Answer Happy

1-) A Lagrangian function is given in the form of
L(q,q',q'') with respect to q,
dq/dt and
d2q/dt2.
a) Starting from Hamilton's principle (assuming zero
variation of the extremes of q
and dq/dt), show that the equation of motion
is
b)Obtain the equations of motion for the Lagrangian function
2-) A point body of mass m moving frictionlessly on the inner
surface of cone
under
the influence of its own weight,
a) Write the Lagrangian equations of motion and find the bond
forces exerted by the surface on the object.
b) Find the angular momentum of the cone when it moves on the
circle, which is the intersection with the z=h plane.
3-) A particle with mass M and +q charge is under the influence
of a homogeneous electric field E0 in the –z-direction
and a homogeneous magnetic field B0 in the same
direction and a gravitational field.
a) Write the Lagrangian function of the motion in Cartesian
coordinates.
b) Write the Lagrangian function of the motion in cylindrical
coordinates.
c) Write down the conserved quantities of motion.
d) Obtain the Euler-Lagrange equations of motion and solve
them.
4-)An object of mass m moves in the force field derived from the
potential
a) Determine the coordinates of the stable equilibrium position of
the object.
b) Find the equations of motion that give the normal modes of small
oscillations around this stable point.