25. (a) The Hamiltonian for a system has the form H = 1/2 (1/q^2 + p^2 q^4). Find the equation of motion for q. (b) Find a canonical transformation that reduces H to the form of a harmonic oscillator. Show that the solution for the transformed variables is such that the equation of motion found in part (a) is satisfied.
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Step 1: Start with the Lagrange equation, which is given by \( \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0 \), where the Lagrangian \( L = p\dot{q} - H \). Show more…
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