Direct Energy, 2018a

310 13.6 Problems
(a) Write the Hamiltonian of Eq. 13.66 as functions of ρ ch instead
of V , so it has the form H ( )
r, ρ ch , dρ ch
dr .
(b) Repeat the above for the Lagrangian of Eq. 13.67.
(c) Find the Euler-Lagrange equation using ρ ch as the generalized
path.
13.3. Verify that
y = 144
t 3
is a solution of the Thomas Fermi equation [46].
(While this solution satises the Thomas Fermi equation, it is not
useful in describing the energy of an atom. In the t → 0 limit, this
solution approaches innity, y(0) →∞. However, in the t → 0 limit,
the solution should approach a constant, y(0) → 1, to correctly describe
the physical behavior of an atom [180].)
13.4. The previous problem discussed that
y = 144
t 3
is a solution of the Thomas Fermi equation. Show that
is not
a solution.
y = 72
t 3
13.5. Prove that the Thomas Fermi equation is nonlinear.