```python
# https://colab.research.google.com/github/kassbohm/tm-snippets/blob/master/ipynb/TM_A/TM_3/lie.ipynb

from sympy import *
from sympy.physics.units import *

t, l = var("t, l")

pprint("\nt=0 s:")
sub_list = [(t, 0*second),]
# pprint("\nt=15 s:")
# sub_list = [(t, 15*second),]

tt = t / (60 * second)
p = 2 * pi * tt
l = 10*cm *(1 + tt)

C, S = cos(p), sin(p)
R = Matrix([
    [C, -S, 0],
    [S,  C, 0],
    [0,  0, 1]
    ])

vx, vy, vz = l*C, l*S, 0

pprint("\n(v'x, v'y, v'z) / (cm/s):")
v = Matrix([vx, vy, vz])
tmp = diff(v, t)
tmp = tmp.subs(sub_list)
tmp /= cm/second
pprint(tmp)


pprint("\n(v'x̄, v'ȳ, v'z̄)")
v = Matrix([l, 0, 0])
w = Matrix([0, 0, diff(p, t)])
tmp = diff(v, t) + w.cross(v)
tmp = tmp.subs(sub_list)
tmp /= cm/second
pprint(tmp)

pprint("\nCheck: (v'x, v'y, v'z) / (cm/s):")
tmp = diff(v, t) + w.cross(v)
tmp = R*tmp
tmp = tmp.subs(sub_list)
tmp /= cm/second
pprint(tmp)

# t=0 s:
#
# (v'x, v'y, v'z) / (cm/s):
# ⎡1/6⎤
# ⎢   ⎥
# ⎢ π ⎥
# ⎢ ─ ⎥
# ⎢ 3 ⎥
# ⎢   ⎥
# ⎣ 0 ⎦
#
# (v'x̄, v'ȳ, v'z̄)
# ⎡1/6⎤
# ⎢   ⎥
# ⎢ π ⎥
# ⎢ ─ ⎥
# ⎢ 3 ⎥
# ⎢   ⎥
# ⎣ 0 ⎦
#
# Check: (v'x, v'y, v'z) / (cm/s):
# ⎡1/6⎤
# ⎢   ⎥
# ⎢ π ⎥
# ⎢ ─ ⎥
# ⎢ 3 ⎥
# ⎢   ⎥
# ⎣ 0 ⎦

```

12ae52c3f0711342913bd6cf9b36d6980d468854
ipynb
Jupyter Notebook
ipynb/TM_A/TM_3/lie.ipynb
kassbohm/tm-snippets
5e0621ba2470116e54643b740d1b68b9f28bff12
MIT
ipynb/TM_A/TM_3/lie.ipynb
kassbohm/tm-snippets
5e0621ba2470116e54643b740d1b68b9f28bff12
MIT
ipynb/TM_A/TM_3/lie.ipynb
kassbohm/tm-snippets
5e0621ba2470116e54643b740d1b68b9f28bff12
MIT
Qwen/Qwen-72B
1. YES
2. YES