One-dimensional Newtonian systems: Conservative and oscillatory systems

 By one-dimensional Newtonian system we mean a particle whose position is determined by a single coordinate x and whose motion is governed by Newton's second law. The system is conservative if the particle is subject to a total force of the form F(x), i.e., a force that depends only on x and not on time or velocity. From Newton's law it then follows that a time-independent (i.e. conserved) total mechanical energy may be defined. Conservation of mechanical energy allows one to express the equation of motion of the particle by a "simple" integral formula.

A one-dimensional oscillatory system may or may not be conservative, depending on whether the oscillation is free (as in simple harmonic motion) or damped. In the latter case the total force contains a velocity-dependent term.

https://eclass.hna.gr/modules/document/file.php/TOM6113/one-dimensional%20newtonian%20systems.pdf