Consider the two-dimensional motion of a particle of mass m with a potential energy U(x,y)=(1/2)k1x2+(1/2)k2y2.
(a) Write down the Lagrangian and Hamiltonian functions, L an H.
(b) Find the Hamilton-Jacobi differential equation in terms of the principle
function S(x,y,a1,a2,t).
(c) Find the general solution of the Hamilton-Jacobi equation defined in (b).
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Consider a particle with mass m in one dimension. It is subject to a time-dependent force F(t),
,
which is assumed to be independent of the canonical coordinates (x,p) and derivable
from an external potential. (a) Find the Lagrangian and Hamiltonian for this
system.
(b) Give the Hamilton-Jacobi equation.
(c) Solve the Hamilton-Jacobi equation using the following separable
"Ansatz" for Hamilton's principal function S(t,x,a)=A(t)x+B(t).
Express A(t) and B(t) in terms of integrals over the arbitrary function F(t).
Determine a in S from the initial conditions x(0)=p(0)=0.
(d) Solve the integrals in part c for the following example: F(t)=F0sinw t.
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Consider a particle with mass m in one dimension moving in a periodic potential.
V(x)=V(x +na), n=-¥,¼,-2,-1,0,1,2,¼,¥, with
.
(a) Give the Hamiltonian of the system and sketch trajectories in phase space for
three different energies: E<V0, E=V0, and E>V0.
(b) Calculate the action variable J(E), and the angle variable q(x,E) for E>V0.
(c) Express the Hamiltonian in terms of J and q in
the limit E®V0.
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In a one-dimensional problem a particle (m=1) moves
in a potential 
, x>0,
V(x)=¥ , x<0.
(a) How does V(x) behave as x®0 and as x®¥? Does the potential have maxima and minima in the region x>0?
(b) For what values of the energy is finite motion possible?
(c) Find xmin and xmax as a function of E
for finite motion.
(d) Find the period of the motion in terms of a definite integral.
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Consider a simple, conservative, mechanical system with a single degree of freedom and Hamiltonian function H=p2/2m+U(q). U(q) has a single minimum, located at q=0, and rises monotonically to infinity as q®±¥ .
(a) Derive a simple expression for the
frequency (cycles per unit time) of the motion as a function of the total energy, n(E).
(b) Show that n can be written as n=¶E/¶J, where J is
the action integral for the system.
(c) Verify the results in parts a and b for the case of the harmonic oscillator.
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