An exercise made me consider the following Lagrangian
$$L = \dot{x}_1^2+\dot{x}_2^2+2 \dot{x}_1 \dot{x}_2 + x_1^2+x_2^2.\tag{1}$$
If I didn't make a mistake the equations of motion should be given by:
$$2x_1 = 2 \ddot{x}_1 + 2 \ddot{x}_2\tag{2}$$
$$2x_2 = 2 \ddot{x}_2 +2 \ddot{x}_1.\tag{3}$$
But this already implies that
$$x_1 = x_2. \tag{4}$$
So this equations of motion seem to impose an additional constraint. Moreover, there should not be a solution for initial conditions, where $x_1 \neq x_2.$ How can it be that the equations of motion already pose a constraint on the initial conditions? Is there some deeper theory behind that as to when this happens? And what does it mean that the lagrangian only has a solutions for very specific constraints?
Best Answer
with
$$L = \dot{x}_1^2+\dot{x_2}^2+2 \dot{x_1} \dot{x_2} + x_1^2+x_2^2$$
you can obtain the "mass matrix"
$$M= \left[ \begin {array}{cc} {\frac {\partial ^{2}}{\partial {{\dot{x}}_{ {1}}}^{2}}}L \left( {\dot{x}}_{{1}},{\dot{x}}_{{2}} \right) &{\frac { \partial ^{2}}{\partial {\dot{x}}_{{2}}\partial {\dot{x}}_{{1}}}}L \left( {\dot{x}}_{{1}},{\dot{x}}_{{2}} \right) \\ { \frac {\partial ^{2}}{\partial {\dot{x}}_{{2}}\partial {\dot{x}}_{{1}}}} L \left( {\dot{x}}_{{1}},{\dot{x}}_{{2}} \right) &{\frac {\partial ^{2}} {\partial {{\dot{x}}_{{2}}}^{2}}}L \left( {\dot{x}}_{{1}},{\dot{x}}_{{2}} \right) \end {array} \right] =\left[ \begin {array}{cc} 2&2\\ 2&2\end {array} \right] $$
thus the determinate of the masse matrix is zero this means that you have "constraint" in the equations of motion (the accelerations are linearly dependent ).
the equations of motion are:
$$\mathbf M\, \begin{bmatrix} \ddot{x}_1 \\ \ddot{x}_2 \\ \end{bmatrix}=\begin{bmatrix} 2\,{x}_1 \\ 2\,{x}_2 \\ \end{bmatrix}\quad\Rightarrow\\ \begin{bmatrix} \ddot{x}_1 \\ \ddot{x}_2 \\ \end{bmatrix}=\mathbf M^{-1}\,\begin{bmatrix} 2\,{x}_1 \\ 2\,{x}_2 \\ \end{bmatrix}$$
and because the determinant of M is zero you don't get any solution.