I am coming from the computer science territory and have not a long trace in mechanics. My background in derivation of the system dynamics could be summarized with utilization of the Lagrange Mechanics but here is a vague point, with which I have confronted, recently. How does the existence of the the jerk term in dynamics of a Mass-Spring-Damper could be justified, where the Lagrange Dynamics just computes up to the 2nd derivatives of the generalized parameters?!...
If one considers a dynamic system, which (from left to right) consists of a spring with constant $k_1$, a mass $m$, a damper with constant $c$ and the other spring with constant $k_2$, all connected together, respectively, the derived dynamics of the system would be declared as: (The junction of spring $k_2$ and damper $c$ is massless)
$$ \frac{cm}{k_2} · \frac{d^{3} x}{d t^{3}} + m \frac{d^{2} x}{d t^{2}} + \left(c + \frac{k-1}{k_2}\right) \frac{dx}{dt}+ k_1 x= 0 $$$$ \frac{cm}{k_2} · \frac{d^{3} x}{d t^{3}} + m \frac{d^{2} x}{d t^{2}} + c \left( 1 +\frac{k_1}{k_2}\right) \frac{dx}{dt}+ k_1 x= 0 $$
Obviously, the Jerk term has been appeared, up there, noticeably.
Would you please guiding me that how such dynamics could be interpreted by either Lagrange Dynamics or Newtonian Method?!
