File:Contour plot of all solutions to Euler's equations.png
![File:Contour plot of all solutions to Euler's equations.png](https://upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Contour_plot_of_all_solutions_to_Euler%27s_equations.png/800px-Contour_plot_of_all_solutions_to_Euler%27s_equations.png?20230427222123)
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[edit]DescriptionContour plot of all solutions to Euler's equations.png |
English: Given fixed angular momentum, for each value of kinetic energy, we have a family of curves for the possible value of angular velocity. The curves are plotted in black, on the orange ellipsoid of fixed angular momentum.
```python import numpy as np import matplotlib.pyplot as plt import scipy
L = 1.0 L2 = L**2 I1, I2, I3 = 1.0, 2.0, 3.0 xmin, xmax, ymin, ymax, zmin, zmax = -L/I1, L/I1, -L/I2, L/I2, -L/I3, L/I3 E1, E2, E3 = 0.5*L2/I1, 0.5*L2/I2, 0.5*L2/I3 def parametric_plot(E, orbit_res=1000): thetas = np.linspace(0, 2*np.pi, orbit_res) if E3 < E and E < E2: rs = np.zeros(orbit_res) zs = np.zeros(orbit_res) for i, theta in enumerate(thetas): invmatrix = scipy.linalg.inv(np.array([[I1 * np.cos(theta)**2 + I2 * np.sin(theta)**2 , I3], [(I1 * np.cos(theta))**2 + (I2 * np.sin(theta))**2, I3**2]])) r2z2 = invmatrix @ np.array([[2*E], [L2]]) rs[i] = np.sqrt(r2z2[0,0]) zs[i] = np.sqrt(r2z2[1,0]) return rs * np.cos(thetas), rs * np.sin(thetas), zs if E2 < E and E < E1: xs = np.zeros(orbit_res) rs = np.zeros(orbit_res) for i, theta in enumerate(thetas): invmatrix = scipy.linalg.inv(np.array([[I1, I2 * np.cos(theta)**2 + I3 * np.sin(theta)**2], [I1, (I2 * np.cos(theta))**2 + (I3 * np.sin(theta))**2]])) x2r2 = invmatrix @ np.array([[2*E], [L2]]) xs[i] = np.sqrt(x2r2[0,0]) rs[i] = np.sqrt(x2r2[1,0]) return xs, rs * np.cos(thetas), rs * np.sin(thetas) fig = plt.figure(figsize = (16,16)) ax = plt.axes(projection='3d') u = np.linspace(0, 2 * np.pi, 100) v = np.linspace(0, np.pi, 100) x = np.outer(np.cos(u), np.sin(v)) y = np.outer(np.sin(u), np.sin(v)) z = np.outer(np.ones(np.size(u)), np.cos(v)) x *= L/I1 y *= L/I2 z *= L/I3 ax.plot_surface(x, y, z, color='orange', alpha=0.3) epsilon=1e-3 for E in np.linspace(E3+epsilon, E1-epsilon, 20): xs, ys, zs = parametric_plot(E) xs *= 1+epsilon ys *= 1+epsilon zs *= 1+epsilon ax.plot3D(xs, ys, zs, linewidth=1, color='k') ax.plot3D(-xs, -ys, -zs, linewidth=1, color='k') ax.axes.set_xlim3d(xmin, xmax) ax.axes.set_ylim3d(ymin, ymax) ax.axes.set_zlim3d(zmin, zmax) ax.set_aspect('equal') ax.set_axis_off() plt.show() ``` |
Date | |
Source | Own work |
Author | Cosmia Nebula |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 22:21, 27 April 2023 | ![]() | 967 × 475 (410 KB) | Cosmia Nebula (talk | contribs) | cropped |
22:19, 27 April 2023 | ![]() | 1,252 × 1,252 (458 KB) | Cosmia Nebula (talk | contribs) | two more orbits | |
22:17, 27 April 2023 | ![]() | 1,252 × 1,252 (438 KB) | Cosmia Nebula (talk | contribs) | Uploaded while editing "Tennis racket theorem" on en.wikipedia.org |
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