- #1

Jaime_mc2

- 8

- 9

Now, imagine the opposite problem: we don't know anything about the system and the potentials involved, but someone gives us the phase space evolution, ##x(t)## and ##p(t)##, for the same initial conditions. Can we get the energy using the hamiltonian formalism?.

From the phase space evolution, we know that ##\dot{y}=-gt = p/m## and ##\dot{p} = -mg##. Then $$ \dot{y}=\dfrac{\partial H}{\partial p} \ \Rightarrow\ H = \dfrac{p^2}{2m} + f(y,t) $$ $$ \dot{p} = -\dfrac{\partial H}{\partial y} = -\dfrac{\partial f}{\partial y} \ \Rightarrow\ f(y,t) = mgy + g(t) $$ Concluding that $$H = \dfrac{p^2}{2m} + mgy + g(t) $$

Apparently, we don't have enough information to determine the form of ##g(t)##. Two questions came to my mind:

- Were the Hamilton's equations integrated correctly? This seems to work when I put ##\dot{y}## as a function of ##p##, but woud it work expressing ##\dot{y}## in terms of other combinations of ##y##, ##p## or ##t##?. When is it mathematically correct to get rid of the time variable to integrate the equations?
- How can we know the expression for ##g(t)##, and how can we know the relation of the found hamiltonian with the energy if we don't have any explicit information about the potentials?.