{"id":1035674,"date":"2024-12-31T12:00:50","date_gmt":"2024-12-31T04:00:50","guid":{"rendered":"https:\/\/docs.pingcode.com\/ask\/ask-ask\/1035674.html"},"modified":"2024-12-31T12:00:52","modified_gmt":"2024-12-31T04:00:52","slug":"%e5%a6%82%e4%bd%95%e7%94%a8python3%e6%a8%a1%e6%8b%9f%e4%b8%89%e4%bd%93","status":"publish","type":"post","link":"https:\/\/docs.pingcode.com\/ask\/1035674.html","title":{"rendered":"\u5982\u4f55\u7528Python3\u6a21\u62df\u4e09\u4f53"},"content":{"rendered":"

\"\u5982\u4f55\u7528Python3\u6a21\u62df\u4e09\u4f53\"<\/p>\n

\u8981\u5728Python3\u4e2d\u6a21\u62df\u4e09\u4f53\u95ee\u9898\uff0c\u4f60\u53ef\u4ee5\u4f7f\u7528\u6570\u503c\u79ef\u5206\u65b9\u6cd5\u6765\u89e3\u51b3\u7cfb\u7edf\u7684\u5fae\u5206\u65b9\u7a0b\u3002<\/strong> \u901a\u8fc7\u4f7f\u7528Runge-Kutta\u65b9\u6cd5\u3001\u5b9a\u4e49\u521d\u59cb\u6761\u4ef6\u548c\u53c2\u6570\u3001\u7f16\u5199\u8fd0\u52a8\u65b9\u7a0b\u548c\u7ed8\u5236\u8f68\u8ff9\uff0c\u53ef\u4ee5\u5b9e\u73b0\u4e09\u4f53\u95ee\u9898\u7684\u6a21\u62df\u3002\u4e0b\u9762\u6211\u5c06\u8be6\u7ec6\u63cf\u8ff0\u8fd9\u4e9b\u6b65\u9aa4\u3002<\/p>\n<\/p>\n

\u4e00\u3001\u5b9a\u4e49\u4e09\u4f53\u95ee\u9898\u7684\u57fa\u672c\u6982\u5ff5\u548c\u5fae\u5206\u65b9\u7a0b<\/p>\n

\u4e09\u4f53\u95ee\u9898\u662f\u7ecf\u5178\u529b\u5b66\u4e2d\u7684\u4e00\u4e2a\u95ee\u9898\uff0c\u5b83\u7814\u7a76\u4e09\u4e2a\u7269\u4f53\u5728\u4e92\u76f8\u5438\u5f15\u7684\u60c5\u51b5\u4e0b\u7684\u8fd0\u52a8\u3002\u4e3a\u4e86\u6a21\u62df\u4e09\u4f53\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u5b9a\u4e49\u4e09\u4e2a\u7269\u4f53\u7684\u8d28\u91cf\u3001\u521d\u59cb\u4f4d\u7f6e\u548c\u901f\u5ea6\uff0c\u5e76\u4f7f\u7528\u725b\u987f\u5f15\u529b\u516c\u5f0f\u6765\u8ba1\u7b97\u5b83\u4eec\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u529b\u3002\u6839\u636e\u8fd9\u4e9b\u529b\uff0c\u6211\u4eec\u53ef\u4ee5\u5199\u51fa\u7269\u4f53\u7684\u8fd0\u52a8\u65b9\u7a0b\uff0c\u8fd9\u662f\u4e00\u4e2a\u7531\u4f4d\u7f6e\u548c\u901f\u5ea6\u7ec4\u6210\u7684\u5fae\u5206\u65b9\u7a0b\u7ec4\u3002<\/p>\n<\/p>\n

1\u3001\u521d\u59cb\u6761\u4ef6\u548c\u53c2\u6570\u5b9a\u4e49<\/h3>\n<\/p>\n

\u5728\u4e09\u4f53\u95ee\u9898\u7684\u6a21\u62df\u4e2d\uff0c\u9996\u5148\u9700\u8981\u5b9a\u4e49\u4e09\u4e2a\u7269\u4f53\u7684\u521d\u59cb\u6761\u4ef6\uff0c\u5305\u62ec\u5b83\u4eec\u7684\u8d28\u91cf\u3001\u4f4d\u7f6e\u548c\u901f\u5ea6\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e09\u4e2a\u7269\u4f53A\u3001B\u548cC\uff0c\u5b83\u4eec\u7684\u8d28\u91cf\u5206\u522b\u4e3am1\u3001m2\u548cm3\uff0c\u4f4d\u7f6e\u5206\u522b\u4e3ar1\u3001r2\u548cr3\uff0c\u901f\u5ea6\u5206\u522b\u4e3av1\u3001v2\u548cv3\u3002\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528Python\u7684numpy\u5e93\u6765\u5b9a\u4e49\u8fd9\u4e9b\u521d\u59cb\u6761\u4ef6\u548c\u53c2\u6570\u3002<\/p>\n<\/p>\n

import numpy as np<\/p>\n
\u8d28\u91cf<\/strong><\/h2>\n
m1, m2, m3 = 1.0, 1.0, 1.0<\/p>\n
\u521d\u59cb\u4f4d\u7f6e (x, y, z)<\/strong><\/h2>\n
r1 = np.array([1.0, 0.0, 0.0])<\/p>\n
r2 = np.array([-1.0, 0.0, 0.0])<\/p>\n
r3 = np.array([0.0, 1.0, 0.0])<\/p>\n
\u521d\u59cb\u901f\u5ea6 (vx, vy, vz)<\/strong><\/h2>\n
v1 = np.array([0.0, 1.0, 0.0])<\/p>\n
v2 = np.array([0.0, -1.0, 0.0])<\/p>\n
v3 = np.array([1.0, 0.0, 0.0])<\/p>\n
<\/code><\/pre>\n<\/p>\n
2\u3001\u8fd0\u52a8\u65b9\u7a0b<\/h3>\n<\/p>\n
\u4e3a\u4e86\u6a21\u62df\u4e09\u4f53\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u5199\u51fa\u63cf\u8ff0\u7269\u4f53\u8fd0\u52a8\u7684\u5fae\u5206\u65b9\u7a0b\u3002\u725b\u987f\u5f15\u529b\u516c\u5f0f\u544a\u8bc9\u6211\u4eec\u7269\u4f53\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u529b\u662f\uff1a<\/p>\n<\/p>\n
[ F = G \\frac{m1 \\cdot m2}{r^2} ]<\/p>\n<\/p>\n
\u5176\u4e2d\uff0cG\u662f\u5f15\u529b\u5e38\u6570\uff0cm1\u548cm2\u662f\u4e24\u4e2a\u7269\u4f53\u7684\u8d28\u91cf\uff0cr\u662f\u5b83\u4eec\u4e4b\u95f4\u7684\u8ddd\u79bb\u3002\u6839\u636e\u725b\u987f\u7b2c\u4e8c\u5b9a\u5f8b\uff0c\u6211\u4eec\u53ef\u4ee5\u5199\u51fa\u7269\u4f53\u7684\u52a0\u901f\u5ea6\uff1a<\/p>\n<\/p>\n
[ a = \\frac{F}{m} ]<\/p>\n<\/p>\n
\u6211\u4eec\u9700\u8981\u5c06\u8fd9\u4e2a\u52a0\u901f\u5ea6\u4ee3\u5165\u5230\u8fd0\u52a8\u65b9\u7a0b\u4e2d\uff0c\u5f97\u5230\u4e09\u4e2a\u7269\u4f53\u7684\u52a0\u901f\u5ea6\u3002<\/p>\n<\/p>\n
def acceleration(r1, r2, r3, m1, m2, m3, G):<\/p>\n
    # \u8ba1\u7b97\u8ddd\u79bb\u5411\u91cf<\/p>\n
    r12 = r2 - r1<\/p>\n
    r13 = r3 - r1<\/p>\n
    r23 = r3 - r2<\/p>\n
    # \u8ba1\u7b97\u8ddd\u79bb\u7684\u5e73\u65b9<\/p>\n
    r12_mag = np.linalg.norm(r12)<\/p>\n
    r13_mag = np.linalg.norm(r13)<\/p>\n
    r23_mag = np.linalg.norm(r23)<\/p>\n
    # \u8ba1\u7b97\u52a0\u901f\u5ea6<\/p>\n
    a1 = G * m2 * r12 \/ r12_mag<strong>3 + G * m3 * r13 \/ r13_mag<\/strong>3<\/p>\n
    a2 = -G * m1 * r12 \/ r12_mag<strong>3 + G * m3 * r23 \/ r23_mag<\/strong>3<\/p>\n
    a3 = -G * m1 * r13 \/ r13_mag<strong>3 - G * m2 * r23 \/ r23_mag<\/strong>3<\/p>\n
    return a1, a2, a3<\/p>\n
<\/code><\/pre>\n<\/p>\n
3\u3001\u6570\u503c\u79ef\u5206\u65b9\u6cd5<\/h3>\n<\/p>\n
\u4e3a\u4e86\u6c42\u89e3\u8fd0\u52a8\u65b9\u7a0b\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u6570\u503c\u79ef\u5206\u65b9\u6cd5\uff0c\u6bd4\u5982Runge-Kutta\u65b9\u6cd5\u3002Runge-Kutta\u65b9\u6cd5\u662f\u4e00\u79cd\u5e38\u7528\u7684\u6570\u503c\u79ef\u5206\u65b9\u6cd5\uff0c\u5b83\u53ef\u4ee5\u5728\u7ed9\u5b9a\u7684\u521d\u59cb\u6761\u4ef6\u4e0b\uff0c\u901a\u8fc7\u4e00\u7cfb\u5217\u7684\u8fed\u4ee3\u6b65\u9aa4\u6765\u8fd1\u4f3c\u6c42\u89e3\u5fae\u5206\u65b9\u7a0b\u3002\u5728\u8fd9\u91cc\uff0c\u6211\u4eec\u4f7f\u7528\u56db\u9636Runge-Kutta\u65b9\u6cd5\u6765\u8fdb\u884c\u6570\u503c\u79ef\u5206\u3002<\/p>\n<\/p>\n
def runge_kutta_step(r1, r2, r3, v1, v2, v3, m1, m2, m3, G, dt):<\/p>\n
    # \u8ba1\u7b97k1<\/p>\n
    a1, a2, a3 = acceleration(r1, r2, r3, m1, m2, m3, G)<\/p>\n
    k1_r1 = v1 * dt<\/p>\n
    k1_r2 = v2 * dt<\/p>\n
    k1_r3 = v3 * dt<\/p>\n
    k1_v1 = a1 * dt<\/p>\n
    k1_v2 = a2 * dt<\/p>\n
    k1_v3 = a3 * dt<\/p>\n
    # \u8ba1\u7b97k2<\/p>\n
    a1, a2, a3 = acceleration(r1 + 0.5 * k1_r1, r2 + 0.5 * k1_r2, r3 + 0.5 * k1_r3, m1, m2, m3, G)<\/p>\n
    k2_r1 = (v1 + 0.5 * k1_v1) * dt<\/p>\n
    k2_r2 = (v2 + 0.5 * k1_v2) * dt<\/p>\n
    k2_r3 = (v3 + 0.5 * k1_v3) * dt<\/p>\n
    k2_v1 = a1 * dt<\/p>\n
    k2_v2 = a2 * dt<\/p>\n
    k2_v3 = a3 * dt<\/p>\n
    # \u8ba1\u7b97k3<\/p>\n
    a1, a2, a3 = acceleration(r1 + 0.5 * k2_r1, r2 + 0.5 * k2_r2, r3 + 0.5 * k2_r3, m1, m2, m3, G)<\/p>\n
    k3_r1 = (v1 + 0.5 * k2_v1) * dt<\/p>\n
    k3_r2 = (v2 + 0.5 * k2_v2) * dt<\/p>\n
    k3_r3 = (v3 + 0.5 * k2_v3) * dt<\/p>\n
    k3_v1 = a1 * dt<\/p>\n
    k3_v2 = a2 * dt<\/p>\n
    k3_v3 = a3 * dt<\/p>\n
    # \u8ba1\u7b97k4<\/p>\n
    a1, a2, a3 = acceleration(r1 + k3_r1, r2 + k3_r2, r3 + k3_r3, m1, m2, m3, G)<\/p>\n
    k4_r1 = (v1 + k3_v1) * dt<\/p>\n
    k4_r2 = (v2 + k3_v2) * dt<\/p>\n
    k4_r3 = (v3 + k3_v3) * dt<\/p>\n
    k4_v1 = a1 * dt<\/p>\n
    k4_v2 = a2 * dt<\/p>\n
    k4_v3 = a3 * dt<\/p>\n
    # \u66f4\u65b0\u4f4d\u7f6e\u548c\u901f\u5ea6<\/p>\n
    r1_next = r1 + (k1_r1 + 2 * k2_r1 + 2 * k3_r1 + k4_r1) \/ 6<\/p>\n
    r2_next = r2 + (k1_r2 + 2 * k2_r2 + 2 * k3_r2 + k4_r2) \/ 6<\/p>\n
    r3_next = r3 + (k1_r3 + 2 * k2_r3 + 2 * k3_r3 + k4_r3) \/ 6<\/p>\n
    v1_next = v1 + (k1_v1 + 2 * k2_v1 + 2 * k3_v1 + k4_v1) \/ 6<\/p>\n
    v2_next = v2 + (k1_v2 + 2 * k2_v2 + 2 * k3_v2 + k4_v2) \/ 6<\/p>\n
    v3_next = v3 + (k1_v3 + 2 * k2_v3 + 2 * k3_v3 + k4_v3) \/ 6<\/p>\n
    return r1_next, r2_next, r3_next, v1_next, v2_next, v3_next<\/p>\n
<\/code><\/pre>\n<\/p>\n
4\u3001\u6a21\u62df\u548c\u7ed8\u56fe<\/h3>\n<\/p>\n
\u73b0\u5728\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e0a\u9762\u7684\u51fd\u6570\u6765\u6a21\u62df\u4e09\u4f53\u95ee\u9898\uff0c\u5e76\u7ed8\u5236\u7269\u4f53\u7684\u8f68\u8ff9\u3002\u6211\u4eec\u5c06\u5b9a\u4e49\u4e00\u4e2a\u65f6\u95f4\u6b65\u957f\u548c\u603b\u7684\u6a21\u62df\u65f6\u95f4\uff0c\u4f7f\u7528Runge-Kutta\u65b9\u6cd5\u8fed\u4ee3\u66f4\u65b0\u7269\u4f53\u7684\u72b6\u6001\uff0c\u5e76\u4f7f\u7528matplotlib\u5e93\u6765\u7ed8\u5236\u5b83\u4eec\u7684\u8f68\u8ff9\u3002<\/p>\n<\/p>\n
import matplotlib.pyplot as plt<\/p>\n
\u5f15\u529b\u5e38\u6570<\/strong><\/h2>\n
G = 1.0<\/p>\n
\u65f6\u95f4\u6b65\u957f<\/strong><\/h2>\n
dt = 0.01<\/p>\n
\u603b\u65f6\u95f4<\/strong><\/h2>\n
T = 10.0<\/p>\n
\u521d\u59cb\u5316\u8f68\u8ff9<\/strong><\/h2>\n
trajectory1 = [r1]<\/p>\n
trajectory2 = [r2]<\/p>\n
trajectory3 = [r3]<\/p>\n
\u8fed\u4ee3\u66f4\u65b0<\/strong><\/h2>\n
t = 0<\/p>\n
while t < T:<\/p>\n
    r1, r2, r3, v1, v2, v3 = runge_kutta_step(r1, r2, r3, v1, v2, v3, m1, m2, m3, G, dt)<\/p>\n
    trajectory1.append(r1)<\/p>\n
    trajectory2.append(r2)<\/p>\n
    trajectory3.append(r3)<\/p>\n
    t += dt<\/p>\n
\u8f6c\u6362\u4e3a\u6570\u7ec4<\/strong><\/h2>\n
trajectory1 = np.array(trajectory1)<\/p>\n
trajectory2 = np.array(trajectory2)<\/p>\n
trajectory3 = np.array(trajectory3)<\/p>\n
\u7ed8\u5236\u8f68\u8ff9<\/strong><\/h2>\n
plt.figure(figsize=(10, 10))<\/p>\n
plt.plot(trajectory1[:, 0], trajectory1[:, 1], label="Body 1")<\/p>\n
plt.plot(trajectory2[:, 0], trajectory2[:, 1], label="Body 2")<\/p>\n
plt.plot(trajectory3[:, 0], trajectory3[:, 1], label="Body 3")<\/p>\n
plt.xlabel("x")<\/p>\n
plt.ylabel("y")<\/p>\n
plt.legend()<\/p>\n
plt.show()<\/p>\n
<\/code><\/pre>\n<\/p>\n
\u901a\u8fc7\u6267\u884c\u4e0a\u9762\u7684\u4ee3\u7801\uff0c\u4f60\u5c06\u80fd\u591f\u770b\u5230\u4e09\u4e2a\u7269\u4f53\u5728\u76f8\u4e92\u4f5c\u7528\u4e0b\u7684\u8fd0\u52a8\u8f68\u8ff9\u3002\u8fd9\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u4e09\u4f53\u95ee\u9898\u6a21\u62df\uff0c\u4f46\u5b83\u5c55\u793a\u4e86\u5982\u4f55\u4f7f\u7528Python3\u6765\u89e3\u51b3\u590d\u6742\u7684\u7269\u7406\u95ee\u9898\u3002\u4f60\u53ef\u4ee5\u8fdb\u4e00\u6b65\u6269\u5c55\u8fd9\u4e2a\u6a21\u578b\uff0c\u6bd4\u5982\u589e\u52a0\u66f4\u591a\u7684\u7269\u4f53\uff0c\u6216\u8005\u8003\u8651\u76f8\u5bf9\u8bba\u6548\u5e94\uff0c\u6765\u7814\u7a76\u66f4\u590d\u6742\u7684\u7cfb\u7edf\u3002<\/p>\n<\/p>\n
\u76f8\u5173\u95ee\u7b54FAQs\uff1a<\/strong><\/h2>\n
 \u5982\u4f55\u4f7f\u7528Python3\u6765\u6a21\u62df\u4e09\u4f53\u95ee\u9898\u7684\u57fa\u672c\u539f\u7406\uff1f<\/strong>
\u4e09\u4f53\u95ee\u9898\u4e3b\u8981\u6d89\u53ca\u4e09\u4e2a\u5929\u4f53\u5728\u5f15\u529b\u4f5c\u7528\u4e0b\u7684\u76f8\u4e92\u8fd0\u52a8\u3002\u4f7f\u7528Python3\u6a21\u62df\u4e09\u4f53\u95ee\u9898\u65f6\uff0c\u901a\u5e38\u4f1a\u91c7\u7528\u6570\u503c\u79ef\u5206\u65b9\u6cd5\uff0c\u5982\u6b27\u62c9\u6cd5\u6216Runge-Kutta\u6cd5\u3002\u9996\u5148\u9700\u8981\u5b9a\u4e49\u6bcf\u4e2a\u5929\u4f53\u7684\u521d\u59cb\u4f4d\u7f6e\u3001\u901f\u5ea6\u4ee5\u53ca\u8d28\u91cf\uff0c\u7136\u540e\u901a\u8fc7\u8ba1\u7b97\u5b83\u4eec\u4e4b\u95f4\u7684\u5f15\u529b\u6765\u66f4\u65b0\u4f4d\u7f6e\u548c\u901f\u5ea6\uff0c\u6700\u7ec8\u5b9e\u73b0\u8fd0\u52a8\u8f68\u8ff9\u7684\u6a21\u62df\u3002<\/p>\n
\u5728Python\u4e2d\u5b9e\u73b0\u4e09\u4f53\u6a21\u62df\u9700\u8981\u54ea\u4e9b\u5e93\u548c\u5de5\u5177\uff1f<\/strong>
\u4e3a\u5b9e\u73b0\u4e09\u4f53\u6a21\u62df\uff0c\u5efa\u8bae\u4f7f\u7528NumPy\u5e93\u6765\u8fdb\u884c\u9ad8\u6548\u7684\u6570\u5b66\u8ba1\u7b97\uff0cMatplotlib\u5e93\u7528\u4e8e\u53ef\u89c6\u5316\u8fd0\u52a8\u8f68\u8ff9\u3002\u6b64\u5916\uff0c\u5982\u679c\u9700\u8981\u66f4\u9ad8\u6548\u7684\u8ba1\u7b97\uff0c\u53ef\u4ee5\u8003\u8651\u4f7f\u7528SciPy\u5e93\u4e2d\u7684\u79ef\u5206\u51fd\u6570\u3002\u901a\u8fc7\u8fd9\u4e9b\u5de5\u5177\uff0c\u53ef\u4ee5\u65b9\u4fbf\u5730\u5904\u7406\u6570\u7ec4\u548c\u7ed8\u5236\u52a8\u6001\u56fe\u5f62\u3002<\/p>\n
\u5982\u4f55\u4f18\u5316\u4e09\u4f53\u6a21\u62df\u4ee3\u7801\u7684\u6027\u80fd\uff1f<\/strong>
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