Question

The Lagrangian of a system is given by \(L = \tfrac{1}{2}\dot{q}^{2} + q\dot{q} - \tfrac{1}{2}q^{2}\). It describes the motion of:

• A. A harmonic oscillator
• B. A damped harmonic oscillator
• C. An anharmonic oscillator
• D. A system with unbound motion

Hint:

The \(q\dot{q}\) piece is a total derivative; the Euler-Lagrange equation reduces to \(\ddot{q}+q=0\).

Answer 1

The Correct Option is A

Step 1: Notice the middle term is a total derivative: $q\dot{q}=\dfrac{d}{dt}\!\left(\tfrac{1}{2}q^{2}\right)$.

Step 2: Adding a total time derivative to a Lagrangian never changes the equations of motion, so we may discard it and use
\[L' = \tfrac{1}{2}\dot{q}^{2} - \tfrac{1}{2}q^{2}.\]
Step 3: This is exactly the standard oscillator Lagrangian $T-V$ with unit mass and unit spring constant, $T=\tfrac{1}{2}\dot{q}^{2}$, $V=\tfrac{1}{2}q^{2}$.

Step 4: Its equation of motion is $\ddot{q}=-q$, i.e. $\ddot{q}+q=0$, with angular frequency $\omega=1$.

Step 5: There is no genuine dissipation (the apparent $\dot{q}$ from the cross term cancels), so the motion is undamped simple harmonic.
\[\boxed{\text{A harmonic oscillator}}\]