def test_SimplePendulumCartesian():
m = 1
g = 1
L = 1
x = Variable("x")
y = Variable("y")
dx = x.dt
dy = y.dt
X = [x, y]
Ek = m*(dx**2 + dy**2)/2
Eu = m*g*y
E = Ek - Eu
timesteps = np.linspace(0, 10, 1000)
x0 = L*np.sin(0.2)
y0 = -L*np.cos(0.2)
IC = {x: (x0, 0),
y: (y0, 0)}
G = [x**2 + y**2 - L**2]
F = [0, 0]
solver = Euler(E, IC, G, F, timesteps)
results = solver.run(timeout=60)
def test_X2():
x = Variable("x")
E = Energy(x**2)
L = Lagrangian(E)
Lvec = L.get()
assert Lvec.shape == (1, )
assert Lvec[0] == -2 * x
def test_XdX():
x = Variable("x")
dx = x.dt
E = Energy(x * dx)
L = Lagrangian(E)
Lvec = L.get()
assert Lvec.shape == (1, )
assert Lvec[0] == 0
def test_dX2():
x = Variable("x")
dx = x.dt
ddx = x.ddt
E = Energy(dx**2)
L = Lagrangian(E)
Lvec = L.get()
assert Lvec.shape == (1, )
assert Lvec[0] == 2 * ddx
def test_cosX():
x = Variable("x")
f = cos(x)
df = -1 * sin(x)
E = Energy(f)
L = Lagrangian(E)
Lvec = L.get()
assert Lvec.shape == (1, )
assert Lvec[0] == -1 * df
def test_DoublePendulumCartesian():
m1 = 1
m2 = 1
g = 1
L1 = 1
L2 = 1
x1 = Variable("x1")
y1 = Variable("y1")
x2 = Variable("x2")
y2 = Variable("y2")
dx1 = x1.dt
dy1 = y1.dt
dx2 = x2.dt
dy2 = y2.dt
X = [x1, y1, x2, y2]
Ek = m1*(dx1**2 + dy1**2)/2
Ek += m2*(dx2**2 + dy2**2)/2
Eu = m1*g*y1 + m2*g*y2
E = Ek - Eu
timesteps = np.linspace(0, 10, 1000)
theta1 = 0.9*np.pi
theta2 = 0.5*np.pi
x10 = L1*np.sin(theta1)
y10 = -L1*np.cos(theta1)
x20 = L2*np.sin(theta2)
y20 = -L2*np.cos(theta2)
IC = {x1: (x10, 0),
y1: (y10, 0),
x2: (x20, 0),
y2: (y20, 0)}
G = [x1**2 + y1**2 - L1**2,
(x2)**2 + (y2)**2 - (L2)**2]
F = [0, 0, 0, 0]
solver = Euler(E, IC, G, F, timesteps)
results = solver.run(timeout=60)
def test_Spring():
m = 1
k = 1
x = Variable("x")
dx = x.dt
X = [x]
Ek = m*(dx**2)/2
Eu = k*x**2/2
E = Ek-Eu
timesteps = np.linspace(0, 10, 100)
IC = {x: (0.3, 0)}
G = []
F = [0]
solver = Euler(E, IC, G, F, timesteps)
result = solver.run(timeout=5)
def test_SimplePendulumAngular():
m = 1
g = 1
L = 1
theta = Variable("theta")
dtheta = theta.dt
X = [theta]
Ek = m*(L**2) *(dtheta**2)/2
Eu = -m*g*L*sp.cos(theta)
E = Ek-Eu
timesteps = np.linspace(0, 10, 100)
IC = {theta: (0.9*np.pi, 0)}
G = []
F = [0]
solver = Euler(E, IC, G, F, timesteps)
result = solver.run(timeout=5)