Consider an object moving in a plane under the influence of some central
force with corresponding potential energy u(r) where r = x2 + y2 in cartesian coordinates.
a. (3 pts) set up the lagrangian for this system (in cartesian coordinates).
b. (2 pts) find the conjugate momenta px and py for this system.
c. (2 pts) set up the hamiltonian, h (x, y, px , py ), for this system.
d. (4 pts) evaluate the poisson bracket h , l where l = xp − yp . what
{z}zyx does this result indicate about the angular momentum?
e. (2 pts) now, set up the lagrangian in polar coordinates for this system.
f. (2 pts) find the conjugate momenta pr and pφ for this system.
g. (2 pts) set up the hamiltonian h r,φ, p , p for this system. (rφ)
h. (4 pts) find the equations of motion from the hamiltonian.
i. (4 pts) evaluate the poisson bracket h, l where l = mr2φ! , ensuring that
everything is in phase space coordinates. what does this result indicate about the angular momentum?
a particle of mass m moves along the x axis under the influence of the potential
u(x)=u0x4e−αx2
where u0 and α> 0 are constants.
a. (4 pts) plot (using a computer) a graph of the potential energy function u(x).
print and attach this plot to the problem. note what values you use for uo
and α. this is meant to give you the shape of the curve.
b. (4 pts) calculate the equilibrium points of the motion and discuss their
stability.
c. (4 pts) calculate the maximum energy of a bound orbit.
d. (4 pts) write an equation for the total energy e in terms of x and x!
e. (5 pts) plot a x! vs x phase portrait of the system. plot several curves in the
regions eemax, and the separatrix curve for e=emax. be sure to label these regions and the separatrix clearly. also, indicate the direction of motion using arrows on the orbits. be sure to note what constants you selected for u0 and α> 0. print this plot and attach it to the problem. you can make the labels of the energy regions and the arrows on the curves by hand.
f. (4 pts) obtain an equation for an orbit very near a point of stable equilibrium and show that the motion is not elliptical. explain based on the equation and phase plot. (hint: think about how you might approximate the exponential piece of the potential energy.)