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 L and find the generalized momentum p = ∂xL/∂ẋ. Find the Hamiltonian 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.)

Solution

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First, the coordinate $x$ looks in the direction of $\vec g$ (gravity acceleration). Now we can decida for this 1D problem since that $x$ is our generalized coordinate. If we write Lagrangian we have:

$T=\dot x^2m/2,\quad U=-mgx, \quad L=T-U=\dot x^2m/2+mgx$

Now we can find generalized momentum like:

$p=\dfrac{\partial L}{\partial\dot x}=m\dot x$

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