Question: How to solve a set or a system of differential equations in maple into a understandable one?

January 08 2011 MapleFans001 5
Maple

0

How to solve a set or a system of differential equations in maple for motion with Schwarzchild metric?

 

************** Schwarzchild metric *****************
with(tensor):
coord := [t, r, theta, Phi]:

g_compts:=array(symmetric,sparse,1..4,1..4):

g_compts[1,1]:=1 - 2*G*M/(r*c^2):
g_compts[2,2]:=-(1 - 2*G*M/(r*c^2))^(-1):
g_compts[3,3]:=-r^2:
g_compts[4,4]:=-(r^2)*sin(theta)^2:

g1 := create([-1,-1], eval(g_compts)):
g1_inv := invert( g1, 'detg' ):

D1g := d1metric( g1, coord ):

Cf1_1 := Christoffel1(D1g):
Cf2_1 := Christoffel2(g1_inv, Cf1_1):
displayGR(Christoffel2,Cf2_1):

*******************************************

i use the equations from mathematical side of motion on surface and then substitute

Christoffel into equations However, i do not know whether it is a correct doing and only one left hand side

as i do not know right hand side F*e when using Schwarzchild metric in physics,

it is different from math side , if you know, please also reply me. Template is follows

*************** from math side motion on surface *************

template := expand((t1+t2+t3+t4)^2);

+ Diff(f1(t), t)^2
+ 2*Diff(f1(t), t)*Diff(f2(t), t)
+ 2*Diff(f1(t), t)*Diff(f3(t), t)
+ 2*Diff(f1(t), t)*Diff(f4(t), t)

+ Diff(f2(t), t$2)
+ 2*Diff(f2(t), t)*Diff(f3(t), t)
+ 2*Diff(f2(t), t)*Diff(f4(t), t)

+ Diff(f3(t), t$2)
+ 2*Diff(f3(t), t)*Diff(f4(t), t)

+ Diff(f4(t), t)^2;

******************************************************

motion_1 :=
Diff(f1(t), t$2)

+ 2*(-G*M/(r*(-r*c^2+2*G*M)))*Diff(f(t), t1)*Diff(f2(t), t) = 0;


motion_2 :=
Diff(f2(t), t$2)

+ (-(-r*c^2+2*G*M)*G*M/(r^3*c^4))*Diff(f1(t), t)^2
+ (G*M/(r*(-r*c^2+2*G*M)))*Diff(f2(t), t$2)
+ ((-r*c^2+2*G*M)/c^2)*Diff(f3(t), t$2)
+ ((-r*c^2+2*G*M)*sin(theta)^2/c^2)*Diff(f4(t), t)^2 = 0;

motion_3 :=
Diff(f3(t), t$2)

+ (-sin(theta)*cos(theta))*Diff(f4(t), t)^2 = 0;

motion_4 :=
Diff(f3(t), t$2)

+ 2*(1/r)*Diff(f2(t), t)*Diff(f4(t), t)

+ 2*(cos(theta)/sin(theta))*Diff(f3(t), t)*Diff(f4(t), t) = 0;

i can solve these equations with dsolve.

However, solution is very complicated. Any other method? And how to analyze the solution?

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