Try:

eqN = {Sec[?[t]/2] (1 - Cos[?[t]] + 2 Cos[2 ?[t]] + 2 (1 + Cos[?[t]]) Derivative[1][?][t]^2) == 
      0, ?[0] == 1};
solN = NDSolve[eqN, ?[t], {t, 0, 4*Pi}, Method -> {"EquationSimplification" -> "Residual"}]
Plot[Evaluate[?[t] /. solN], {t, 0, 4*Pi}]

1:

 eq = {Sec[?[t]/2] (1 - Cos[?[t]] + 2 Cos[2 ?[t]] + 2 (1 + Cos[?[t]]) Derivative[1][?][t]^2) == 0};
 INIT = Derivative[1][?][0] /. Solve[eq /. t -> 0 /. ?[0] -> 1, Derivative[1][?][0]](*A second initial conditions *)

Then we have: $\theta '(0)=-\sqrt{\frac{-1+\cos (1)-2 \cos (2)}{2 (1+\cos (1))}}$ and $\theta '(0)=\sqrt{\frac{-1+\cos (1)-2 \cos (2)}{2 (1+\cos (1))}}$

 sol = ParametricNDSolve[{eq, ?[0] == 1, ?'[0] == a}, ?, {t, 0, 4*Pi}, {a}, Method -> {"EquationSimplification" -> "Residual"}];
 Plot[{?[(INIT[[1]])][t] /. sol, ?[(INIT[[2]])][t] /. sol}, {t, 0, 4*Pi}]

2:

 eq = {Sec[?[t]/2] (1 - Cos[?[t]] + 2 Cos[2 ?[t]] + 2 (1 + Cos[?[t]]) Derivative[1][?][t]^2) == 0};
 INIT = Derivative[1][?][0] /. Solve[eq /. t -> 0 /. ?[0] -> 1, Derivative[1][?][0]](*A second initial conditions *)
  sol = ParametricNDSolve[{D[eq,t], ?[0] == 1, ?'[0] == a}, ?, {t, 0, 4*Pi}, {a}];
  Plot[{?[(INIT[[1]])][t] /. sol, ?[(INIT[[2]])][t] /. sol}, {t, 0, 4*Pi}]

Here is the spinning top model in WSM. I sent it spinning and moving in one direction and then hit it with an off center force at 2 seconds to send it off precessing.

Here is the animation:

Ox,

What version of MMA are you running? On my version the NDSolve completes for all 4 seconds using the manually formed DAE formulation.

You are correct that you specified the problem in the form of f(x,x',t)==0. However, NDSolve goes through a series of steps in solving the problem. One step ("EquationSimplification") is to solve your equation again to put it in a new form. The default form is x'=f(x,t). When it does this, it gets a Sqrt on the right hand side which leads to your numerical problem. A second alternative is to use "Residual". This option takes your equation and substitutes new variables for each of the derivatives so you have an expression g(x,newx,t)==0 where the newx's are the derivatives in the original expression. Now this problem is a DAE -- you have an algebraic equation relating the x's and the newx's and you have differential equations relating x'[t] == newx[t].

If you want to view "under the hood" you can use the following commands to see what MMA is actually doing:

Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
Needs["DifferentialEquations`NDSolveUtilities`"];

state = First@
   NDSolve`ProcessEquations[eqN, {\[Theta][t]}, {t}, 
    Method -> {"EquationSimplification" -> "Residual"}];
state["NumericalFunction"]["FunctionExpression"]

Out[25]= Function[{t, \[Theta], \[Theta]$12339105}, {(1 - 
     Cos[\[Theta]] + 2 \[Theta]$12339105^2 (1 + Cos[\[Theta]]) + 
     2 Cos[2 \[Theta]]) Sec[\[Theta]/2]}]

state = First@NDSolve`ProcessEquations[eqN, {\[Theta][t]}, {t}];
state["NumericalFunction"]["FunctionExpression"]

Out[27]= Function[{t, \[Theta]}, {-(
   Sqrt[-1 + Cos[\[Theta]] - 2 Cos[2 \[Theta]]]/Sqrt[
   2 + 2 Cos[\[Theta]]])}]

Note that the residual case adds a new variable, the original EquationSimplification results in a Sqrt[].

(as an aside: Note that you only get one branch of the solution using the DAE approach and there is a second branch that you can get by adding a (consistent) initial condition to the problem (see the example that Mariusz pointed to in the NDSolve docs under "Method" option.)

Regards

Neil

Thanks for making this!

Although I personally find solving DEs are more intuitive for a simple problem like this, I think WSM will save a lot of time in many other situations.

Hello

I'm ordinary user not Jedi Master in Mathematica. I only know how much it is written in the documentation. About "Residual" find here: (NDSolve->Options->Method->EquationSimplification)

Regards MI

This maybe off topic but, from what I see on the Systemmodeler webpage, it's don't have a physics interface built-in that one can draw, say, a ellipsoid and specify the gravity and fix point. So how does one simulate this? Do we have to manually put in the differential equation?