Práctica 12: Geodésicas

Primera forma fundamental

EE[X_][u_, v_] := 
      Simplify[D[X[x, y], x] . D[X[x, y], x] /. {x -> u, 
            y -> v}]; FF[X_][u_, v_] := 
      Simplify[D[X[x, y], x] . D[X[x, y], y] /. {x -> u, 
            y -> v}];
GG[X_][u_, v_] := 
        Simplify[
          D[X[x, y], y] . D[X[x, y], y] /. {x -> u, 
              y -> v}];
W[X_][u_, v_] := 
      Simplify[EE[X][u, v] GG[X][u, v] - FF[X][u, 
              v]^2]

Símbolos de Christoffel

G111[X_][u_, v_] := 
        Simplify[(GG[X][uu, vv] D[EE[X][uu, vv], uu] - 
                  2  FF[X][uu, vv] D[FF[X][uu, vv], 
                      uu] +  
FF[X][uu, 
                      vv] D[EE[X][uu, vv], vv])/(2 W[X][uu, 
                    vv]) /. {uu -> u, 
              vv -> v}];
G121[X_][u_, v_] := 
        Simplify[(GG[X][uu, vv] D[EE[X][uu, vv], vv] - FF[X][
                      uu, vv] D[GG[X][uu, vv], uu])/(2 W[X][uu, 
                    vv]) /. {uu -> u, 
              vv -> v}] ;
G221[X_][u_, v_] := 
        Simplify[(2 GG[X][uu, vv] D[FF[X][uu, vv], 
                      vv] - GG[X][uu, vv] D[GG[X][uu, vv], 
                      uu] - FF[X][uu, vv] D[GG[X][uu, vv], 
                      vv])/(2 W[X][uu, vv]) /. {uu -> u, 
              vv -> v}];
G112[X_][u_, v_] := 
        Simplify[(2 EE[X][uu, vv] D[FF[X][uu, vv], 
                      uu] - EE[X][uu, vv] D[EE[X][uu, vv], 
                      vv] - FF[X][uu, vv] D[EE[X][uu, vv], 
                      uu])/(2 W[X][uu, vv]) /. {uu -> u, 
              vv -> v}];
G122[X_][u_, v_] := 
        Simplify[(EE[X][uu, vv] D[GG[X][uu, vv], uu] - FF[X][
                      uu, vv] D[EE[X][uu, vv], vv])/(2 W[X][uu, 
                    vv]) /. {uu -> u, 
              vv -> v}];
G222[X_][u_, v_] := 
        Simplify[(EE[X][uu, v] D[GG[X][uu, vv], vv] - 
                  2  FF[X][uu, vv] D[FF[X][uu, vv], 
                      vv] +  
FF[X][uu, 
                      vv] D[GG[X][uu, vv], uu])/(2 W[X][uu, 
                    vv]) /. {uu -> u, vv -> v}];

Ecuación de las geodésicas

En la esfera

X[u_, v_] := {Cos[u] Cos[v],  Cos[u] Sin[v], Sin[u]}

s1 = 
      NDSolve[{u''[
                s]  + 
u'[s]^2 G111[X][u[s], 
                  v[s]] + 
              2 u'[s] v'[s] G121[X][u[s], 
                  v[s]] +  v'[s]^2 G221[X][u[s], v[s]] == 
            0, v''[
                s]  + 
u'[s]^2 G112[X][u[s], 
                  v[s]] + 
              2 u'[s] v'[s] G122[X][u[s], 
                  v[s]] +  v'[s]^2 G222[X][u[s], v[s]] == 0, 
          u[0] == 0, 
          v[0] == 0, u'[0] == -1, v'[0] == 
            1}, {u[s], v[s]}, {s, -4, 4}]

d1 = 
      ParametricPlot3D[
        Evaluate[X[u[s], v[s]] /. s1], {s, -4, 4}]

d2 = 
        ParametricPlot3D[
          Evaluate[X[u, v]], {u, -Pi/2, Pi/2}, {v, 0, 
            2  Pi}];

Show[d1, d2, ViewPoint -> {1, -2, 1}]

En el cilindro

Y[u_, v_] := {Cos[v], Sin[v], u}

s2 = 
      NDSolve[{u''[
                s]  + 
u'[s]^2 G111[Y][u[s], 
                  v[s]] + 
              2 u'[s] v'[s] G121[Y][u[s], 
                  v[s]] +  v'[s]^2 G221[Y][u[s], v[s]] == 
            0, v''[
                s]  + 
u'[s]^2 G112[Y][u[s], 
                  v[s]] + 
              2 u'[s] v'[s] G122[Y][u[s], 
                  v[s]] +  v'[s]^2 G222[Y][u[s], v[s]] == 0, 
          u[0] == 0, 
          v[0] == 0, u'[0] == 1, v'[0] == 5}, {u[
            s], v[s]}, {s, -4, 4}]

d3 = 
      ParametricPlot3D[Evaluate[Y[u[s], v[s]] /. s2], {s, -2, 2}, 
        PlotPoints -> 100]

d4 = 
        ParametricPlot3D[
          Evaluate[Y[u, v]], {u, -2, 0}, {v, 0, 2  Pi}];

Show[d3, d4]

En el paraboloide

Z[u_, v_] := {u Cos[v], u Sin[v], u^2}

s3 = 
      NDSolve[{u''[
                s]  + 
u'[s]^2 G111[Z][u[s], 
                  v[s]] + 
              2 u'[s] v'[s] G121[Z][u[s], 
                  v[s]] +  v'[s]^2 G221[Z][u[s], v[s]] == 
            0, v''[
                s]  + 
u'[s]^2 G112[Z][u[s], 
                  v[s]] + 
              2 u'[s] v'[s] G122[Z][u[s], 
                  v[s]] +  v'[s]^2 G222[Z][u[s], v[s]] == 0, 
          u[0] == 1, 
          v[0] == 0, u'[0] ==  .5, v'[0] == 4}, {u[
            s], v[s]}, {s, -5, 5}]

d5 = 
      ParametricPlot3D[Evaluate[Z[u[s], v[s]] /. s3], {s, -5, 5}, 
        PlotRange -> All]

d6 = 
        ParametricPlot3D[
          Evaluate[Z[u, v]], {u, 0, 3}, {v, 0, 2  Pi}];

Show[d5, d6, ViewPoint -> {1, 5, 1}]