91Ó°ÊÓ

Show that the geodesics on a circular cylinder (with elements parallel to the z axis) are helics $\mathit{a}\mathit{z}\mathbf{+}\mathit{b}\mathit{θ}\mathbf{=}\mathit{c}$, where a,b,c are constants depending on the given endpoints.(Hint: Use cylindrical coordinates) Note that the equation $\mathit{a}\mathit{z}\mathbf{+}\mathit{b}\mathit{θ}\mathbf{=}\mathit{c}$includes the circles $\mathit{z}\mathbf{=}$const.(for $\mathit{b}\mathbf{=}\mathbf{0}$), straight lines $\mathit{θ}\mathbf{=}$const.(for $\mathit{a}\mathbf{=}\mathbf{0}$), and the special heclices $\mathit{a}\mathit{z}\mathbf{+}\mathit{b}\mathit{θ}\mathbf{=}\mathbf{0}$.

For small vibrations, find the characteristic frequencies and the characteristic modes of vibration of the coupled pendulums shown. All motion takes place in a single vertical plane. Assume the spring is unstretched when both pendulums hang vertically and take the spring constant as$\mathbf{k}\mathbf{=}\mathbf{mg}\mathbf{/}\mathbf{l}$to simplify the algebra. Hints: Write the kinetic and potential energies in terms of the rectangular coordinates of the masses relative to their positions hanging at rest. Don’t forget the gravitational potential energies. Then write the rectangular coordinates $\mathit{x}$and $\mathit{y}$in terms of $\mathit{θ}$and $\mathit{Ï•}$, and for small vibrations approximate $\mathbf{²õ¾\pm ²Ôθ}\mathbf{=}\mathbf{θ}\mathbf{,}\mathbf{³¦´Ç²õθ}\mathbf{=}\mathbf{1}\mathbf{-}{\mathbf{θ}}^{\mathbf{2}}\mathbf{/}\mathbf{2}$, and similar equations for $\mathit{Ï•}$.

Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.

1.${\mathbf{∫}}_{{\mathbf{x}}_{\mathbf{1}}}^{{\mathbf{x}}_{\mathbf{2}}}\sqrt{\mathbf{x}}\sqrt{\mathbf{1}\mathbf{+}\mathbf{y}{\mathbf{\text{'}}}^{\mathbf{2}}}\mathit{d}\mathit{x}$

Write theθLagrange equation for a particle moving in a plane ifV=V(r) (that

is, a central force). Use theθequation to show that:

(a) The angular momentum r×mvis constant.

(b) The vector r sweeps out equal areas in equal times (Kepler’s second law).

Set up Lagrange’s equations in cylindrical coordinates for a particle of mass $\mathbf{m}$in a potential field $\mathbf{V}\left(r,\mathrm{θ},z\right)$. Hint: $\mathit{v}\mathbf{=}\frac{\mathbf{d}\mathbf{s}}{\mathbf{d}\mathbf{t}}$; writein cylindrical coordinates.

In the brachistochrone problem, show that if the particle is given an initial velocity${\mathbf{v}}_{\mathbf{0}}\mathbf{≠}\mathbf{0}$, the path of minimum time is still a cycloid.

Change the independent variable to simplify the Euler equation, and then find a first integral of it.

$\mathbf{4}\mathbf{.}\mathbf{}\mathbf{}\underset{{\mathbf{x}}_{\mathbf{1}}}{\overset{{\mathbf{x}}_{\mathbf{2}}}{\mathbf{∫}}}\mathit{y}\sqrt{\mathbf{y}{\mathbf{\text{'}}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}\mathit{d}\mathit{x}$

Two particles each of mass m are connected by an (inextensible) string of length I. One particle moves on a horizontal table (assume no friction), The string passes through a hole in the table and the particle at the lower end moves up and down along a vertical line. Find the Lagrange equations of motion of the particles. Hint: Let the coordinates of the particle on the table be r and $\mathit{θ}$, and let the coordinate of the other particle be z. Eliminate one variable from $\mathit{L}\left(usingr+\left|z\right|=I\right)$and write two Lagrange equations.