Given a particle of mass m moving in a one-dimensional potential, the Hamiltonian of the system is expressed as \( H = \frac{p^2}{2m} + \frac{1}{2}kx^2 \ \). The corresponding Lagrangian for this system is:
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\( -\frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 \ \)
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\( \frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2 \ \)
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\(\frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 \ \)
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\( -\frac{1}{2}m\dot{x}^2 + \frac{1}{2}kx^2 \ \)
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