Converting a second order n x n system into a first order 2n x 2n system

Question

Say I have the following second order 7 x 7 system of equations:

• $x_1'' = 10(x_2- x_1- 1)$
• $x_2'' = 10(x_3- 2x_2+ x_1)$
• $x_3'' = 10(x_4- 2x_3+ x_2)$
• $x_4'' = 10(x_5- 2x_4+ x_3)$
• $x_5'' = 10(x_6- 2x_5+ x_4)$
• $x_6'' = 10(x_7- 2x_6+ x_5)$
• $x_7'' = 10(x_6- x_7)$.

How would I convert this second order 7 x 7 system into a first order 14 x 14 system using the additional equations $v_j = x'_j$, where $j = 1, 2, 3, ..., 7$?

Answer 1

You would define variables $x_8,x_9\dots x_{14}$, add equations $x_i'=x_{i+7}$ and rewrite your existing equations to look like $x_1''=x_8'=10(x_2-x_1-1)$

Answer 2

Just have the 7 equations $v_j = x'_j$ and the seven equations with $v'_j$ equal to the right hand sides of the ones you have there.

If the $x_j$ were positions, all this would do is to say, instead of giving 7 differential equations for relating the accelerations and positions, you get 7 equations relating the velocities with the rate of change of the positions $x' = v$ (which are indeed identical) and the 7 equations relating the change in velocities with the position.

You can see this is a bit artificial but absolutely true.