Solution Found!
Consider a mass m constrained to move in a vertical line
Chapter 13, Problem 13.2(choose chapter or problem)
Consider a mass m constrained to move in a vertical line under the influence of gravity. Using the coordinate x measured vertically down from a convenient origin O, write down the Lagrangian \(\mathcal{L}\) and find the generalized momentum \(p=\partial \mathcal{L} / \partial \dot{x}\). Find the Hamiltonian \(\mathcal{H}\) as a function of x and p, and write down Hamilton's equations of motion. (It is too much to hope with a system this simple that you would learn anything new by using the Hamiltonian approach, but do check that the equations of motion make sense.)
Questions & Answers
QUESTION:
Consider a mass m constrained to move in a vertical line under the influence of gravity. Using the coordinate x measured vertically down from a convenient origin O, write down the Lagrangian \(\mathcal{L}\) and find the generalized momentum \(p=\partial \mathcal{L} / \partial \dot{x}\). Find the Hamiltonian \(\mathcal{H}\) as a function of x and p, and write down Hamilton's equations of motion. (It is too much to hope with a system this simple that you would learn anything new by using the Hamiltonian approach, but do check that the equations of motion make sense.)
ANSWER:Step 1 of 3
First, the coordinate x looks in the direction of \(\vec{g}\) (gravity acceleration).
Now we can decide for this 1D problem since x is our generalized coordinate If we write Lagrangian we have:
\(T=\dot{x}^{2} m / 2, \quad U=-m g x, \quad L=T-U=\dot{x^2} m / 2+m g x\)
Now we can find generalized momentum like:
\(p=\frac{\partial L}{\partial \dot{x}}=m \dot{x}\)