Solutions for a quadratic system of equations

[greypilgrim]

Hi,

I'm looking for the real solutions to the system
\begin{array}{rcl} 1 & = & v_1+v_2+v_3+v_4+v_5 \\ 1 & = & v_1^2+v_2^2+v_3^2+v_4^2+v_5^2 \end{array}

Background: I'm looking at a Newton's cradle with 5 balls, each of mass $m=1$. The first ball is pulled away and let go such that it hits the others with velocity $v=1$. Above equations are conservation of momentum and energy. I know that Newton's cradle is more complicated and restricts the solutions much more, I just want to find out which dynamics are allowed from momentum and energy conservation only.

Some solutions I've found:

1. $v_1=v_2=v_3=v_4=0,v_5=1$
2. $v_1=-\frac{1}{3},v_2=v_3=0,v_4=v_5=\frac{2}{3}$
3. $v_1=-\frac{1}{2},v_2=0,v_3=v_4=v_5=\frac{1}{2}$
4. $v_1=-\frac{3}{5},v_2=v_3=v_4=v_5=\frac{2}{5}$

And of course all permutations of these solutions.

Are there more? If so, how can I find them?

 

[fresh_42]

You can substitute $v_1$ in the second equation with the first one which leaves you with one equation with four variables. This defines you a 4-dimensional manifold in I suppose $ℝ^5$. I guess something parabolic or so. I have no idea how to imagine it. The number of solutions, however, is infinite., i.e. you can't write them down.

 

[greypilgrim]

What if I make the situation more physical by imposing the inequality$$v_1\leq\ v_2\leq\ v_3\leq\ v_4\leq\ v_5,$$
i.e. no ball can overtake its neighbours? Are there still infinitely many solutions?

 

[fresh_42]

What if I make the situation more physical by imposing the inequality$$v_1\leq\ v_2\leq\ v_3\leq\ v_4\leq\ v_5,$$
i.e. no ball can overtake its neighbours? Are there still infinitely many solutions?

That's like cutting an - what is quadrant in the 5th dimension? - out and consider it. As long as the velocities are continuous you need more equations to get a unique answer. You can probably have further limitations and find some numerical solutions by using an appropriate software tool.
But can't you assume an elastic collision where the full energy is transmitted? I don't know where to read about the physics of it. On the Wiki-page of Newton's cradle there are some numbers and descriptions that might help you.

 

[mfig]

You could try a numerical solver. MATLAB's fsolve finds two different solutions from what you found by simply varying the initial guess.