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7/31/2019 Noethet Theorem Invariants
1/70
Noethers Theorem and Invariants
for time-dependentHamilton-Lagrange Systems
Jurgen Struckmeier
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
Noethers Theorem . 2/30
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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.
Noethers Theorem . 3/30
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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
Noethers Theorem . 3/30
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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
.
Noethers Theorem . 4/30
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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
.
The mapping q q is hereby uniquely determined
qi =
dqidt =
dqi + didt + d =
qi + i
1 + = qi + i qi + O(
2
) ,
Noethers Theorem . 4/30
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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
.
The mapping q q is hereby uniquely determined
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
.
Noethers Theorem . 4/30
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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+ . . . = qi+iqi
+ . . .
(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
) ,
which means that to first order the variation qi is
qi = i qi
.
Noethers Theorem . 4/30
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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 .
Noethers Theorem . 5/30
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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 .
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
) .
Noethers Theorem . 5/30
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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 .
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
) .
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 + . . .
Noethers Theorem . 5/30
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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 .
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
) .
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 + . . .
Noethers Theorem . 5/30
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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
Noethers Theorem . 6/30
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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
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
Noethers Theorem . 6/30
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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
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
Noethers Theorem . 6/30
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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
Noethers Theorem . 7/30
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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
Noethers Theorem . 7/30
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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
Noethers Theorem . 8/30
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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
Noethers Theorem . 8/30
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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
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).
Noethers Theorem . 10/30
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.
Noethers Theorem . 10/30
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.
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.
Noethers Theorem . 10/30
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.
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. The invariant cannot ease the problem of
integrating the systems equations of motion.
Noethers Theorem . 10/30
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.
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. 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
Noethers Theorem . 11/30
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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
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
Noethers Theorem . 12/30
(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 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
Noethers Theorem . 12/30
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.
Noethers Theorem . 13/30
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.
The other invariants I2, I3 are associated with(t) = const.
Noethers Theorem . 13/30
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.
The other invariants I2, I3 are associated with(t) = const.
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.
Noethers Theorem . 13/30
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.
The other invariants I2, I3 are associated with(t) = const.
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.
Noethers Theorem . 13/30
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
Invariant:I = (t) H 1
2(t)pq+ 1
4(t) q2
with (t) a solution of the auxiliary equation...(t) + 42(t) (t) + 4 (t) = 0
Noethers Theorem . 14/30
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
Invariant:I = (t) H 1
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
Noethers Theorem . 15/30
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
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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2D linear oscillator
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
Noethers Theorem . 16/30
2D linear oscillator
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2D linear oscillator
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.
Noethers Theorem . 16/30
Isotropic 2D Oscillator
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Isotropic 2D Oscillator
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
Isotropic 2D oscillator, isotropic initial conditions
py = 0, y > 0
Noethers Theorem . 17/30
Isotropic 2D Oscillator
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Isotropic 2D Oscillator
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
Isotropic 2D oscillator, isotropic initial conditions
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
Noethers Theorem . 17/30
Anisotropic 2D Oscillator
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Anisotropic 2D Oscillator
-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
Anisotropic 2D oscillator
py = 0, y > 0
Noethers Theorem . 18/30
Anisotropic 2D Oscillator
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Anisotropic 2D Oscillator
-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
Anisotropic 2D oscillator
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
Noethers Theorem . 18/30
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
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specializes to
i
x2i
... + 42x + 4xx
+ y2i
... + 42y + 4yy
+ z2i ... + 42z + 4zz +
j=i
c1rij = 0 .
Noethers Theorem . 20/30
Here, the 3rd-order auxiliary equation for (t)specializes to
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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.
Noethers Theorem . 20/30
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.
Noethers Theorem . 21/30
80
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-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.
Noethers Theorem . 22/30
We summarize:
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First time that an invariant Ihas been derivedfor such a high-dimensional time-dependentnon-linear system.
Noethers Theorem . 23/30
We summarize:
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First time that an invariant Ihas been derivedfor such a high-dimensional time-dependentnon-linear system.
Price: the auxiliary equation for (t) depends onall particle coordinates.
Noethers Theorem . 23/30
We summarize:
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First time that an invariant Ihas been derivedfor such a high-dimensional time-dependentnon-linear system.
Price: the auxiliary equation for (t) depends onall particle coordinates.
(t) reflects the collective evolution of the
N-particle system.
Noethers Theorem . 23/30
We summarize:
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First time that an invariant Ihas been derivedfor such a high-dimensional time-dependentnon-linear system.
Price: the auxiliary equation for (t) depends onall particle coordinates.
(t) reflects the collective evolution of the
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.
Noethers Theorem . 23/30
We summarize:
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First time that an invariant Ihas been derivedfor such a high-dimensional time-dependentnon-linear system.
Price: the auxiliary equation for (t) depends onall particle coordinates.
(t) reflects the collective evolution of the
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.
Noethers Theorem . 23/30
Verification of simulations
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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.
Noethers Theorem . 24/30
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.
Noethers Theorem . 25/30
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.
Ifq(t) and p(t) follow from computersimulations, the canonical equations are only
approximately satisfied because of the generallylimited accuracy of numerical methods.
Noethers Theorem . 25/30
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.
Ifq(t) and p(t) follow from computersimulations, the canonical equations are only
approximately satisfied because of the generallylimited accuracy of numerical methods.
; If the canonical equations and the auxiliary
equation are integrated numerically, the quantityI is no longer strictly constant.
Noethers Theorem . 25/30
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.
Ifq(t) and p(t) follow from computersimulations, the canonical equations are only
approximately satisfied because of the generallylimited accuracy of numerical methods.
; If the canonical equations and the auxiliary
equation are integrated numerically, the quantityI is no longer strictly constant.
[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.
Noethers Theorem . 25/30
The invariant I is a time integral of the auxiliaryequation ifq(t) and p(t) are time integrals of the
i l i
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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.
; If the canonical equations and the auxiliary
equation are integrated numerically, the quantityI is no longer strictly constant.
[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.
Noethers Theorem . 25/30
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.
Noethers Theorem . 26/30
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.
Noethers Theorem . 27/30
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.
Noethers Theorem . 28/30
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.
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.
Noethers Theorem . 28/30
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.
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.
Noethers Theorem . 28/30
This coupling is the reason why the strategy to
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simplify the solution of a given dynamical systemby finding invariants generally does not work.
Noethers Theorem . 2 /30
This coupling is the reason why the strategy to
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simplify the solution of a given dynamical systemby finding invariants generally does not work.
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).
Noethers Theorem . 2 /30
This coupling is the reason why the strategy to
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simplify the solution of a given dynamical systemby finding invariants generally does not work.
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