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{\bf INTERMEDIATE MECHANICS II --- PHY 4936}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 3}\\%
\vspace {3mm} (September 16, 2011) \\
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\item{(10)} Verify the solution of Problem 3 (a), p.11 of Landau-Lifschitz
along the following steps: Calculate as function of the generalized
coordinate $\phi$

10.1 $\dot x$ and $\dot y$.

10.2 $\dot x^2$ and $\dot y^2$.

10.3 Eliminate the term with $\dot\phi\,\sin(\phi-\gamma t)$ from $L$.

Due Friday, September 16 before class (10 points).
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\item{(11a)} Calculate explicitly
$$ \delta_x L\ {\rm for\ the\ Lagrangian\ of\
two\ harmonically\ bound\ particles\ in\ 1D}$$
$$L=\frac{m_1}{2}(\dot{{x}}_1)^2+\frac{m_2}{2}(\dot{x}_2)^2
   -k\,(x_1-x_2)^2\ .$$
Is there a conservation law? Due in class (2 points).

\item{(11b)} Repeat the previous excercise for the harmonic ocillator
in 1D $$L=\frac{m_1}{2}(\dot{x})^2 - \frac{k}{2}\,x^2\ .$$
Due in class (2 points).
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\item{(12a)} Use the Einstein summation convention in the following.
Show that
$$ L_i = \epsilon_{ijk}\,x_j\,p_k = [\vec{r}\times\vec{p}\,]_i~~
   {\rm holds\ for}~~i=1,2,3.$$
Due in class (2 points). 

\item{(12b)} Express $\epsilon_{12k}\,\epsilon_{lmk}$ in terms 
of Kronecker delta and then $\epsilon_{ijk}\,\epsilon_{lmk}$.
Due in class (2 points). 
\bigskip

\item{(13)} Derive the answers of problems~1 and~2, Landau-Lifschitz, p.21.
Due Monday, September 26 before class (10 points).

\item{(14)} Derive the solutions of problems 1 and 2, Landau-Lifshitz p.24
            Due Friday, September 30 (6 points).

\item{(15)} Compute the solution for the equation of motion
$$ m\,\ddot x\ =\ \frac{k}{x^3}\,,~~k>0 $$
with $t=0$ initial values $x_0>0$ and $\dot x_0=0$.
Due Monday, October 3 before class (10 points).

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