{
"cells": [
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import scipy as sc\n",
"import scipy.integrate as integrate"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"# constants\n",
"G = 1\n",
"M = 1"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [
{
"ename": "TypeError",
"evalue": "only length-1 arrays can be converted to Python scalars",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn[12], line 26\u001b[0m\n\u001b[1;32m 24\u001b[0m x0 \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39marray([\u001b[38;5;241m1\u001b[39m, \u001b[38;5;241m0\u001b[39m, \u001b[38;5;241m0\u001b[39m, \u001b[38;5;241m0.1\u001b[39m])\n\u001b[1;32m 25\u001b[0m t \u001b[38;5;241m=\u001b[39m np\u001b[38;5;241m.\u001b[39mlinspace(\u001b[38;5;241m0\u001b[39m, \u001b[38;5;241m10\u001b[39m, \u001b[38;5;241m1000\u001b[39m)\n\u001b[0;32m---> 26\u001b[0m x \u001b[38;5;241m=\u001b[39m \u001b[43mf\u001b[49m\u001b[43m(\u001b[49m\u001b[43mx0\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mt\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mintegrate\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mquad_vec\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 28\u001b[0m \u001b[38;5;66;03m# plot the solution\u001b[39;00m\n\u001b[1;32m 29\u001b[0m plt\u001b[38;5;241m.\u001b[39mplot(x[:, \u001b[38;5;241m0\u001b[39m], x[:, \u001b[38;5;241m1\u001b[39m])\n",
"Cell \u001b[0;32mIn[12], line 20\u001b[0m, in \u001b[0;36mf\u001b[0;34m(x, t, integrator)\u001b[0m\n\u001b[1;32m 19\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mf\u001b[39m(x: np\u001b[38;5;241m.\u001b[39mndarray, t: \u001b[38;5;28mfloat\u001b[39m, integrator: \u001b[38;5;28mcallable\u001b[39m) \u001b[38;5;241m-\u001b[39m\u001b[38;5;241m>\u001b[39m np\u001b[38;5;241m.\u001b[39mndarray:\n\u001b[0;32m---> 20\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mintegrator\u001b[49m\u001b[43m(\u001b[49m\u001b[43mdf\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mx\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mt\u001b[49m\u001b[43m)\u001b[49m\n",
"File \u001b[0;32m~/.local/share/virtualenvs/projects-X-9bmgL6/lib/python3.12/site-packages/scipy/integrate/_quad_vec.py:225\u001b[0m, in \u001b[0;36mquad_vec\u001b[0;34m(f, a, b, epsabs, epsrel, norm, cache_size, limit, workers, points, quadrature, full_output, args)\u001b[0m\n\u001b[1;32m 105\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mquad_vec\u001b[39m(f, a, b, epsabs\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m1e-200\u001b[39m, epsrel\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m1e-8\u001b[39m, norm\u001b[38;5;241m=\u001b[39m\u001b[38;5;124m'\u001b[39m\u001b[38;5;124m2\u001b[39m\u001b[38;5;124m'\u001b[39m, cache_size\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m100e6\u001b[39m,\n\u001b[1;32m 106\u001b[0m limit\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m10000\u001b[39m, workers\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m1\u001b[39m, points\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mNone\u001b[39;00m, quadrature\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mNone\u001b[39;00m, full_output\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mFalse\u001b[39;00m,\n\u001b[1;32m 107\u001b[0m \u001b[38;5;241m*\u001b[39m, args\u001b[38;5;241m=\u001b[39m()):\n\u001b[1;32m 108\u001b[0m \u001b[38;5;250m \u001b[39m\u001b[38;5;124mr\u001b[39m\u001b[38;5;124;03m\"\"\"Adaptive integration of a vector-valued function.\u001b[39;00m\n\u001b[1;32m 109\u001b[0m \n\u001b[1;32m 110\u001b[0m \u001b[38;5;124;03m Parameters\u001b[39;00m\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 223\u001b[0m \n\u001b[1;32m 224\u001b[0m \u001b[38;5;124;03m \"\"\"\u001b[39;00m\n\u001b[0;32m--> 225\u001b[0m a \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mfloat\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43ma\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 226\u001b[0m b \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mfloat\u001b[39m(b)\n\u001b[1;32m 228\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m args:\n",
"\u001b[0;31mTypeError\u001b[0m: only length-1 arrays can be converted to Python scalars"
]
}
],
"source": [
"### linearized ode\n",
"\n",
"def A(x: np.ndarray) -> np.ndarray:\n",
" # assuming 2d data\n",
" r = x[:x.size//2]\n",
" v = x[x.size//2:]\n",
" r_norm = np.linalg.norm(r)\n",
" return np.array([\n",
" [0, 0, 1, 0],\n",
" [0, 0, 0, 1],\n",
" [-G*M/r_norm**3 + 3*G*M*r[0]**2/r_norm**5, 3*G*M*r[0]*r[1]/r_norm**5, 0, 0],\n",
" [3*G*M*r[0]*r[1]/r_norm**5, -G*M/r_norm**3 + 3*G*M*r[1]**2/r_norm**5, 0, 0]\n",
" ])\n",
"\n",
"def df(x: np.ndarray, t) -> np.ndarray:\n",
" return A(x) @ x\n",
"\n",
"\n",
"def f(x: np.ndarray, t: float, integrator: callable) -> np.ndarray:\n",
" integrate.simpson()\n",
" return integrator(df, x, t)\n",
"\n",
"\n",
"# solve the ode for some initial conditions\n",
"x0 = np.array([1, 0, 0, 0.1])\n",
"t = np.linspace(0, 10, 1000)\n",
"x = f(x0, t, integrate.quad_vec)\n",
"\n",
"# plot the solution\n",
"plt.plot(x[:, 0], x[:, 1])\n",
"plt.show()\n"
]
}
],
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