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 Ham

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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.