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Set up the Hamiltonian and Hamilton's equations for a
Chapter 13, Problem 13.9(choose chapter or problem)
Set up the Hamiltonian and Hamilton's equations for a projectile of mass m, moving in a vertical plane and subject to gravity but no air resistance. Use as your coordinates x measured horizontally and y measured vertically up. Comment on each of the four equations of motion.
Questions & Answers
QUESTION:
Set up the Hamiltonian and Hamilton's equations for a projectile of mass m, moving in a vertical plane and subject to gravity but no air resistance. Use as your coordinates x measured horizontally and y measured vertically up. Comment on each of the four equations of motion.
ANSWER:Step 1 of 3
The Lagrangian is \(\mathcal{L}=T-U=\frac{1}{2} m\left(\dot{x}^{2}+\dot{y}^{2}\right)-m g y\), and the generalized momentum has components
\(P_{x}=\partial \mathcal{L} / \partial \dot{x}=m \dot{x}\) and \(P_{y}=\partial \mathcal{L} / \partial \dot{y}=m \dot{y} \).
Therefore, the Hamiltonian is
\(\mathcal{H}=\mathbf{p} . \dot{\mathbf{r}}-\mathcal{L}=\left(P_{x}^{2}+P_{y}^{2}\right) / 2 m+m g y \).