Word Count : 346

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The Hamiltonian for a particle in a gravitational field is given as: \[ H = \frac{p^2}{2m} + mgx \] Where: - \( p \) is the momentum, - \( m \) is the mass of the particle, - \( g \) is the gravitational acceleration, - \( x \) is the position of the particle. ### Step 1: Determining the equations of motion To determine the equations of motion, we use Hamilton's equations of motion: 1. \( \frac{d\hat{x}}{dt} = \frac{\partial H}{\partial p} \) 2. \( \frac{d\hat{p}}{dt} ____ _____ _________ ______ ___ ______.
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