What are the equations for the modeling?

Question

We have a system of spring-mass-spring-mass-spring-mass and required to write the motion equation of the described system.

If I apply hooke's law to the system I got the following :

Are these the correct equation form?

Answer 1

The movement equations can be derived from the Lagrangian $L = T - U$ with

$$ \cases{ U = m g \sum_{k=1}^n x_k\\ T =\frac 12 m\sum_{k=1}^n \dot x_k^2+\frac 12 k\sum_{k=1}^n(x_{k-1}-x_k)^2 } $$

with $x_0 = 0$

Answer 2

if you apply Newton's second law on $x_1,x_2,x_3$, you should get (with $l_0$ the caracterisitc length of the spring) (and neglecting gravity) :

$$m\ddot{x_1}=-k(2x_1-x_2)\\m\ddot{x_2}=-k(2x_2-x_3)\\m\ddot{x_3}=-k(x_3-x_2-l_0) $$