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import numpy as np |
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class MidPoint(): |
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def __init__(self, j): |
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self.j = j |
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def initial(self): |
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j = self.j |
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h, b, n, u = 0.1, None, None, None |
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x, y, z, px, py, pz = None, 0.1, 0.001, 0.01, None, 0.0001 |
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xa, ya, za, pxa, pya, pza = None, None, None, None, None, None |
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if j == 1: |
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b = 5.0 / 4 |
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n = b / (1 + b) ** 2 |
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u = 1.0 / (1.0 / b + b + 2.0) |
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x = 10.0 |
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py = 0.5 |
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elif j == 2: |
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b = 3.0 / 4 |
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n = b / (1 + b) ** 2 |
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u = 1.0 / (1.0 / b + b + 2.0) |
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x = 8.3 |
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py = 0.6 |
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elif j == 3: |
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b = 3.0 / 2 |
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x = 12.0 |
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py = 0.4 |
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elif j == 4: |
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b = 7.0 / 4 |
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n = b / (1 + b) ** 2 |
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u = 1.0 / (1.0 / b + b + 2.0) |
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x = 15.0 |
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py = 0.35 |
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elif j == 5: |
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b = 1.0 |
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n = b / (1 + b) ** 2 |
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u = 1.0 / (1.0 / b + b + 2.0) |
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x = 18.0 |
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py = 0.3 |
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elif j == 6: |
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b = 3.0 / 5 |
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n = b / (1 + b) ** 2 |
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u = 1.0 / (1.0 / b + b + 2.0) |
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x = 20.0 |
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py = 0.25 |
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elif j == 7: |
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b = 5.0 / 7 |
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n = b / (1 + b) ** 2 |
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u = 1.0 / (1.0 / b + b + 2.0) |
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x = 22.0 |
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py = 0.22 |
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elif j == 8: |
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b = 2.0 |
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x = 26.0 |
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py = 0.2 |
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elif j == 9: |
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b = 0.5 |
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n = b / (1 + b) ** 2 |
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u = 1.0 / (1.0 / b + b + 2.0) |
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x = 30.0 |
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y = 0.5 |
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z = 0.1 |
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pz = 0.01 |
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elif j == 10: |
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b = 5.0 |
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n = b / (1 + b) ** 2 |
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u = 1.0 / (1.0 / b + b + 2.0) |
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x = 35.0 |
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y = 2.0 |
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z = 0.1 |
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pz = 0.03 |
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py = 0.15 |
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xa, ya, za, pxa, pya, pza = x, y, z, px, py, pz |
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return j, h, b, n, x, y, z, xa, ya, za, px, py, pz, pxa, pya, pza |
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def f(self, x, y, z, px, py, pz, b): |
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n = b / (1 + b) ** 2 |
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u = 1 / (1 / b + b + 2) |
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ht = px ** 2 / 2 + py ** 2 / 2 + pz ** 2 / 2 |
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hv = -1 / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) |
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h1pn = 1 / (2 * x ** 2 + 2 * y ** 2 + 2 * z ** 2) - (((u + 3) * (px ** 2 + py ** 2 + pz ** 2)) / 2 + (u * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2)) ** 2) / 2) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + ((3 * u) / 8 - 1 / 8) * ( |
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px ** 2 + py ** 2 + pz ** 2) ** 2 |
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e = ht + hv + h1pn |
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vnpx = px |
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v1pnpx = 4 * px * ((3 * n) / 8 - 1 / 8) * (px ** 2 + py ** 2 + pz ** 2) - (px * (n + 3) + (n * x * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2))) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2)) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (1 / 2) |
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vpx = vnpx + v1pnpx |
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vnpy = py |
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v1pnpy = 4 * py * ((3 * n) / 8 - 1 / 8) * (px ** 2 + py ** 2 + pz ** 2) - (py * (n + 3) + (n * y * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2))) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2)) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (1 / 2) |
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vpy = vnpy + v1pnpy |
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vnpz = pz |
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v1pnpz = 4 * pz * ((3 * n) / 8 - 1 / 8) * (px ** 2 + py ** 2 + pz ** 2) - (pz * (n + 3) + (n * z * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2))) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2)) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (1 / 2) |
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vpz = vnpz + v1pnpz |
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vnx = x / (x ** 2 + y ** 2 + z ** 2) ** (3 / 2) |
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v1pnx = (x * ((n * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2)) ** 2) / 2 + ( |
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(n + 3) * (px ** 2 + py ** 2 + pz ** 2)) / 2)) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2) - (4 * x) / (2 * x ** 2 + 2 * y ** 2 + 2 * z ** 2) ** 2 + (n * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2)) * ((px * x ** 2) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (3 / 2) - px / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * x * y) / ( |
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x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2) + ( |
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pz * x * z) / ( |
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x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2))) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (1 / 2) |
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vx = vnx + v1pnx |
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vny = y / (x ** 2 + y ** 2 + z ** 2) ** (3 / 2) |
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v1pny = (y * ((n * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2)) ** 2) / 2 + ( |
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(n + 3) * (px ** 2 + py ** 2 + pz ** 2)) / 2)) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2) - (4 * y) / (2 * x ** 2 + 2 * y ** 2 + 2 * z ** 2) ** 2 + (n * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2)) * ((py * y ** 2) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (3 / 2) - py / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (px * x * y) / ( |
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x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2) + ( |
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pz * y * z) / ( |
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x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2))) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (1 / 2) |
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vy = vny + v1pny |
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vnz = z / (x ** 2 + y ** 2 + z ** 2) ** (3 / 2) |
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v1pnz = (z * ((n * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2)) ** 2) / 2 + ( |
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(n + 3) * (px ** 2 + py ** 2 + pz ** 2)) / 2)) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2) - (4 * z) / (2 * x ** 2 + 2 * y ** 2 + 2 * z ** 2) ** 2 + (n * ( |
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(px * x) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (py * y) / (x ** 2 + y ** 2 + z ** 2) ** ( |
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1 / 2) + (pz * z) / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2)) * ((pz * z ** 2) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (3 / 2) - pz / (x ** 2 + y ** 2 + z ** 2) ** (1 / 2) + (px * x * z) / ( |
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x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2) + ( |
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py * y * z) / ( |
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x ** 2 + y ** 2 + z ** 2) ** ( |
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3 / 2))) / ( |
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x ** 2 + y ** 2 + z ** 2) ** (1 / 2) |
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vz = vnz + v1pnz |
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return vx, vy, vz, vpx, vpy, vpz, e |
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def implecitsymplectic(self, h, x, y, z, px, py, pz, b): |
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x1 = x |
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y1 = y |
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z1 = z |
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px1 = px |
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py1 = py |
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pz1 = pz |
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num = 0 |
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d = 1.1 |
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while (d > 1e-16 and num < 1000): |
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px2 = px |
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py2 = py |
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pz2 = pz |
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x2 = x |
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y2 = y |
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z2 = z |
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x = (x1 + x) / 2 |
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y = (y1 + y) / 2 |
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z = (z1 + z) / 2 |
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px = (px1 + px) / 2 |
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py = (py1 + py) / 2 |
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pz = (pz1 + pz) / 2 |
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vx, vy, vz, vpx, vpy, vpz, e = self.f(x, y, z, px, py, pz, b) |
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x = x1 + h * vpx |
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y = y1 + h * vpy |
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z = z1 + h * vpz |
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px = px1 - h * vx |
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py = py1 - h * vy |
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pz = pz1 - h * vz |
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d = np.sqrt(np.abs(x - x2) + np.abs(y - y2) + np.abs(z - z2) + np.abs(px - px2) + np.abs(py - py2) + np.abs( |
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pz - pz2)) |
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num += 1 |
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return x, y, z, px, py, pz |
