# analytic solution to non-linear system of equations.

In one physics problem, I got the following system of non-linear equations that I need to solve for $v_e$, $v_s$ and $\omega$:

$$m_p v_a = m_s v_s + m_p v_e$$

$$\frac{l}{2} m_p v_a = \frac{l}{2} m_p v_e + I \omega$$

$$\frac{1}{2} m_p v_a^2 = \frac{1}{2} m_p v_e^2 + \frac{1}{2} m_s v_s^2 + \frac{1}{2} I \omega^2$$

I know how to solve this with substitution and a lot of scratch paper.

With linear systems, one can just derive the matrix and use gauss-jordan / reduced row echolon form and then the solution is directly apparent.

Is there something handy for non-linear equations as well?

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Looks quite like something you can use Gröbner bases for, in the general case. – J. M. Dec 31 '11 at 0:26
Gröbner bases can be viewed as a generalization of Gaussian Elimination for systems of polynomial equations. – user2468 Dec 31 '11 at 0:35
It looks like what I need, I just think it is a little over my head currently. I'll read into it. – Martin Ueding Dec 31 '11 at 15:42

For your example, you have a "nearly" linear system, because only one equation is quadratic. In that case, you can express all solutions of the linear part of the system in the form $x_0+\alpha x_h$, substitute that expression into the quadratic equation and solve the resulting equation for $\alpha$.
If you apply this technique to your system, you get $v_e=v_a-\alpha$, $v_s=\frac{m_p}{m_s}\alpha$ and $\omega=\frac{lm_p}{2I}\alpha$. If you substitute this into the quadratic equation $$m_p (v_a^2-v_e^2) = m_s v_s^2 + I \omega^2$$ you get
$$m_p (2v_a-\alpha)\alpha = \frac{m_p^2}{m_s}\alpha^2 + \frac{l^2m_p^2}{4I}\alpha^2$$ One solution of this quadratic equation is obviously $\alpha=0$, but I guess you are more interested in the other solution. Assuming $m_p\alpha\neq0$, we can divide by $m_p\alpha$ to get $2v_a-\alpha = \frac{m_p}{m_s}\alpha + \frac{l^2m_p}{4I}\alpha$. It's easy to solve this equation for $\alpha$.