Deriving the Hamiltonian of a system given the Lagrangian

[astroholly]

Homework Statement

Derive the Hamiltonian, H([q][/1], [q][/2], [p][/1], [p][/1]), of a system that has the Lagrngian, L(q_1, q_2, \dot{q_1}, \dot{q_2}) = \dot{q_1}^2 + 0.5 \dot{q_2}^2 + 3q_1^2 + \dot{q_1} * \dot{q_2}

Relevant Equations

H(q_1, q_2, p_1, p_2) = sum over i (p_i \dot{q_i}) - L

L(q_1, q_2, \dot{q_1}, \dot{q_2}) = \dot{q_1}^2 + 0.5 \dot{q_2}^2 + 3q_1^2 + \dot{q_1} * \dot{q_2}

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?

 

[Hill]

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.

 

[PhDeezNutz]

^^^ 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"