What is Legendre transformation of a function? Express the Legendre transformation of a Lagrangian to obtain the Hamilton's function H.

(ii) The Hamiltonian of a simple harmonic oscillator is given as:

Show that the velocity vector is tangential to the curve defined by the Hamiltonian. Draw a labeled diagram of the velocity vector for the phase trajectory for energy E.

(i) The Legendre transformation of a function is a mathematical operation that transforms a function from one set of variables to another. For example, in classical mechanics, the Legendre transformation is used to move from the Lagrangian formulation to the Hamiltonian formulation. The Lagrangian \( L(q, \dot{q}, t) \) depends on the generalized coordinates \( q \) and generalized velocities \( \dot{q} \), while the Hamiltonian \( H(q, p, t) \) depends on the generalized coordinates \( q \) and generalized momenta \( p \). 

To obtain the Hamiltonian from the Lagrangian, the generalized momentum is defined as:

\[
p = \frac{\partial L}{\partial \dot{q}}
\]

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