symbolic params q_1_1=
q
1
,
1
p_1_1=
p
1
,
1
q_2_1=
q
2
,
1
p_2_1=
p
2
,
1
q_3_1=
q
3
,
1
p_3_1=
p
3
,
1
Hamiltonian
H
=
1
2
p
1
,
1
⋅
p
1
,
1
+
1
2
p
2
,
1
⋅
p
2
,
1
+
1
2
p
3
,
1
⋅
p
3
,
1
+
0.05
(
(
(
q
1
,
1
−
q
2
,
1
)
(
q
1
,
1
−
q
2
,
1
)
−
1
)
2
+
(
(
q
1
,
1
−
q
3
,
1
)
(
q
1
,
1
−
q
3
,
1
)
−
1
)
2
+
(
(
q
2
,
1
−
q
1
,
1
)
(
q
2
,
1
−
q
1
,
1
)
−
1
)
2
+
(
(
q
2
,
1
−
q
3
,
1
)
(
q
2
,
1
−
q
3
,
1
)
−
1
)
2
+
(
(
q
3
,
1
−
q
1
,
1
)
(
q
3
,
1
−
q
1
,
1
)
−
1
)
2
+
(
(
q
3
,
1
−
q
2
,
1
)
(
q
3
,
1
−
q
2
,
1
)
−
1
)
2
)
Evolution Equations:
∂
q
1
,
1
∂
t
=
p
1
,
1
∂
p
1
,
1
∂
t
=
1
4
(
0.8
q
2
,
1
+
0.8
q
3
,
1
−
1.6
q
1
,
1
)
∂
q
2
,
1
∂
t
=
p
2
,
1
∂
p
2
,
1
∂
t
=
1
4
(
0.8
q
1
,
1
+
0.8
q
3
,
1
−
1.6
q
2
,
1
)
∂
q
3
,
1
∂
t
=
p
3
,
1
∂
p
3
,
1
∂
t
=
1
4
(
0.8
q
1
,
1
+
0.8
q
2
,
1
−
1.6
q
3
,
1
)
...in a vector:
[
∂
q
1
,
1
∂
t
∂
p
1
,
1
∂
t
∂
q
2
,
1
∂
t
∂
p
2
,
1
∂
t
∂
q
3
,
1
∂
t
∂
p
3
,
1
∂
t
]
=
[
p
1
,
1
1
4
(
0.8
q
2
,
1
+
0.8
q
3
,
1
−
1.6
q
1
,
1
)
p
2
,
1
1
4
(
0.8
q
1
,
1
+
0.8
q
3
,
1
−
1.6
q
2
,
1
)
p
3
,
1
1
4
(
0.8
q
1
,
1
+
0.8
q
2
,
1
−
1.6
q
3
,
1
)
]
...linearized:
[
∂
q
1
,
1
∂
t
∂
p
1
,
1
∂
t
∂
q
2
,
1
∂
t
∂
p
2
,
1
∂
t
∂
q
3
,
1
∂
t
∂
p
3
,
1
∂
t
]
=
[
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
]
[
q
1
,
1
p
1
,
1
q
2
,
1
p
2
,
1
q
3
,
1
p
3
,
1
]
1st order approximate derivative for Forward Euler
[
1
Δ
t
(
−
q
1
,
1
(
t
)
+
q
1
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
p
1
,
1
(
t
)
+
p
1
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
q
2
,
1
(
t
)
+
q
2
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
p
2
,
1
(
t
)
+
p
2
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
q
3
,
1
(
t
)
+
q
3
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
p
3
,
1
(
t
)
+
p
3
,
1
(
t
+
Δ
t
)
)
]
=
[
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
]
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
=
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
+
Δ
t
⋅
[
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
]
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
=
(
[
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
]
+
Δ
t
⋅
[
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
]
)
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
=
[
1
Δ
t
0
0
0
0
0
1
0
0
0
0
0
0
1
Δ
t
0
0
0
0
0
1
0
0
0
0
0
0
1
Δ
t
0
0
0
0
0
1
]
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
=
[
1
0.1
0
0
0
0
0
1
0
0
0
0
0
0
1
0.1
0
0
0
0
0
1
0
0
0
0
0
0
1
0.1
0
0
0
0
0
1
]
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
1st order approximate derivative for Backwards Euler
[
1
Δ
t
(
−
q
1
,
1
(
t
)
+
q
1
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
p
1
,
1
(
t
)
+
p
1
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
q
2
,
1
(
t
)
+
q
2
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
p
2
,
1
(
t
)
+
p
2
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
q
3
,
1
(
t
)
+
q
3
,
1
(
t
+
Δ
t
)
)
1
Δ
t
(
−
p
3
,
1
(
t
)
+
p
3
,
1
(
t
+
Δ
t
)
)
]
=
[
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
−
Δ
t
⋅
[
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
=
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
(
[
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
]
−
Δ
t
⋅
[
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
]
)
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
=
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
[
1
−
Δ
t
0
0
0
0
0
1
0
0
0
0
0
0
1
−
Δ
t
0
0
0
0
0
1
0
0
0
0
0
0
1
−
Δ
t
0
0
0
0
0
1
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
=
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
[
q
1
,
1
(
t
+
Δ
t
)
p
1
,
1
(
t
+
Δ
t
)
q
2
,
1
(
t
+
Δ
t
)
p
2
,
1
(
t
+
Δ
t
)
q
3
,
1
(
t
+
Δ
t
)
p
3
,
1
(
t
+
Δ
t
)
]
=
[
1
−
0.1
0
0
0
0
0
1
0
0
0
0
0
0
1
−
0.1
0
0
0
0
0
1
0
0
0
0
0
0
1
−
0.1
0
0
0
0
0
1
]
−
1
[
q
1
,
1
(
t
)
p
1
,
1
(
t
)
q
2
,
1
(
t
)
p
2
,
1
(
t
)
q
3
,
1
(
t
)
p
3
,
1
(
t
)
]
Gnuplot
Produced by GNUPLOT 5.4 patchlevel 4
-200
-150
-100
-50
0
50
100
150
200
0
100
200
300
400
500
600
700
800
900
1000
1100
q1,1
q
1,1
p1,1
p
1,1
q2,1
q
2,1
p2,1
p
2,1
q3,1
q
3,1
p3,1
p
3,1