Lagrangian and Hamiltonian Methods Practice Questions | GRE Subject Test Physics

Lagrangian and Hamiltonian Methods

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A particle moves in one dimension with action \[ S=\int_{t_1}^{t_2}\left(\frac{1}{2}m\dot{x}^2 - V(x)\right)\,dt. \] The endpoint times \(t_1,t_2\) are fixed, but the endpoint positions \(x(t_1)\) and \(x(t_2)\) are free to vary. What additional condition must the extremal path satisfy at the endpoints?

Select one option.

\(\dot{x}(t_1)=\dot{x}(t_2)=0\)
\(V(x(t_1))=V(x(t_2))\)
\(\dfrac{\partial L}{\partial \dot{x}}=0\) at \(t_1\) and \(t_2\)
\(\dfrac{\partial L}{\partial x}=0\) at \(t_1\) and \(t_2\)
No additional condition is required

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