Motivating Example
Consider a system which consists of two masses $m_1$ and $m_2$ at positions $x_1$ and $x_2$ respectively joined together by a rigid rod of negligible mass and length $l$ . We have the Lagrangian $$ L = \frac{1}{2} m_1 \dot{x}_1^2 + \frac{1}{m} m_2 \dot{x}_2^2 $$ and the obvious constraint $$ x_2 - x_1 - l = 0 \label{con1}\tag{1} $$ which yields the equations of motion: \begin{align} m_1 \ddot{x}_1 + \lambda &= 0 \\ m_2 \ddot{x}_2 - \lambda &= 0 \end{align} for some function of time $\lambda$. These are the expected equations of motion with $\lambda$ playing the role of tension in the rod.
What stops us from choosing instead the constraint $$ (x_2 - x_1 - l)^2 = 0 \label{con2}\tag{2} $$ which has exactly the same solutions in $x_1 , x_2$ as \ref{con1}? We then get the equations of motion: \begin{align} m_1 \ddot{x} + 2\lambda (x_2 - x_1 - l) &= 0 \\ m_2 \ddot{x} - 2\lambda (x_2 - x_1 - l) &= 0 \end{align} These equations of motion look like what we would expect if $m_1$ and $m_2$ were joined by a spring with spring constant $2\lambda$.
My Question
Above I obtained physically distinct equations of motion by using different constraints with identical solution sets.
If the solution set of a constraint doesn't uniquely determine the physics, what does? How do I pick the "right" constraint? What information do I have to use beyond the solution set of the constraint?
