Phisics/Atom_collaps.ipynb

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   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots, LinearAlgebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "cd(s\"C:\\Users\\User\\Desktop\\Mycop\\plots\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "https://doi.org/10.1119/10.0000189\n",
    "\n",
    "$$\n",
    "\\vec \\lambda = \\frac{\\vec n + (\\gamma-1)(\\vec n\\cdot\\vec\\beta)\\vec \\beta + \\gamma\\vec\\beta}{\\gamma(1 + \\vec\\beta\\cdot\\vec n)}\n",
    "$$\n",
    "\n",
    "$$\n",
    "\\vec E = \\frac{\\Delta\\vec r/\\gamma^2 + \\Delta\\vec r(\\vec R\\cdot\\vec a) - \\vec a(\\vec R\\cdot\\Delta\\vec r)}{\\left((t-t^\\prime)-\\vec R\\cdot\\vec\\beta\\right)^3}\n",
    "$$\n",
    "\n",
    "$\\Delta\\vec r = \\vec R - \\vec r_0 - (t-t^\\prime)\\vec\\beta$  \n",
    "$t - t^\\prime = r = \\sqrt{x^2+y^2}$  \n",
    "$t^\\prime = t - \\sqrt{x^2+y^2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "function E(r, t)\n",
    "    r -= x₀(t)\n",
    "    t′ = t - hypot(r...)\n",
    "    β = v₀ₓ(t′)\n",
    "    a = a₀ₓ(t′)\n",
    "    γ = 1/sqrt(1 - β ⋅ β)\n",
    "    Δr = r - (t-t′)*β\n",
    "    vec_E =  ( Δr/γ^2 + Δr*(r ⋅ a) - a*(r ⋅ Δr) ) / ( t-t′ - r ⋅ β )^3\n",
    "    hypot(vec_E...)\n",
    "end\n",
    "\n",
    "function λ(t, t′, ϕ)\n",
    "    n = [cos(ϕ) sin(ϕ)]\n",
    "    β = v₀ₓ(t′) \n",
    "    nβ = n ⋅ β\n",
    "    γ = 1/sqrt(1-β ⋅ β)\n",
    "    vec_λ = ( n + (γ-1)*nβ*β + γ*β ) / ( γ*(1+nβ) )\n",
    "    return x₀(t′) + (t-t′)*vec_λ\n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bremsstrahlung"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# двукомпонентные вектора\n",
    "x₀(t) = begin\n",
    "    [t<=0 ? t/√2 :\n",
    "    0<t<=1 ? 1 + t/√2 - sqrt(1+t^2) :\n",
    "    1 - 1/√2 0.0]\n",
    "end\n",
    "\n",
    "v₀ₓ(t) = begin\n",
    "    [t<=0 ? 1.0/√2 :\n",
    "    0<t<=1 ? 1.0/√2 - t/sqrt(1+t^2) : 0.0 0.0]\n",
    "end\n",
    "\n",
    "a₀ₓ(t) = begin\n",
    "    [0<t<=1 ? -1.0/sqrt(1+t^2)^3 : 0.0 0.0]\n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Oscillator"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [],
   "source": [
    "x₀(t)  = [ 0.0 t<0 ? 0.0 : 0.8/π * sin(π*t) ]\n",
    "v₀ₓ(t) = [ 0.0 t<0 ? 0.0 : 0.8cos(π*t) ]\n",
    "a₀ₓ(t) = [ 0.0 t<0 ? 0.0 : -0.8π * sin(π*t) ];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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       "  1407.64,366.846 1410.7,366.846 1413.76,366.846 1416.82,366.846 1419.88,366.846 1422.94,366.846 1426,366.846 1429.05,366.846 1432.11,366.846 1435.17,366.846 \n",
       "  1438.23,366.846 1441.29,366.846 1444.35,366.846 1447.41,366.846 1450.46,366.846 1453.52,366.846 1456.58,366.846 1459.64,366.846 1462.7,366.846 1465.76,366.846 \n",
       "  1468.82,366.846 1471.88,366.846 1474.93,366.846 1477.99,366.846 1481.05,366.846 1484.11,366.846 1487.17,366.846 1490.23,366.846 1493.29,366.846 1496.34,366.846 \n",
       "  1499.4,366.846 1502.46,366.846 1505.52,366.846 1508.58,366.846 1511.64,366.846 1514.7,366.846 1517.75,366.846 1520.81,366.846 1523.87,366.846 1526.93,366.846 \n",
       "  1529.99,366.846 1533.05,366.846 1536.11,366.846 1539.16,366.846 1542.22,366.846 1545.28,366.846 1548.34,366.846 1551.4,366.846 1554.46,366.846 1557.52,366.846 \n",
       "  1560.57,366.846 1563.63,366.846 1566.69,366.846 1569.75,366.846 1572.81,366.846 1575.87,366.846 1578.93,366.846 1581.99,366.846 1585.04,366.846 1588.1,366.846 \n",
       "  1591.16,366.846 1594.22,366.846 1597.28,366.846 1600.34,366.846 1603.4,366.846 1606.45,366.846 1609.51,366.846 1612.57,366.846 1615.63,366.846 1618.69,366.846 \n",
       "  1621.75,366.846 1624.81,366.846 1627.86,366.846 1630.92,366.846 1633.98,366.846 1637.04,366.846 1640.1,366.846 1643.16,366.846 1646.22,366.846 1649.27,366.846 \n",
       "  1652.33,366.846 1655.39,366.846 1658.45,366.846 1661.51,366.846 1664.57,366.846 1667.63,366.846 1670.68,366.846 1673.74,366.846 1676.8,366.846 1679.86,366.846 \n",
       "  1682.92,366.846 1685.98,366.846 1689.04,366.846 1692.1,366.846 1695.15,366.846 1698.21,366.846 1701.27,366.846 1704.33,366.846 1707.39,366.846 1710.45,366.846 \n",
       "  1713.51,366.846 1716.56,366.846 1719.62,366.846 1722.68,366.846 1725.74,366.846 1728.8,366.846 1731.86,366.846 1734.92,366.846 1737.97,366.846 1741.03,366.846 \n",
       "  1744.09,366.846 1747.15,366.846 1750.21,366.846 1753.27,366.846 1756.33,366.846 1759.38,366.846 1762.44,366.846 1765.5,366.846 1768.56,366.846 1771.62,366.846 \n",
       "  1774.68,366.846 1777.74,366.846 1780.8,366.846 1783.85,366.846 1786.91,366.846 1789.97,366.846 1793.03,366.846 1796.09,366.846 1799.15,366.846 1802.21,366.846 \n",
       "  1805.26,366.846 1808.32,366.846 1811.38,366.846 1814.44,366.846 1817.5,366.846 1820.56,366.846 1823.62,366.846 1826.67,366.846 1829.73,366.846 1832.79,366.846 \n",
       "  1835.85,366.846 1838.91,366.846 1841.97,366.846 1845.03,366.846 1848.08,366.846 1851.14,366.846 1854.2,366.846 1857.26,366.846 1860.32,366.846 1863.38,366.846 \n",
       "  1866.44,366.846 1869.49,366.846 1872.55,366.846 1875.61,366.846 1878.67,366.846 1881.73,366.846 1884.79,366.846 1887.85,366.846 1890.91,366.846 1893.96,366.846 \n",
       "  1897.02,366.846 1900.08,366.846 1903.14,366.846 1906.2,366.846 1909.26,366.846 1912.32,366.846 1915.37,366.846 1918.43,366.846 1921.49,366.846 1924.55,366.846 \n",
       "  1927.61,366.846 1930.67,366.846 1933.73,366.846 1936.78,366.846 1939.84,366.846 1942.9,366.846 1945.96,366.846 1949.02,366.846 1952.08,366.846 1955.14,366.846 \n",
       "  1958.19,366.846 1961.25,366.846 1964.31,366.846 1967.37,366.846 1970.43,366.846 1973.49,366.846 1976.55,366.846 1979.6,366.846 1982.66,366.846 1985.72,366.846 \n",
       "  1988.78,366.846 1991.84,366.846 1994.9,366.846 1997.96,366.846 2001.02,366.846 2004.07,366.846 2007.13,366.846 2010.19,366.846 2013.25,366.846 2016.31,366.846 \n",
       "  2019.37,366.846 2022.43,366.846 2025.48,366.846 2028.54,366.846 2031.6,366.846 2034.66,366.846 2037.72,366.846 2040.78,366.846 2043.84,366.846 2046.89,366.846 \n",
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      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tt = -1:0.01:2\n",
    "ploter(fun) = plot(tt, vcat(fun.(tt)...), lab = \"$(fun)\")\n",
    "\n",
    "plot(ploter(x₀), ploter(v₀ₓ), ploter(a₀ₓ), line = 3 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = 3.0\n",
    "X = range(-4, stop = 4, length = 128)\n",
    "Y = range(-4, stop = 4, length = 128)  \n",
    "Z = [ E([x y], t ) for y in Y, x in X ];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1623.1534708088388"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "maximum(Z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "heatmap(X, Y, log10.(Z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      ]
     },
     "execution_count": 203,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "heatmap(X, Y, clamp.(log10.(Z), -2, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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     "execution_count": 204,
     "metadata": {},
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   ],
   "source": [
    "heatmap(X, Y, clamp.(log10.(Z), -2, 2), # oбрезание\n",
    "    legend = false, ticks = nothing, border = :none, size = (512, 256), c = :devon  ) # :grays\n",
    "\n",
    "for fi in π/6:π/6:2π\n",
    "    zz = vcat( [λ(t, tt, fi) for tt in t-10:0.1:t ]... );\n",
    "    plot!(zz[:,1], zz[:,2], line = (1, :yellow), )\n",
    "end\n",
    "scatter!([x₀(t)[1]],[x₀(t)[2]], c = :red)\n",
    "vline!([0.0], c = :green, line = (1, :dash))\n",
    "yaxis!((Y[1], Y[end]))\n",
    "xaxis!((X[1], X[end]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 19.196218 seconds (262.95 M allocations: 6.260 GiB, 4.57% gc time)\n"
     ]
    }
   ],
   "source": [
    "@time for (i, t) in enumerate(range(0, stop = 8, length = 64)) # картинки для гифки\n",
    "    Z = [ E([x y], t) for y in Y, x in X ];\n",
    "    heatmap(X, Y, clamp.(log10.(Z), -2, 2), \n",
    "        legend = false, ticks = nothing, border = :none, size = (512, 256)  ) # \n",
    "    # , c = :grays\n",
    "\n",
    "    for fi in π/6:π/6:2π # t-10 на сколько назад во времени\n",
    "        zz = vcat( [λ(t, tt, fi) for tt in t-10:0.01:t ]... )  # 0.01 четкость линий поля\n",
    "        plot!(zz[:,1], zz[:,2], line = (1, :yellow), )\n",
    "    end\n",
    "    scatter!([x₀(t)[1]],[x₀(t)[2]], c = :red) # , title = \"t = $(round(t, digits = 1))\"\n",
    "    #vline!([0.0], c = :green, line = (1, :dash))\n",
    "    yaxis!((Y[1], Y[end]))\n",
    "    xaxis!((X[1], X[end]))\n",
    "\n",
    "    savefig(\"img$i.png\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Синхротрон\n",
    "\n",
    "https://cyberleninka.ru/article/n/problemy-teorii-atoma-vodoroda-i-fotonnoe-vzaimodeystvie-elektrona-i-protona\n",
    "\n",
    "https://cyberleninka.ru/article/n/sinhrotron-kak-makroatom"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "function λ₂(t, t′, ϕ)\n",
    "    β = radi*omeg\n",
    "    γ = 1/sqrt(1-β^2)\n",
    "    ψt = omeg*t′ + 2atan( β-tan( 0.5γ*omeg*t′ + atan( γ*( β-tan(0.5ϕ) ) ) )/γ )\n",
    "    \n",
    "    return x₀(t′) + (t-t′)*[cos(ψt) sin(ψt)]\n",
    "end;\n",
    "\n",
    "radi = 0.2\n",
    "omeg = 0.4/radi\n",
    "peri = 2π/omeg\n",
    "\n",
    "x₀(t) = radi*[cos(omeg*t) sin(omeg*t)]\n",
    "v₀ₓ(t) = radi*omeg*[-sin(omeg*t) cos(omeg*t)]\n",
    "a₀ₓ(t) = -omeg^2 * x₀(t);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = range(-4, stop = 4, length = 128)\n",
    "Y = range(-4, stop = 4, length = 128) \n",
    "\n",
    "t = 3.0\n",
    "Z = [ E([x y], t ) for y in Y, x in X ];\n",
    "heatmap(X, Y, log10.(Z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
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       "  1576.39,1551.31 1581.7,1549.32 1586.9,1547.32 1592.02,1545.31 1597.02,1543.28 1601.93,1541.24 1606.73,1539.18 1611.42,1537.11 1616,1535.02 1620.47,1532.92 \n",
       "  1624.81,1530.81 1629.04,1528.68 1633.14,1526.54 1637.12,1524.38 1640.96,1522.21 1644.67,1520.03 1648.25,1517.83 1651.68,1515.62 1654.97,1513.4 1658.11,1511.17 \n",
       "  1661.11,1508.92 1663.94,1506.67 1666.62,1504.4 1669.14,1502.12 1671.49,1499.83 1673.67,1497.52 1675.68,1495.21 1677.51,1492.88 1679.16,1490.55 1680.62,1488.2 \n",
       "  1681.9,1485.85 1682.98,1483.48 1683.86,1481.11 1684.54,1478.72 1685.02,1476.33 1685.28,1473.93 1685.34,1471.52 1685.17,1469.1 1684.79,1466.67 1684.17,1464.24 \n",
       "  1683.33,1461.8 1682.26,1459.35 1680.94,1456.89 1679.39,1454.43 1677.6,1451.96 1675.55,1449.49 1673.26,1447.01 1670.71,1444.52 1667.9,1442.03 1664.84,1439.53 \n",
       "  1661.51,1437.03 1657.92,1434.52 1654.06,1432.01 1649.93,1429.5 1645.53,1426.98 1640.86,1424.46 1635.92,1421.94 1630.71,1419.41 1625.22,1416.88 1619.45,1414.35 \n",
       "  1613.42,1411.82 1607.11,1409.28 1600.53,1406.75 1593.68,1404.21 1586.56,1401.67 1579.18,1399.13 1571.53,1396.6 1563.63,1394.06 1555.47,1391.52 1547.06,1388.98 \n",
       "  1538.41,1386.45 1529.51,1383.91 1520.39,1381.38 1511.03,1378.85 1501.46,1376.32 1491.68,1373.79 1481.69,1371.26 1471.51,1368.74 1461.14,1366.22 1450.59,1363.71 \n",
       "  1439.88,1361.2 1429.02,1358.69 1418.01,1356.19 1406.87,1353.69 1395.6,1351.19 1384.23,1348.7 1372.77,1346.22 1361.22,1343.74 1349.6,1341.27 1337.93,1338.8 \n",
       "  1326.21,1336.34 1314.46,1333.89 1302.7,1331.44 1290.94,1329 1279.19,1326.57 1267.46,1324.15 1255.78,1321.73 1244.15,1319.32 1232.58,1316.92 1221.1,1314.52 \n",
       "  1209.71,1312.14 1198.42,1309.76 1187.25,1307.4 1176.22,1305.04 1165.32,1302.69 1154.57,1300.36 1143.99,1298.03 1133.58,1295.71 1123.36,1293.4 1113.32,1291.1 \n",
       "  1103.49,1288.82 1093.86,1286.54 1084.46,1284.27 1075.27,1282.02 1066.31,1279.78 1057.59,1277.54 1049.1,1275.32 1040.87,1273.12 1032.87,1270.92 1025.13,1268.73 \n",
       "  1017.65,1266.56 1010.42,1264.4 1003.44,1262.25 996.728,1260.12 990.273,1258 984.077,1255.89 978.14,1253.79 972.462,1251.7 967.039,1249.63 961.872,1247.58 \n",
       "  956.958,1245.53 952.295,1243.5 947.88,1241.48 943.709,1239.48 939.78,1237.49 936.088,1235.51 932.631,1233.55 929.403,1231.6 926.402,1229.67 923.621,1227.74 \n",
       "  921.057,1225.84 918.705,1223.94 916.56,1222.06 914.617,1220.2 912.871,1218.35 911.317,1216.51 909.951,1214.69 908.766,1212.88 907.757,1211.08 906.92,1209.3 \n",
       "  906.249,1207.54 905.74,1205.78 905.386,1204.05 905.184,1202.32 905.127,1200.61 905.212,1198.92 905.433,1197.23 905.785,1195.57 906.263,1193.91 906.864,1192.27 \n",
       "  907.582,1190.65 908.413,1189.03 909.352,1187.44 910.396,1185.85 911.539,1184.28 912.779,1182.72 914.111,1181.18 915.53,1179.65 917.035,1178.13 918.62,1176.63 \n",
       "  920.282,1175.13 922.018,1173.66 923.824,1172.19 925.697,1170.74 927.635,1169.3 929.633,1167.87 931.69,1166.46 933.802,1165.05 935.966,1163.66 938.18,1162.29 \n",
       "  940.442,1160.92 942.749,1159.57 945.098,1158.22 947.487,1156.89 949.914,1155.57 952.378,1154.26 954.875,1152.97 957.405,1151.68 959.965,1150.4 962.553,1149.14 \n",
       "  965.168,1147.88 967.808,1146.64 970.471,1145.41 973.157,1144.18 975.863,1142.97 978.588,1141.76 981.331,1140.56 984.091,1139.38 986.865,1138.2 989.655,1137.03 \n",
       "  992.457,1135.87 995.271,1134.72 998.096,1133.57 1000.93,1132.44 1003.78,1131.31 1006.63,1130.19 1009.49,1129.07 1012.36,1127.97 1015.23,1126.87 1018.11,1125.77 \n",
       "  1020.99,1124.69 1023.87,1123.61 1026.76,1122.53 1029.66,1121.46 1032.55,1120.4 1035.44,1119.34 1038.34,1118.29 1041.23,1117.24 1044.13,1116.2 1047.03,1115.16 \n",
       "  1049.92,1114.12 1052.81,1113.09 1055.7,1112.07 1058.59,1111.04 1061.47,1110.02 1064.36,1109 1067.24,1107.99 1070.11,1106.97 1072.98,1105.96 1075.85,1104.96 \n",
       "  1078.71,1103.95 1081.57,1102.94 1084.42,1101.94 1087.26,1100.94 1090.1,1099.93 1092.93,1098.93 1095.76,1097.93 1098.58,1096.93 1101.39,1095.93 1104.2,1094.93 \n",
       "  1107,1093.92 1109.79,1092.92 1112.57,1091.91 1115.34,1090.91 1118.11,1089.9 1120.87,1088.89 1123.61,1087.88 1126.35,1086.86 1129.08,1085.85 1131.8,1084.83 \n",
       "  1134.51,1083.81 1137.21,1082.78 1139.9,1081.75 1142.58,1080.72 1145.24,1079.68 1147.9,1078.64 1150.54,1077.59 1153.17,1076.54 1155.79,1075.49 1158.4,1074.43 \n",
       "  1161,1073.36 1163.58,1072.29 1166.15,1071.21 1168.7,1070.13 1171.25,1069.04 1173.77,1067.95 1176.29,1066.84 1178.79,1065.74 1181.27,1064.62 1183.74,1063.5 \n",
       "  1186.19,1062.37 1188.63,1061.23 1191.05,1060.08 1193.45,1058.93 1195.84,1057.77 1198.21,1056.6 1200.56,1055.42 1202.9,1054.24 1205.21,1053.04 1207.51,1051.84 \n",
       "  1209.79,1050.62 1212.05,1049.4 1214.29,1048.17 1216.51,1046.93 1218.7,1045.68 1220.88,1044.42 1223.03,1043.14 1225.17,1041.86 1227.28,1040.57 1229.37,1039.27 \n",
       "  1231.43,1037.96 1233.47,1036.63 1235.49,1035.3 1237.48,1033.96 1239.45,1032.6 1241.39,1031.23 1243.3,1029.85 1245.19,1028.46 1247.05,1027.06 1248.89,1025.65 \n",
       "  1250.7,1024.22 1252.47,1022.79 1254.22,1021.34 1255.94,1019.88 1257.64,1018.41 1259.3,1016.92 1260.92,1015.43 1262.52,1013.92 1264.09,1012.4 1265.62,1010.86 \n",
       "  1267.12,1009.32 1268.59,1007.76 1270.03,1006.19 1271.43,1004.61 1272.79,1003.01 1274.12,1001.4 1275.42,999.78 1276.67,998.146 1277.89,996.5 1279.08,994.841 \n",
       "  1280.22,993.169 1281.33,991.484 1282.4,989.787 1283.43,988.077 1284.42,986.354 1285.38,984.619 1286.29,982.87 1287.16,981.11 1287.99,979.336 1288.78,977.55 \n",
       "  1289.52,975.751 1290.23,973.94 1290.89,972.116 1291.51,970.279 1292.09,968.43 1292.62,966.569 1293.11,964.695 1293.56,962.809 1293.96,960.91 1294.32,959 \n",
       "  1294.63,957.077 1294.9,955.142 1295.13,953.196 1295.31,951.237 1295.45,949.266 1295.55,947.284 1295.6,945.29 1295.6,943.285 1295.57,941.268 1295.49,939.24 \n",
       "  1295.37,937.2 1295.2,935.15 1294.99,933.088 1294.75,931.016 1294.46,928.932 1294.13,926.838 1293.76,924.734 1293.35,922.619 1292.9,920.493 1292.42,918.358 \n",
       "  1291.9,916.212 1291.34,914.057 1290.75,911.892 1290.13,909.717 1289.47,907.533 1288.79,905.34 1288.07,903.137 1287.33,900.926 1286.56,898.706 1285.76,896.477 \n",
       "  1284.95,894.24 1284.11,891.994 1283.25,889.74 1282.37,887.478 1281.48,885.209 1280.58,882.932 1279.66,880.647 1278.73,878.356 1277.8,876.057 1276.86,873.752 \n",
       "  1275.92,871.439 1274.98,869.121 1274.03,866.796 1273.1,864.465 1272.17,862.128 1271.25,859.785 1270.34,857.437 1269.44,855.084 1268.56,852.726 1267.7,850.363 \n",
       "  1266.87,847.995 1266.05,845.623 1265.26,843.246 1264.5,840.866 1263.77,838.482 1263.07,836.094 1262.41,833.703 1261.78,831.309 1261.19,828.911 1260.64,826.512 \n",
       "  1260.13,824.109 1259.67,821.705 1259.25,819.298 1258.88,816.89 1258.55,814.48 1258.27,812.069 1258.05,809.656 1257.87,807.243 1257.74,804.829 1257.66,802.415 \n",
       "  1257.64,800 \n",
       "  \"/>\n",
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       "  2484.01,1386.21 2466.2,1394.07 2448.27,1402.02 2430.24,1410.06 2412.12,1418.17 2393.94,1426.34 2375.71,1434.56 2357.44,1442.82 2339.16,1451.11 2320.89,1459.41 \n",
       "  2302.65,1467.72 2284.44,1476.02 2266.3,1484.31 2248.23,1492.56 2230.27,1500.77 2212.41,1508.94 2194.69,1517.04 2177.12,1525.06 2159.72,1533 2142.49,1540.85 \n",
       "  2125.47,1548.59 2108.66,1556.22 2092.07,1563.73 2075.73,1571.11 2059.64,1578.35 2043.81,1585.44 2028.27,1592.38 2013.01,1599.15 1998.06,1605.76 1983.41,1612.19 \n",
       "  1969.08,1618.45 1955.08,1624.52 1941.41,1630.4 1928.09,1636.09 1915.1,1641.58 1902.47,1646.88 1890.2,1651.97 1878.28,1656.85 1866.72,1661.53 1855.53,1666.01 \n",
       "  1844.7,1670.27 1834.24,1674.33 1824.14,1678.18 1814.41,1681.82 1805.04,1685.25 1796.04,1688.47 1787.4,1691.49 1779.11,1694.31 1771.19,1696.92 1763.61,1699.34 \n",
       "  1756.38,1701.55 1749.5,1703.57 1742.95,1705.4 1736.74,1707.04 1730.86,1708.5 1725.31,1709.77 1720.07,1710.86 1715.15,1711.78 1710.53,1712.53 1706.22,1713.1 \n",
       "  1702.2,1713.52 1698.47,1713.77 1695.02,1713.87 1691.85,1713.81 1688.95,1713.61 1686.31,1713.26 1683.93,1712.77 1681.8,1712.14 1679.91,1711.38 1678.26,1710.49 \n",
       "  1676.84,1709.48 1675.65,1708.35 1674.67,1707.1 1673.9,1705.73 1673.34,1704.26 1672.98,1702.68 1672.82,1701 1672.84,1699.22 1673.04,1697.35 1673.42,1695.38 \n",
       "  1673.97,1693.33 1674.68,1691.19 1675.55,1688.97 1676.58,1686.67 1677.75,1684.29 1679.07,1681.84 1680.53,1679.32 1682.13,1676.73 1683.85,1674.08 1685.7,1671.37 \n",
       "  1687.66,1668.59 1689.75,1665.76 1691.94,1662.87 1694.25,1659.93 1696.65,1656.94 1699.16,1653.91 1701.76,1650.82 1704.46,1647.69 1707.24,1644.52 1710.11,1641.31 \n",
       "  1713.07,1638.06 1716.1,1634.77 1719.2,1631.44 1722.38,1628.09 1725.63,1624.7 1728.95,1621.27 1732.33,1617.82 1735.77,1614.34 1739.27,1610.83 1742.82,1607.3 \n",
       "  1746.43,1603.74 1750.09,1600.16 1753.8,1596.55 1757.56,1592.93 1761.36,1589.28 1765.2,1585.61 1769.09,1581.92 1773.01,1578.22 1776.97,1574.49 1780.96,1570.75 \n",
       "  1784.99,1567 1789.04,1563.23 1793.13,1559.44 1797.24,1555.64 1801.38,1551.83 1805.54,1548 1809.73,1544.16 1813.93,1540.31 1818.16,1536.45 1822.41,1532.58 \n",
       "  1826.67,1528.7 1830.95,1524.8 1835.24,1520.9 1839.55,1516.99 1843.87,1513.06 1848.19,1509.13 1852.53,1505.19 1856.88,1501.25 1861.23,1497.29 1865.59,1493.33 \n",
       "  1869.96,1489.36 1874.33,1485.38 1878.7,1481.39 1883.08,1477.4 1887.45,1473.4 1891.83,1469.4 1896.2,1465.39 1900.57,1461.37 1904.94,1457.35 1909.31,1453.32 \n",
       "  1913.67,1449.29 1918.03,1445.25 1922.38,1441.21 1926.72,1437.16 1931.05,1433.1 1935.38,1429.04 1939.69,1424.98 1944,1420.91 1948.29,1416.84 1952.57,1412.77 \n",
       "  1956.84,1408.69 1961.1,1404.6 1965.34,1400.52 1969.56,1396.43 1973.77,1392.33 1977.97,1388.23 1982.14,1384.13 1986.3,1380.03 1990.44,1375.92 1994.55,1371.81 \n",
       "  1998.65,1367.7 2002.73,1363.59 2006.78,1359.47 2010.81,1355.35 2014.82,1351.23 2018.8,1347.11 2022.76,1342.99 2026.69,1338.86 2030.6,1334.74 2034.48,1330.61 \n",
       "  2038.33,1326.49 2042.15,1322.36 2045.95,1318.23 2049.71,1314.11 2053.44,1309.98 2057.14,1305.86 2060.81,1301.73 2064.45,1297.61 2068.05,1293.49 2071.61,1289.37 \n",
       "  2075.14,1285.25 2078.64,1281.14 2082.1,1277.03 2085.52,1272.92 2088.9,1268.82 2092.24,1264.72 2095.54,1260.62 2098.8,1256.53 2102.02,1252.45 2105.19,1248.37 \n",
       "  2108.32,1244.3 2111.41,1240.23 2114.45,1236.17 2117.44,1232.12 2120.39,1228.08 2123.29,1224.05 2126.14,1220.02 2128.94,1216.01 2131.69,1212.01 2134.38,1208.01 \n",
       "  2137.03,1204.03 2139.62,1200.07 2142.15,1196.11 2144.63,1192.17 2147.05,1188.24 2149.41,1184.33 2151.71,1180.44 2153.95,1176.56 2156.13,1172.7 2158.25,1168.86 \n",
       "  2160.3,1165.03 2162.29,1161.23 2164.21,1157.45 2166.07,1153.69 2167.85,1149.95 2169.56,1146.24 2171.2,1142.55 2172.77,1138.88 2174.27,1135.25 2175.68,1131.64 \n",
       "  2177.02,1128.06 2178.28,1124.51 2179.46,1120.99 2180.56,1117.5 2181.57,1114.05 2182.5,1110.63 2183.34,1107.25 2184.1,1103.9 2184.76,1100.59 2185.33,1097.33 \n",
       "  2185.81,1094.1 2186.19,1090.92 2186.47,1087.78 2186.65,1084.68 2186.73,1081.64 2186.71,1078.64 2186.58,1075.69 2186.35,1072.79 2186,1069.94 2185.55,1067.15 \n",
       "  2184.98,1064.42 2184.29,1061.74 2183.49,1059.12 2182.57,1056.57 2181.52,1054.07 2180.35,1051.65 2179.06,1049.28 2177.63,1046.99 2176.07,1044.76 2174.38,1042.61 \n",
       "  2172.55,1040.53 2170.59,1038.52 2168.49,1036.6 2166.24,1034.75 2163.85,1032.98 2161.31,1031.29 2158.62,1029.69 2155.78,1028.17 2152.78,1026.74 2149.63,1025.4 \n",
       "  2146.33,1024.14 2142.86,1022.99 2139.23,1021.92 2135.44,1020.95 2131.48,1020.08 2127.35,1019.31 2123.06,1018.63 2118.6,1018.06 2113.96,1017.59 2109.16,1017.22 \n",
       "  2104.17,1016.96 2099.02,1016.8 2093.69,1016.75 2088.18,1016.8 2082.5,1016.97 2076.64,1017.24 2070.61,1017.62 2064.39,1018.11 2058.01,1018.7 2051.44,1019.41 \n",
       "  2044.71,1020.22 2037.8,1021.15 2030.72,1022.18 2023.47,1023.31 2016.06,1024.55 2008.48,1025.9 2000.73,1027.35 1992.83,1028.9 1984.78,1030.55 1976.57,1032.3 \n",
       "  1968.21,1034.14 1959.71,1036.08 1951.07,1038.11 1942.3,1040.22 1933.4,1042.42 1924.38,1044.7 1915.24,1047.06 1905.99,1049.49 1896.64,1051.99 1887.18,1054.56 \n",
       "  1877.65,1057.19 1868.02,1059.88 1858.33,1062.62 1848.56,1065.41 1838.74,1068.24 1828.87,1071.11 1818.96,1074.02 1809.01,1076.95 1799.04,1079.9 1789.05,1082.87 \n",
       "  1779.06,1085.86 1769.07,1088.85 1759.09,1091.84 1749.13,1094.83 1739.2,1097.8 1729.31,1100.77 1719.47,1103.71 1709.68,1106.63 1699.95,1109.52 1690.3,1112.37 \n",
       "  1680.72,1115.18 1671.24,1117.95 1661.85,1120.66 1652.56,1123.33 1643.38,1125.93 1634.32,1128.47 1625.38,1130.95 1616.56,1133.36 1607.89,1135.69 1599.35,1137.95 \n",
       "  1590.96,1140.13 1582.71,1142.22 1574.62,1144.24 1566.69,1146.16 1558.92,1148 1551.31,1149.75 1543.87,1151.4 1536.6,1152.97 1529.5,1154.43 1522.57,1155.81 \n",
       "  1515.81,1157.09 1509.23,1158.27 1502.82,1159.36 1496.59,1160.35 1490.54,1161.24 1484.66,1162.04 1478.95,1162.75 1473.42,1163.36 1468.05,1163.88 1462.86,1164.3 \n",
       "  1457.84,1164.64 1452.99,1164.88 1448.3,1165.04 1443.77,1165.11 1439.4,1165.1 1435.19,1165 1431.14,1164.82 1427.24,1164.57 1423.49,1164.23 1419.89,1163.82 \n",
       "  1416.43,1163.33 1413.11,1162.78 1409.93,1162.15 1406.88,1161.46 1403.97,1160.7 1401.18,1159.88 1398.52,1159 1395.97,1158.06 1393.55,1157.06 1391.24,1156.01 \n",
       "  1389.04,1154.9 1386.95,1153.75 1384.97,1152.55 1383.09,1151.3 1381.3,1150.01 1379.61,1148.67 1378.02,1147.3 1376.51,1145.88 1375.09,1144.43 1373.75,1142.95 \n",
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       "  2330.65,161.157 2333.47,171.946 2336.43,182.938 2339.53,194.124 2342.78,205.495 2346.16,217.042 2349.66,228.755 2353.28,240.622 2357.02,252.634 2360.86,264.779 \n",
       "  2364.79,277.045 2368.82,289.419 2372.93,301.89 2377.12,314.444 2381.37,327.07 2385.68,339.752 2390.04,352.478 2394.44,365.234 2398.87,378.007 2403.32,390.782 \n",
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       "  2375.6,763.862 2372.01,760.871 2368.42,757.848 2364.81,754.793 2361.2,751.708 2357.57,748.594 2353.94,745.454 2350.3,742.287 2346.65,739.096 2342.99,735.882 \n",
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       "  2264.79,664.84 2261,661.361 2257.21,657.879 2253.41,654.394 2249.61,650.907 2245.8,647.418 2241.98,643.929 2238.16,640.439 2234.33,636.95 2230.5,633.461 \n",
       "  2226.65,629.974 2222.8,626.488 2218.95,623.005 2215.09,619.525 2211.22,616.048 2207.34,612.575 2203.45,609.107 2199.56,605.643 2195.66,602.184 2191.75,598.731 \n",
       "  2187.84,595.285 2183.91,591.845 2179.98,588.412 2176.04,584.987 2172.1,581.571 2168.14,578.162 2164.17,574.763 2160.2,571.373 2156.22,567.993 2152.23,564.623 \n",
       "  2148.23,561.265 2144.22,557.917 2140.2,554.581 2136.17,551.258 2132.14,547.947 2128.09,544.649 2124.04,541.365 2119.98,538.095 2115.9,534.839 2111.82,531.598 \n",
       "  2107.73,528.373 2103.63,525.163 2099.53,521.971 2095.41,518.794 2091.28,515.636 2087.15,512.495 2083,509.372 2078.85,506.269 2074.69,503.185 2070.52,500.12 \n",
       "  2066.34,497.077 2062.16,494.054 2057.96,491.052 2053.76,488.073 2049.55,485.116 2045.33,482.183 2041.11,479.273 2036.87,476.388 2032.63,473.527 2028.39,470.692 \n",
       "  2024.14,467.883 2019.88,465.101 2015.61,462.346 2011.34,459.619 2007.07,456.92 2002.79,454.251 1998.5,451.612 1994.22,449.004 1989.92,446.426 1985.63,443.881 \n",
       "  1981.33,441.369 1977.03,438.89 1972.73,436.445 1968.42,434.035 1964.12,431.661 1959.81,429.324 1955.5,427.024 1951.2,424.762 1946.89,422.539 1942.59,420.356 \n",
       "  1938.29,418.214 1933.99,416.113 1929.7,414.055 1925.41,412.041 1921.12,410.071 1916.84,408.146 1912.57,406.267 1908.31,404.436 1904.05,402.654 1899.8,400.92 \n",
       "  1895.56,399.238 1891.33,397.606 1887.12,396.028 1882.91,394.503 1878.72,393.033 1874.54,391.619 1870.38,390.263 1866.23,388.965 1862.1,387.727 1857.99,386.55 \n",
       "  1853.89,385.435 1849.82,384.384 1845.77,383.397 1841.74,382.477 1837.73,381.624 1833.75,380.841 1829.8,380.127 1825.87,379.486 1821.97,378.918 1818.1,378.424 \n",
       "  1814.26,378.007 1810.45,377.667 1806.67,377.406 1802.93,377.226 1799.22,377.129 1795.55,377.115 1791.92,377.186 1788.33,377.344 1784.79,377.591 1781.28,377.927 \n",
       "  1777.82,378.355 1774.4,378.876 1771.03,379.492 1767.7,380.204 1764.43,381.014 1761.2,381.922 1758.03,382.932 1754.92,384.043 1751.85,385.258 1748.84,386.578 \n",
       "  1745.89,388.005 1743,389.539 1740.17,391.182 1737.4,392.934 1734.69,394.799 1732.04,396.775 1729.46,398.865 1726.94,401.069 1724.49,403.389 1722.1,405.824 \n",
       "  1719.79,408.375 1717.54,411.044 1715.36,413.83 1713.25,416.734 1711.21,419.755 1709.25,422.895 1707.35,426.152 1705.53,429.527 1703.77,433.019 1702.09,436.628 \n",
       "  1700.48,440.352 1698.94,444.191 1697.47,448.143 1696.08,452.208 1694.75,456.384 1693.49,460.668 1692.3,465.059 1691.18,469.556 1690.12,474.154 1689.13,478.853 \n",
       "  1688.2,483.648 1687.33,488.538 1686.52,493.518 1685.77,498.586 1685.08,503.737 1684.44,508.968 1683.85,514.274 1683.31,519.652 1682.81,525.096 1682.36,530.603 \n",
       "  1681.96,536.166 1681.58,541.782 1681.25,547.445 1680.94,553.15 1680.67,558.891 1680.42,564.663 1680.19,570.46 1679.98,576.276 1679.79,582.105 1679.6,587.943 \n",
       "  1679.43,593.782 1679.26,599.617 1679.08,605.442 1678.91,611.251 1678.73,617.039 1678.54,622.799 1678.34,628.527 1678.12,634.215 1677.89,639.86 1677.62,645.455 \n",
       "  1677.34,650.995 1677.02,656.476 1676.67,661.893 1676.29,667.24 1675.87,672.514 1675.41,677.71 1674.91,682.825 1674.37,687.854 1673.78,692.795 1673.14,697.643 \n",
       "  1672.45,702.395 1671.71,707.05 1670.92,711.604 1670.07,716.056 1669.17,720.403 1668.21,724.643 1667.2,728.775 1666.13,732.798 1665,736.712 1663.81,740.514 \n",
       "  1662.57,744.204 1661.27,747.783 1659.91,751.251 1658.5,754.606 1657.02,757.851 1655.5,760.984 1653.91,764.008 1652.28,766.922 1650.58,769.729 1648.84,772.428 \n",
       "  1647.05,775.022 1645.2,777.512 1643.31,779.899 1641.36,782.186 1639.37,784.373 1637.34,786.464 1635.26,788.459 1633.15,790.361 1630.99,792.172 1628.79,793.894 \n",
       "  1626.55,795.53 1624.28,797.081 1621.98,798.549 1619.64,799.938 1617.27,801.249 1614.88,802.485 1612.45,803.647 1610,804.739 1607.53,805.762 1605.03,806.718 \n",
       "  1602.51,807.611 1599.97,808.442 1597.42,809.213 1594.84,809.926 1592.25,810.584 1589.65,811.189 1587.04,811.743 1584.41,812.247 1581.78,812.704 1579.14,813.116 \n",
       "  1576.48,813.484 1573.83,813.811 1571.17,814.098 1568.5,814.347 1565.84,814.559 1563.17,814.736 1560.5,814.881 1557.83,814.994 1555.16,815.076 1552.5,815.13 \n",
       "  1549.83,815.157 1547.18,815.158 1544.52,815.134 1541.87,815.087 1539.23,815.018 1536.6,814.927 1533.97,814.817 1531.35,814.689 1528.74,814.542 1526.14,814.379 \n",
       "  1523.55,814.201 1520.97,814.007 1518.4,813.8 1515.84,813.579 1513.29,813.347 1510.76,813.103 1508.23,812.848 1505.72,812.584 1503.22,812.31 1500.74,812.028 \n",
       "  1498.26,811.738 1495.8,811.44 1493.36,811.136 1490.92,810.826 1488.51,810.51 1486.1,810.189 1483.71,809.863 1481.33,809.533 1478.97,809.2 1476.62,808.863 \n",
       "  1474.29,808.523 1471.97,808.18 1469.66,807.836 1467.37,807.489 1465.09,807.142 1462.82,806.793 1460.57,806.443 1458.33,806.093 1456.11,805.742 1453.9,805.391 \n",
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       "  1430.43,801.605 1428.36,801.272 1426.31,800.943 1424.27,800.616 1422.24,800.292 1420.22,799.971 1418.21,799.654 1416.21,799.34 1414.22,799.03 1412.24,798.724 \n",
       "  1410.27,798.422 1408.3,798.124 1406.35,797.83 1404.4,797.54 1402.46,797.256 1400.53,796.975 1398.6,796.7 1396.68,796.429 1394.77,796.164 1392.87,795.903 \n",
       "  1390.97,795.648 1389.08,795.399 1387.19,795.155 1385.31,794.917 1383.43,794.685 1381.56,794.458 1379.69,794.238 1377.82,794.025 1375.96,793.817 1374.11,793.616 \n",
       "  1372.25,793.422 1370.4,793.235 1368.55,793.055 1366.71,792.882 1364.86,792.716 1363.02,792.558 1361.18,792.407 1359.34,792.264 1357.5,792.129 1355.66,792.002 \n",
       "  1353.83,791.883 1351.99,791.773 1350.15,791.671 1348.32,791.577 1346.48,791.493 1344.64,791.418 1342.8,791.351 1340.96,791.295 1339.12,791.247 1337.27,791.21 \n",
       "  1335.43,791.182 1333.58,791.164 1331.73,791.156 1329.87,791.159 1328.01,791.172 1326.15,791.196 1324.29,791.231 1322.42,791.277 1320.55,791.334 1318.68,791.402 \n",
       "  1316.8,791.482 1314.92,791.574 1313.03,791.677 1311.14,791.793 1309.24,791.921 1307.34,792.061 1305.43,792.214 1303.52,792.379 1301.6,792.557 1299.67,792.749 \n",
       "  1297.74,792.953 1295.81,793.171 1293.86,793.402 1291.91,793.647 1289.96,793.905 1288,794.178 1286.03,794.464 1284.05,794.765 1282.06,795.079 1280.07,795.409 \n",
       "  1278.07,795.752 1276.07,796.11 1274.05,796.483 1272.03,796.87 1270,797.273 1267.96,797.69 1265.91,798.122 1263.86,798.569 1261.79,799.031 1259.72,799.508 \n",
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       "  1866.7,17.7568 1877.52,27.3907 1888.5,37.1017 1899.62,46.8812 1910.87,56.7203 1922.23,66.6101 1933.69,76.5412 1945.23,86.5044 1956.83,96.4902 1968.49,106.489 \n",
       "  1980.17,116.491 1991.88,126.487 2003.58,136.466 2015.27,146.421 2026.93,156.339 2038.54,166.213 2050.09,176.033 2061.56,185.789 2072.94,195.473 2084.2,205.075 \n",
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       "  2040.34,411.768 2034.93,409.829 2029.5,407.888 2024.07,405.945 2018.63,404.001 2013.19,402.057 2007.74,400.112 2002.29,398.168 1996.83,396.226 1991.36,394.284 \n",
       "  1985.9,392.345 1980.42,390.408 1974.95,388.474 1969.47,386.544 1963.98,384.618 1958.49,382.696 1953,380.78 1947.51,378.868 1942.01,376.963 1936.51,375.064 \n",
       "  1931.01,373.171 1925.5,371.286 1920,369.409 1914.49,367.539 1908.98,365.678 1903.46,363.826 1897.95,361.983 1892.43,360.151 1886.92,358.328 1881.4,356.516 \n",
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       "<circle clip-path=\"url(#clip612)\" cx=\"1257.64\" cy=\"800\" r=\"14\" fill=\"#ff0000\" fill-rule=\"evenodd\" fill-opacity=\"1\" stroke=\"#000000\" stroke-opacity=\"1\" stroke-width=\"3.2\"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V₀ = v₀ₓ(0)\n",
    "V² = V₀ ⋅ V₀\n",
    "Vn = normalize(V₀)\n",
    "\n",
    "heatmap(X, Y, clamp.(log10.(Z), -1, 4), # oбрезание\n",
    "    legend = false, ticks = nothing, border = :none, size = (600, 400), c = :devon  ) # :grays\n",
    "\n",
    "for alph in π/6:π/6:2π\n",
    "    nn = [cos(alph) sin(alph)]\n",
    "    fvec = (sqrt(1-V²)-1) * (Vn ⋅ nn) * Vn + nn\n",
    "    fi = angle(fvec[1] + im*fvec[2])\n",
    "    \n",
    "    Vv = V₀ ⋅ [cos(fi) sin(fi)]\n",
    "    μ = Vv + sqrt( 1-V² + Vv^2 )\n",
    "    fvec = V₀ - μ*[cos(fi) sin(fi)]\n",
    "    fi = angle(fvec[1] + im*fvec[2])\n",
    "    \n",
    "    zz = vcat( [λ(t, tt, fi) for tt in t-6:0.01:t ]... );\n",
    "    plot!(zz[:,1], zz[:,2], line = (1, :yellow), )\n",
    "end\n",
    "scatter!([x₀(t)[1]],[x₀(t)[2]], c = :red)\n",
    "yaxis!((Y[1], Y[end]))\n",
    "xaxis!((X[1], X[end]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  5.974609 seconds (80.24 M allocations: 1.845 GiB, 4.32% gc time)\n"
     ]
    }
   ],
   "source": [
    "@time for (i, t) in enumerate(range(0, stop = 3, length = 16)) # картинки для гифки\n",
    "    Z = [ E([x y], t) for y in Y, x in X ];\n",
    "    heatmap(X, Y, clamp.(log10.(Z), -1, 4), # oбрезание\n",
    "        legend = false, ticks = nothing, border = :none, size = (600, 400)  )\n",
    "\n",
    "    for alph in π/4:π/4:2π\n",
    "        nn = [cos(alph) sin(alph)]\n",
    "        fvec = (sqrt(1-V²)-1) * (Vn ⋅ nn) * Vn + nn\n",
    "        fi = angle(fvec[1] + im*fvec[2])\n",
    "\n",
    "        Vv = V₀ ⋅ [cos(fi) sin(fi)]\n",
    "        μ = Vv + sqrt( 1-V² + Vv^2 )\n",
    "        fvec = V₀ - μ*[cos(fi) sin(fi)]\n",
    "        fi = angle(fvec[1] + im*fvec[2])\n",
    "\n",
    "        zz = vcat( [λ(t, tt, fi) for tt in t-6:0.005:t ]... );\n",
    "        plot!(zz[:,1], zz[:,2], line = (1, :yellow), )\n",
    "    end\n",
    "    scatter!([x₀(t)[1]],[x₀(t)[2]], c = :red)\n",
    "    yaxis!((Y[1], Y[end]))\n",
    "    xaxis!((X[1], X[end]))\n",
    "\n",
    "    savefig(\"img$i.png\")\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Атом Резерфорда"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "rk4 (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function rk4(f, x, y, h)\n",
    "    k1 = h * f(x       , y        )\n",
    "    k2 = h * f(x + 0.5h, y + 0.5k1)\n",
    "    k3 = h * f(x + 0.5h, y + 0.5k2)\n",
    "    k4 = h * f(x +    h, y +    k3)\n",
    "\n",
    "    return y + (k1 + 2*(k2 + k3) + k4)/6.0\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.187691249014857e6"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "const nm = 1e-9 # m\n",
    "const ps = 1e-12 # s\n",
    "const fs = 1e-15 # s\n",
    "\n",
    "const c = 299792458 # m/s\n",
    "const ħ = 1.0546e-34 # J*s\n",
    "const e = 1.6022e-19 # Kl\n",
    "const mₑ = 9.1094e-31 # kg\n",
    "const ε₀ = 8.8542e-12 # F/m\n",
    "\n",
    "const k = 1/(4π*ε₀) # 8.987552e9 # N*m^2/q^2\n",
    "const rₑ = k*e^2 / (mₑ*c^2) # 2.81794e-15 # m\n",
    "const a₀ = 0.052917720859nm # 4π*ε₀*ħ^2 / (mₑ*q^2)\n",
    "const α = 1/137.036 # e^2/(ħ*c)\n",
    "const vₑ = α*c # 2187 km/c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "15.56096513768266"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tfall = a₀^3/(4c*rₑ^2) # время падения\n",
    "tfall / ps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\dot r = - \\frac 4 3 \\frac{r_0^2}{r^2} c \\\\\n",
    "\\dot\\theta = \\sqrt{\\frac{r_0}{r^3}}c\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "function solver(cron, r_θ::Array{Float64,2})\n",
    "    \n",
    "    st = step(cron)\n",
    "    syst(t, r) = begin\n",
    "        [ -4c/3 * (rₑ/r[1])^2    c*sqrt(abs(rₑ/r[1]^3)) ]\n",
    "    end\n",
    "    \n",
    "    vcat( [ r_θ = rk4(syst, t, r_θ, st) for t in cron ]... ) \n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "256×2 Array{Float64,2}:\n",
       " 5.28489e-11   2509.06\n",
       " 5.27799e-11   5023.03\n",
       " 5.27107e-11   7541.94\n",
       " 5.26414e-11  10065.8\n",
       " 5.25718e-11  12594.7\n",
       " 5.25021e-11  15128.6\n",
       " 5.24322e-11  17667.6\n",
       " 5.23621e-11  20211.6\n",
       " 5.22918e-11  22760.8\n",
       " 5.22213e-11  25315.2\n",
       " 5.21506e-11  27874.7\n",
       " 5.20798e-11  30439.4\n",
       " 5.20087e-11  33009.4\n",
       " ⋮            \n",
       " 1.88765e-11      1.01252e6\n",
       " 1.83201e-11      1.02455e6\n",
       " 1.77277e-11      1.03716e6\n",
       " 1.70929e-11      1.05044e6\n",
       " 1.6407e-11       1.06451e6\n",
       " 1.56584e-11      1.07954e6\n",
       " 1.48304e-11      1.09575e6\n",
       " 1.38979e-11      1.11347e6\n",
       " 1.28201e-11      1.13322e6\n",
       " 1.15221e-11      1.15592e6\n",
       " 9.83819e-12      1.1835e6\n",
       " 7.20892e-12      1.22193e6"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tgrid = range(0, stop = tfall-0.1ps, length = 256);\n",
    "Z = solver(tgrid, [a₀ 0.0]) # начальные радиус и угол"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "194475.9872841647"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Z[end]/2π # число полных оборотов"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
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       "  1776.51,501.641 1758.78,476.243 1739.75,451.774 1719.47,428.301 1697.99,405.889 1675.36,384.604 1651.65,364.505 1626.91,345.651 1601.2,328.098 1574.6,311.897 \n",
       "  1547.16,297.098 1518.97,283.746 1490.09,271.883 1460.61,261.547 1430.61,252.772 1400.16,245.589 1369.36,240.022 1338.27,236.095 1307,233.824 1275.63,233.22 \n",
       "  1244.25,234.293 1212.95,237.046 1181.82,241.476 1150.95,247.577 1120.43,255.338 1090.35,264.742 1060.8,275.768 1031.87,288.39 1003.65,302.577 976.222,318.291 \n",
       "  949.672,335.494 924.084,354.138 899.537,374.173 876.111,395.545 853.881,418.192 832.919,442.052 813.296,467.056 795.076,493.132 778.322,520.202 763.091,548.188 \n",
       "  749.438,577.004 737.412,606.564 727.056,636.778 718.412,667.552 711.514,698.791 706.39,730.397 703.066,762.269 701.558,794.305 701.88,826.402 704.038,858.455 \n",
       "  708.034,890.359 713.861,922.007 721.508,953.294 730.957,984.114 742.185,1014.36 755.162,1043.93 769.851,1072.72 786.209,1100.63 804.189,1127.55 823.734,1153.4 \n",
       "  844.786,1178.08 867.277,1201.49 891.134,1223.55 916.281,1244.17 942.635,1263.28 970.106,1280.81 998.602,1296.67 1028.03,1310.8 1058.27,1323.16 1089.24,1333.67 \n",
       "  1120.82,1342.3 1152.89,1349.01 1185.34,1353.76 1218.06,1356.52 1250.91,1357.28 1283.78,1356.02 1316.54,1352.74 1349.07,1347.43 1381.24,1340.12 1412.92,1330.81 \n",
       "  1444,1319.53 1474.34,1306.33 1503.82,1291.23 1532.33,1274.3 1559.74,1255.58 1585.94,1235.14 1610.81,1213.06 1634.25,1189.42 1656.17,1164.3 1676.45,1137.81 \n",
       "  1695,1110.04 1711.75,1081.09 1726.6,1051.1 1739.49,1020.17 1750.36,988.429 1759.14,956.013 1765.78,923.056 1770.25,889.695 1772.51,856.076 1772.55,822.341 \n",
       "  1770.34,788.641 1765.88,755.125 1759.19,721.942 1750.28,689.246 1739.17,657.186 1725.9,625.913 1710.53,595.575 1693.11,566.318 1673.71,538.285 1652.41,511.615 \n",
       "  1629.29,486.443 1604.47,462.897 1578.03,441.102 1550.12,421.172 1520.84,403.218 1490.33,387.339 1458.75,373.628 1426.23,362.167 1392.94,353.028 1359.04,346.273 \n",
       "  1324.7,341.951 1290.09,340.102 1255.39,340.751 1220.78,343.912 1186.44,349.585 1152.57,357.758 1119.33,368.404 1086.93,381.482 1055.53,396.938 1025.32,414.705 \n",
       "  996.483,434.7 969.178,456.828 943.575,480.98 919.832,507.032 898.099,534.849 878.517,564.285 861.217,595.177 846.317,627.357 833.926,660.641 824.137,694.838 \n",
       "  817.03,729.749 812.67,765.165 811.104,800.871 812.367,836.647 816.473,872.267 823.418,907.505 833.182,942.129 845.726,975.911 860.99,1008.62 878.897,1040.04 \n",
       "  899.349,1069.93 922.231,1098.09 947.407,1124.31 974.725,1148.39 1004.01,1170.14 1035.08,1189.38 1067.72,1205.97 1101.72,1219.74 1136.84,1230.58 1172.82,1238.38 \n",
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       "  1030.56,567.387 1003.87,598.662 981.065,633.056 962.555,670.118 948.705,709.335 939.826,750.139 936.162,791.91 937.883,833.985 945.078,875.663 957.743,916.214 \n",
       "  975.78,954.894 998.986,990.953 1027.05,1023.65 1059.55,1052.27 1095.95,1076.14 1135.62,1094.63 1177.78,1107.22 1221.6,1113.45 1266.11,1113 1310.29,1105.68 \n",
       "  1353.06,1091.44 1393.27,1070.42 1429.78,1042.96 1461.43,1009.57 1487.15,970.999 1505.92,928.214 1516.86,882.397 1519.28,834.944 1512.69,787.449 1496.93,741.676 \n",
       "  1472.13,699.516 1438.86,662.937 1398.12,633.905 1351.42,614.29 1300.79,605.75 1248.82,609.576 1198.65,626.521 1153.89,656.575 1118.53,698.715 1096.74,750.599 \n",
       "  1092.49,808.22 1108.98,865.509 1147.68,913.896 1206.52,941.861 1276.1,934.609 1328,874.724 \n",
       "  \"/>\n",
       "<path clip-path=\"url(#clip880)\" d=\"\n",
       "M1987.15 216.178 L2280.16 216.178 L2280.16 95.2176 L1987.15 95.2176  Z\n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
       "<polyline clip-path=\"url(#clip880)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
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       "  \"/>\n",
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      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R, Θ = Z[:, 1]/a₀, Z[:, 2]/5e4 # радиус нормировать на бора, а углы спресовать\n",
    "plot(Θ, R, proj = :polar, line = 2 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "R, Θ = Z[:, 1]/a₀, Z[:, 2]/5e4\n",
    "Xs = R.*cos.(Θ)\n",
    "Ys = R.*sin.(Θ);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in 4:4:length(R) # картинки для гифки\n",
    "    plot(Xs[1:i], Ys[1:i], line = 2, leg = false, ticks = nothing, border = :none, size = (780, 440) )\n",
    "    xaxis!((-1.0,1.0)); yaxis!((-1.0,1.0));\n",
    "    scatter!( [Xs[i]], [Ys[i]], m = (4, :blue) )\n",
    "    scatter!( [0.0], [0.0],  m = (60, :red) )\n",
    "    savefig(\"img$i.png\")\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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