Deriving the Hamiltonian of a system given the Lagrangian

I have found the Hamiltonian to be ##H = L - 6 (q_1)^2## using the method below:

1. Find momenta using δL/δ\dot{q_i}
2. Apply Hamiltonian equation: H = sum over i (p_i \dot{q_i}) - L 3(q_1)^2. Simplifying result by combining terms
4. Comparing the given Lagrangian to the resulting Hamiltonian I found H =\dot{q_1}^2 + 0.5 \dot{q_2}^2 + \dot{q_1} * \dot{q_2} - 3q_1^2 = L - 6(q_1)^2 This is wrong because my Hamiltonian should be in terms of generalised coordinate and momentum only: H(q_1, q_2, p_1, p_2). What am I neglecting?

From the equations for ##p_i## you could find ##\dot{q_i}##. Substituting these everywhere, you get the expression in terms of ##q_i## and ##p_i## only.

^^^ That

The Hamiltonian is supposed to be a function of just the ##q_i##'s and ##p_i##'s so once you find the conjugate momenta via

##p_i = \frac{\partial L}{\partial \dot{q}_i}## you need to re-arange and solve for each of the ##\dot{q}_i##'s and then plug it into

##H = \sum_{i=1}^{2} p_i \dot{q}_i - L##

You need to re-express ##L## in terms of the phase space variables ##q_i##'s and ##p_i##'s as well....after all that is the "second part of the hamiltonian"