## A Harmonic Oscillator in a Potential Energy Proportional to the Square of the Second Derivative of the Coordinate with Respect to Time

**Authors:** Faisal Amin Yassein Abdelmohssin

In search to construct a Lagrangian functional of a damped harmonic oscillator I thought to study higher derivatives of coordinates with respect to time in the Lagrangian of a simple harmonic oscillator by adding a term proportional to the square of the second derivative of the coordinate with respect to time in its Lagrangian. In Newtonian mechanics a damping term is added directly to the equation of motion of a simple harmonic oscillator, whereas in Lagrangian and Hamiltonian mechanics (Analytical Mechanics as opposed to Vectroial Mechanics of Newton) adding a term to the Lagrangian of the simple harmonic oscillator wouldn’t reveal whether the term is a damping driving or a forced driving agent until one study the solutions of the equation of motion.
Here, The Euler-Lagrange and equation of motion of a harmonic oscillator in a potential energy proportional to the square of the second derivative of the coordinate with respect to time have been formulated and discussed. The equation of motion is derived from Euler-Lagrange equation by performing the partial derivatives on the Lagrangian functional of the second variation of the calculus of variations.

**Comments:** 12 Pages.

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### Submission history

[v1] 2017-09-17 06:15:04

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