Objectif

Passer en coordonnées sphériques dans le lagrangien libre.

Initialisations

import warnings

warnings.filterwarnings("ignore")

import sympy as sp

sp.init_printing()

t = sp.Symbol("t")
t

$\displaystyle t$

r = sp.Function("r")(t)
r

$\displaystyle r{\left(t \right)}$

sp.diff(r)

$\displaystyle \frac{d}{d t} r{\left(t \right)}$

theta = sp.Function("\\theta")(t)
theta

$\displaystyle \theta{\left(t \right)}$

phi = sp.Function("\\phi")(t)
phi

$\displaystyle \phi{\left(t \right)}$

Construction du changement de coordonnées.

x = r * sp.cos(theta) * sp.cos(phi)
x

$\displaystyle r{\left(t \right)} \cos{\left(\phi{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)}$

y = r * sp.cos(theta) * sp.sin(phi)
y

$\displaystyle r{\left(t \right)} \sin{\left(\phi{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)}$

z = r * sp.sin(theta)
z

$\displaystyle r{\left(t \right)} \sin{\left(\theta{\left(t \right)} \right)}$

Construction du Lagrangien en cartésien

m, vx, vy, vz = sp.symbols("m v_x v_y v_z")
vx

$\displaystyle v_{x}$

L = m * (vx ** 2 + vy ** 2 + vz ** 2) / 2
L

$\displaystyle \frac{m \left(v_{x}^{2} + v_{y}^{2} + v_{z}^{2}\right)}{2}$

Substitutions

xd, yd, zd = x.diff(), y.diff(), z.diff()
xd

$\displaystyle - r{\left(t \right)} \sin{\left(\phi{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)} \frac{d}{d t} \phi{\left(t \right)} - r{\left(t \right)} \sin{\left(\theta{\left(t \right)} \right)} \cos{\left(\phi{\left(t \right)} \right)} \frac{d}{d t} \theta{\left(t \right)} + \cos{\left(\phi{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)} \frac{d}{d t} r{\left(t \right)}$

res = L.subs(
    {
        vx: xd,
        vy: yd,
        vz: zd
    }
).expand()
res

$\displaystyle \frac{m r^{2}{\left(t \right)} \sin^{2}{\left(\phi{\left(t \right)} \right)} \sin^{2}{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} \theta{\left(t \right)}\right)^{2}}{2} + \frac{m r^{2}{\left(t \right)} \sin^{2}{\left(\phi{\left(t \right)} \right)} \cos^{2}{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} \phi{\left(t \right)}\right)^{2}}{2} + \frac{m r^{2}{\left(t \right)} \sin^{2}{\left(\theta{\left(t \right)} \right)} \cos^{2}{\left(\phi{\left(t \right)} \right)} \left(\frac{d}{d t} \theta{\left(t \right)}\right)^{2}}{2} + \frac{m r^{2}{\left(t \right)} \cos^{2}{\left(\phi{\left(t \right)} \right)} \cos^{2}{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} \phi{\left(t \right)}\right)^{2}}{2} + \frac{m r^{2}{\left(t \right)} \cos^{2}{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} \theta{\left(t \right)}\right)^{2}}{2} - m r{\left(t \right)} \sin^{2}{\left(\phi{\left(t \right)} \right)} \sin{\left(\theta{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)} \frac{d}{d t} \theta{\left(t \right)} \frac{d}{d t} r{\left(t \right)} - m r{\left(t \right)} \sin{\left(\theta{\left(t \right)} \right)} \cos^{2}{\left(\phi{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)} \frac{d}{d t} \theta{\left(t \right)} \frac{d}{d t} r{\left(t \right)} + m r{\left(t \right)} \sin{\left(\theta{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)} \frac{d}{d t} \theta{\left(t \right)} \frac{d}{d t} r{\left(t \right)} + \frac{m \sin^{2}{\left(\phi{\left(t \right)} \right)} \cos^{2}{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} r{\left(t \right)}\right)^{2}}{2} + \frac{m \sin^{2}{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} r{\left(t \right)}\right)^{2}}{2} + \frac{m \cos^{2}{\left(\phi{\left(t \right)} \right)} \cos^{2}{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} r{\left(t \right)}\right)^{2}}{2}$

l = sp.Symbol("l")

final = res.subs(r.diff(), 0).subs(r, l).trigsimp()
final

$\displaystyle \frac{l^{2} m \left(\cos^{2}{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} \phi{\left(t \right)}\right)^{2} + \left(\frac{d}{d t} \theta{\left(t \right)}\right)^{2}\right)}{2}$

Equations différentielles

e1 = sp.Eq(final.diff(theta.diff()).diff(t) - final.diff(theta), 0)
e1

$\displaystyle l^{2} m \sin{\left(\theta{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)} \left(\frac{d}{d t} \phi{\left(t \right)}\right)^{2} + l^{2} m \frac{d^{2}}{d t^{2}} \theta{\left(t \right)} = 0$

e2 = sp.Eq(final.diff(phi.diff()).diff(t) - final.diff(phi), 0)
e2

$\displaystyle - 2 l^{2} m \sin{\left(\theta{\left(t \right)} \right)} \cos{\left(\theta{\left(t \right)} \right)} \frac{d}{d t} \phi{\left(t \right)} \frac{d}{d t} \theta{\left(t \right)} + l^{2} m \cos^{2}{\left(\theta{\left(t \right)} \right)} \frac{d^{2}}{d t^{2}} \phi{\left(t \right)} = 0$

Exercice

Rajouter la force de gravitation.