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: Apply the Euler-Lagrange equation \(\dfrac{d}{dt}\!\left(\dfrac{\partial L}{\partial \dot{q}}\right) - \dfrac{\partial L}{\partial q} = 0\).

Step 2: Momentum term: \(\dfrac{\partial L}{\partial \dot{q}} = \dot{q} + q\), so \(\dfrac{d}{dt}\!\left(\dot{q}+q\right) = \ddot{q} + \dot{q}\).

Step 3: Force term: \(\dfrac{\partial L}{\partial q} = \dot{q} - q\).

Step 4: Combine:
\[\ddot{q} + \dot{q} - (\dot{q} - q) = 0 \;\Rightarrow\; \ddot{q} + q = 0.\]
Step 5: The \(q\dot{q} = \tfrac{1}{2}\dfrac{d}{dt}(q^{2})\) term is a total time derivative and drops out, leaving simple harmonic motion.
\[\boxed{\ddot{q} + q = 0 \;\text{(harmonic oscillator)}}\]