How do I get the numerical solution of this equation?

Question

So I have a rather non-trivial differential equation. I can't solve it analytically so I want to solve it numerically using NDSolve. But the thing is that the output just becomes what I had as input, the command does not execute.

So, I have a Lagrangian given by: L2 = r^3 (((1/Pi)*Sqrt[(1^4)*(Sin[x[r]]^4) + (Pi*(800) - x[r] + (1/2)*Sin[2*x[r]])^2])*(1 + ((1/2)*(x'[r]^2))) + (1/Pi)*(Pi*(800) - x[r] + (1/2) Sin[2*x[r]]))

With some help from below (<3) I get:

eq={D[D[L2, x'[r]], r] == D[L2, x[r]], x[0] == 0, x'[0] == 0}

Then I use NDSolve to get

NDSolve[eq, x, {r, 0, Infinity}]

Where the range of x[r] is from zero to Pi and range of r is from zero to infinity.

But I get some errors:

1: "Power:Infinite expression $\frac{1}{0^3}$ encountered." 2: "Infinity: Indeterminate expression 0.π ComplexInfinity encountered." 3: "NDSolve: Encountered non-numerical value for a derivative atr==0."

I know that some people have solved this Lagrangian numerically in a paper so should be possible.

Are there any workarounds I could do to these error messages? Many thanks in advance!

Answer 1

I assume that x[t] in the definition of L2 is a typo and should read x[r].

With this assumption, you can simply define L2 as:

L2 = r^3 ((V2[x[r], r])*(1 + ((1/2)*(x'[r]^2))) + (1/Pi)*(Pi*(800) - 
       x[r] + (1/2) Sin[2*x[r]]))

The equations then read:

eq={D[D[L2, x'[r]], r] == D[L2, x[r]], x[0] == 0, x'[0] == 0}

Furthermore, the syntax of NDSolveis wrong. {x[r], 0, Pi} is not needed as x[r] is the searched for solution and its range is given by the range of r.

Note further that the equations can not be solved without specifying V2.

The command to solve the ODE then reads:

NDSolve[eq, x, {r, 0, Infinity}]

But to solve it, you need V2