H(p, q, t) = 12

p2x + p

2

y

+ V(q, t) , V(q, t) = 1

2

2x(t) q

2

x + 2

y(t) q2

y

Invariant:

I = (t) H 12

(t) (pxqx + pyqy) +1

4(t)

q2x + q

2

y

The auxiliary equation for this case reads

...

q2x + q2

y

+ 4

2xq

2

x + 2

yq2

y

+ 4

xxq

2

x + yyq2

y

= 0




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Hamiltonian:

H(p, q, t) = 12

p2x + p

2

y

+ V(q, t) , V(q, t) = 1

2

2x(t) q

2

x + 2

y(t) q2

y

Invariant:

I = (t) H 12

(t) (pxqx + pyqy) +1

4(t)

q2x + q

2

y

The auxiliary equation for this case reads

...

q2x + q2

y

+ 4

2xq

2

x + 2

yq2

y

+ 4

xxq

2

x + yyq2

y

= 0

Non-autonomous system: a solution (t) = const does not exist.

2D system: the auxiliary equation depends on q(t). The auxiliary equation couples the degrees of freedom.

The solution (t) may be unstable even ifqx(t) and qy(t) are stable.


Isotropic 2D Oscillator


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0

1

2

3

4

5

6

7

0 20 40 60 80 100

(t)

t

Isotropic 2D oscillator, isotropic initial conditions

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

y

x


py = 0, y > 0




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0

1

2

3

4

5

6

7

0 20 40 60 80 100

(t)

t


2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

y

x


py = 0, y > 0

0

1

2

3

4

5

6

7

0 20 40 60 80 100

(t)

t

Isotropic 2D oscillator, anisotropic initial conditions

1

1.5

2

2.5

3

-4 -3 -2 -1 0 1 2 3 4

y

x

Isotropic 2D oscillator, anisotropic initial conditions

py = 0, y > 0


Anisotropic 2D Oscillator


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-700

-600

-500

-400

-300

-200

-100

0

100

200

0 50 100 150 200 250

(t)

t

Anisotropic 2D oscillator

3.5

3.51

3.52

3.53

3.54

3.55

-8 -6 -4 -2 0 2 4 6 8

y

x


py = 0, y > 0




45/70


-700

-600

-500

-400

-300

-200

-100

0

100

200

0 50 100 150 200 250

(t)

t


3.5

3.51

3.52

3.53

3.54

3.55

-8 -6 -4 -2 0 2 4 6 8

y

x


py = 0, y > 0

-60000

-40000

-20000

0

20000

40000

60000

0 50 100 150 200 250

(t)

t

Slightly anisotropic 2D oscillator

1

1.2

1.4

1.6

1.8

22.2

2.4

2.6

2.8

3

-8 -6 -4 -2 0 2 4 6 8

y

x

Slightly anisotropic 2D oscillator

py = 0, y > 0


Coulomb-interacting particles


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a g pa

Hamiltonian:

H =N

i=11

2

p2x,i + p

2

y,i + p2

z,i

+ V

x, y, z, t

The effective potential V contained herein be given by

Vx, y, z, t = 12N

i=1

2x

(t) x2

i

+ 2

y

(t) y2

i

+ 2

z

(t) z2

i

+ j=i

c1

rij

with c1 = q2/40m and r

2ij = (xi xj)2 + (yi yj)2 + (zi zj)2.

The invariant has the usual form

I = (t) H 12

Ni=1

xi px,i + yi py,i + zi pz,i

+ 1

4

Ni=1

x2i + y2

i + z2

i

Noethers Theorem . 1 /30

Here, the 3rd-order auxiliary equation for (t)specializes to


47/70

specializes to

i

x2i

... + 42x + 4xx

+ y2i

... + 42y + 4yy

+ z2i ... + 42z + 4zz +

j=i

c1rij = 0 .


Here, the 3rd-order auxiliary equation for (t)specializes to


48/70

specializes to

i

x2i

... + 42x + 4xx

+ y2i

... + 42y + 4yy

+ z2i ... + 42z + 4zz +

j=i

c1rij = 0 .

Non-autonomous system: a solution (t) = constdoes not exist.

3N-degree-of-freedom system: the auxiliaryequation depends on x, y, and z.

The particular evolution of(t) characterizes thedynamical behavior of the system as a whole.

(t) may be unstable even if the dynamicalsystem itself is stable. Then, the hyper-surfaceI= const becomes increasingly distorted.


1.4


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-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

(t/)

Cells (t/)

(t) as stable solution of the auxiliary equation for 0 = 45, = 9.


80


50/70

-80

-60

-40

-20

0

20

40

60

0 1 2 3 4 5 6 7 8 9 10

(t/)

Cells (t/)

(t) as unstable solution of the auxiliary equation for 0 = 60, = 15.


We summarize:


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First time that an invariant Ihas been derivedfor such a high-dimensional time-dependentnon-linear system.


We summarize:


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Price: the auxiliary equation for (t) depends onall particle coordinates.


We summarize:


53/70



(t) reflects the collective evolution of the

N-particle system.


We summarize:


54/70




N-particle system. Example: stability analysis of the auxiliaryequation for the time-dependent linear oscillator;

regimes of oscillatory versus chaotic timeevolution of charged particle beam envelopes.


We summarize:


55/70




N-particle system. Example: stability analysis of the auxiliaryequation for the time-dependent linear oscillator;

regimes of oscillatory versus chaotic timeevolution of charged particle beam envelopes.

Further investigations on this issue are necessary.


Verification of simulations


56/70

We return the general case and recapitulate Noethers theorem:

I = (t)

n

i=11

2p2i + V(q, t)

1

2(t)

n

i=1qipi +

1

4(t)

n

i=1q2i

is an invariant for a system whose time evolution follows from

qi = pi , pi +V(q, t)

qi

= 0 , i = 1, . . . , n

and for (t) a solution of the linear third-order auxiliary equation

...

ni=1

q2i + 4

V + 12

ni=1

qi Vqi

+ 4Vt = 0 .

This can also be shown directly if we evaluate dI/dt and insert the auxiliary

equation. The remaining terms vanish exactly if the canonical equations hold.


The invariant I is a time integral of the auxiliaryequation ifq(t) and p(t) are time integrals of the


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( ) ( )canonical equations.




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canonical equations.

Ifq(t) and p(t) follow from computersimulations, the canonical equations are only

approximately satisfied because of the generallylimited accuracy of numerical methods.




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; If the canonical equations and the auxiliary

equation are integrated numerically, the quantityI is no longer strictly constant.




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[I(t) I(0)]/I(0): relative deviation of the

calculated I(t) from the exact invariant I(0). ;A posteriori error estimation for the simulation.



i l i


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[I(t) I(0)]/I(0): relative deviation of the

calculated I(t) from the exact invariant I(0). ;A posteriori error estimation for the simulation.

; Generalization of the accuracy test H= const,which applies for autonomous systems only.


8

10

500 i l i i l


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-2

0

2

4

6

8

0 1 2 3 4 5 6 7 8 9 10

10

7

I/I0

Cells (t/)

500 simulation particles

1000 simulation particles

Relative invariant error I/I0 for 3D simulations of a charged particle beam

with different numbers of macro-particles.


8

10


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-2

0

2

4

6

8

0 1 2 3 4 5 6 7 8 9 10

10

4

I/I0

Cells (t/)

Relative invariant error I/I0 for a 3D simulation of a charged particle beam

with a systematic 5 %-error in the space-charge force calculations.


Conclusions


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For explicitly time-dependent Hamiltoniansystems, invariants I that depend on the canonicalvariables only do not exist. All invariants are

associated with solutions (t) of an auxiliaryequation.


Conclusions


65/70



Apart from very specific cases, the auxiliary

equation for (t) depends on the canonicalcoordinates q(t). This induces a coupling of theauxiliary equation to the canonical equations.


Conclusions


66/70



Apart from very specific cases, the auxiliary

equation for (t) depends on the canonicalcoordinates q(t). This induces a coupling of theauxiliary equation to the canonical equations.

The coupled (2n + 3)-system of2n first-ordercanonical equations and the 3 first-order auxiliaryequations determines both the systems time

evolution and its symmetries.


This coupling is the reason why the strategy to


67/70

simplify the solution of a given dynamical systemby finding invariants generally does not work.




68/70


In the (2n + 3)-dimensional space, the invariant I

represents a hyper-surface on which the motiontakes place. It can be interpreted as the energysurface that includes the energy flow into and

from the Hamiltonian system H(p, q, t).




69/70


In the (2n + 3)-dimensional space, the invariant I

represents a hyper-surface on which the motiontakes place. It can be interpreted as the energysurface that includes the energy flow into and

from the Hamiltonian system H(p, q, t). The solution (t) of the auxiliary equation may

be unstable even if the dynamical system itself is

stable. The stability analysis of the the auxiliaryequation may provide a direct method to classifya Hamiltonian system with respect to chaotic and

non-chaotic behavior.Noethers Theorem . 2 /30

Publications85


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Phys. Rev. Lett.85

, 3830 (2000) Phys. Rev. E 64, 026503 (2001) Ann. Phys. (Leipzig) 11, 15 (2002) Habilitation thesis (submitted Dec. 2001) This talk is available under

http://www.gsi.de/ struck

Noethers Theorem 30/30

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Noethet Theorem Invariants