% ============================================================================
% AUTO-GENERATED STUDENT LATEX TEMPLATE
% This file is generated from the canonical assignment source.
% All selected QABank files have been inlined and all solution environments
% have been removed.  It is therefore self-contained and safe to distribute.
% Students may type their work directly into this copy.
% Do not edit this generated file when maintaining the canonical assignment.
% ============================================================================

\documentclass[11pt]{article}
% >>> BEGIN EXPANDED INPUT: AssignmentCommon.tex
% AssignmentCommon.tex
% Shared formatting and reusable question-bank machinery for PHYS V1100.
% Intended to be input from PSet*.tex and Exam*.tex master files.

\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,mathtools,bm}
\usepackage{enumitem}
\usepackage{etoolbox}
\usepackage{xparse}
\usepackage{comment}
\usepackage{xcolor}
\usepackage{keyval}
\usepackage{fancyhdr}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning,decorations.pathmorphing}

% -----------------------------------------------------------------------------
% Global switches controlled by the master assignment/exam file.
% -----------------------------------------------------------------------------
\newtoggle{ANS}
\newtoggle{HINTS}
\newtoggle{SPACE}

% -----------------------------------------------------------------------------
% Guided multipart questions.
% QA is deliberately a list environment: each part can scaffold the next part.
% Two levels are defined for occasional subparts.
% -----------------------------------------------------------------------------
\newlist{QA}{enumerate}{2}
\setlist[QA,1]{
  label=(\alph*),
  ref=\alph*,
  leftmargin=*,
  itemsep=0.9em,
  topsep=0.5em,
  parsep=0pt
}
\setlist[QA,2]{
  label=(\roman*),
  ref=\roman*,
  leftmargin=*,
  itemsep=0.45em,
  topsep=0.35em,
  parsep=0pt
}

% Semantic wrapper for the common stem of a question.  This is intentionally a
% normal environment rather than a body-capturing macro, so bank questions remain
% compatible with figures, listings, and other complex LaTeX content.
\newenvironment{question}{}{}

% Configure visibility after the master file sets ANS and HINTS.  The comment
% package skips hidden bodies completely and remains robust for long solutions,
% nested lists, figures, and verbatim-style material.
\newcommand{\ConfigureQAVisibility}{%
  \iftoggle{ANS}
  {%
    \specialcomment{answer}
      {\par\smallskip\begingroup\small\color{red}\noindent\textbf{Solution.}\quad}
      {\par\endgroup\smallskip}%
  }
  {\excludecomment{answer}}%
  \iftoggle{HINTS}
  {%
    \specialcomment{hint}
      {\par\smallskip\begingroup\small\color{black!65}\noindent\textit{Hint.}\quad}
      {\par\endgroup\smallskip}%
  }
  {\excludecomment{hint}}%
}


% Optional response space for exam-style student copies.  Bank questions may
% place \AnswerSpace{<number of baseline lines>} immediately after an answer
% environment.  The same source then supports three output modes:
%   question only:        ANS=false, SPACE=false
%   question + space:     ANS=false, SPACE=true
%   question + solution:  ANS=true,  SPACE=false
% If ANS and SPACE are both true, the solution takes precedence and no blank
% response space is inserted.  The line counts can be adjusted manually now and
% estimated automatically from rendered solutions in a future build tool.
\newcommand{\AnswerSpace}[1]{%
  \iftoggle{ANS}{}{%
    \iftoggle{SPACE}{%
      \par\smallskip
      \vspace{#1\baselineskip}%
      \par
    }{}%
  }%
}

% -----------------------------------------------------------------------------
% Question selection from the bank.
% Assignment-specific metadata belongs here, not inside the bank file.
%
% -----------------------------------------------------------------------------
\newcounter{problem}

\makeatletter
\define@key{QAUse}{label}{\def\QAUseLabel{#1}}
\define@key{QAUse}{points}{\def\QAUsePoints{#1}}

\newcommand{\QuestionHeader}[1]{%
  \par\bigskip
  \noindent
  {\large\bfseries Problem #1}%
  \ifdefempty{\QAUseLabel}{}{\hfill\fbox{\strut\sffamily\small \QAUseLabel}}%
  \ifdefempty{\QAUsePoints}{}{\hspace{0.75em}{\small[\QAUsePoints\ points]}}%
  \par\smallskip
  \hrule
  \medskip
}

% Begin a question without specifying where its body comes from.  This public
% wrapper is also used by the student-template generator after it inlines a bank
% question into a single self-contained source file.
\NewDocumentCommand{\BeginQuestion}{O{}}{%
  \def\QAUseLabel{}%
  \def\QAUsePoints{}%
  \setkeys{QAUse}{#1}%
  \refstepcounter{problem}%
  \QuestionHeader{\theproblem}%
}

\makeatother

% Convenience notation used repeatedly in the mechanics assignments.
\newcommand{\vect}[1]{\bm{#1}}
\newcommand{\ones}{\vect{1}}

% <<< END EXPANDED INPUT: AssignmentCommon.tex

% Student version: ANS=false, SPACE=false. Set HINTS true for the hinted copy
% and ANS true for the instructor solution copy.
\settoggle{ANS}{false}
\settoggle{SPACE}{false}
\settoggle{HINTS}{false}
\ConfigureQAVisibility

\begin{document}

\begin{center}
  {\Large\bfseries Problem Set 5}\\[2mm]
  {\large PHYS V1100: Analytical Dynamics}\\
  Fall 2026\\[1mm]
  Professor Mark Shattuck\\[1mm]
  \textbf{Due: Monday, October 26}
\end{center}

\medskip
\noindent
This set develops continuum field mechanics and completes the central-force unit.  In several places the physical system is specified but the useful description is not.  Choose coordinates, conserved quantities, boundary conditions, or checks that expose the mechanics before beginning detailed algebra.  Predictions and independent checks are part of the solution where requested.

\noindent
Show enough reasoning that another person can follow your argument.  Keep boundary terms until their physical meaning is clear, distinguish continuum approximations from separate linearization assumptions, and treat configuration and complete dynamical state as different concepts.  Computation is not required for this set.  In the written-plus-oral problem, all students complete the written portion; only the designated presenter prepares the oral extension.

\bigskip

% Field action with a mechanically active boundary term.
% =====================================================================
% FLATTENED QUESTION: QABank/Q09_02.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q09_02
% Title: A continuum field with an elastic boundary
% Status: ready
% Introduced: 9
% BestFit: continuum fields; variational formulation; boundary conditions
% Topics: field action; boundary energy; field variation; natural boundary condition; complete state; continuum approximation; linearization
% Type: modeling / variational derivation / interpretation
% Difficulty: medium
% Computational: no
% Source: original
% Tags: continuum field; action; boundary term; elastic boundary; natural boundary condition; state; continuum approximation; small slope
% Notes: Students construct the action rather than receiving it. The endpoint spring makes the spatial boundary term mechanically active. Continuum smoothness and geometric/constitutive linearization are kept separate.
% ============================================================

\begin{question}
\textbf{A continuum field with an elastic boundary.}
Consider a one-dimensional continuum field $\phi(x,t)$ on
\[
  0\le x\le \ell
\]
with bulk Lagrangian density
\[
  \boxed{
  \mathcal L
  =\frac12\rho\phi_t^2
  -\frac12\mathcal T\phi_x^2.}
\]
The left endpoint is fixed,
\[
  \phi(0,t)=0,
\]
while the right endpoint is attached to a linear spring of stiffness $k_b$.  The other end of this spring is fixed, so its extension is $\phi(\ell,t)$.

Variations vanish at the initial and final times:
\[
  \delta\phi(x,t_1)=\delta\phi(x,t_2)=0.
\]

\begin{QA}

\item \textbf{Construct the model before varying it.}
Before doing any calculation, predict what boundary behavior you expect in the two limits
\[
  k_b=0,
  \qquad
  k_b\to\infty.
\]
Then construct the complete action for the field plus the endpoint spring.  Explain why the spring contribution is a boundary term in the Lagrangian rather than part of the integral over $x$.



\item \textbf{Vary the action and keep the boundary term.}
Carry out the field variation at fixed spacetime coordinates,
\[
  \delta x=0,
  \qquad
  \delta t=0.
\]
Integrate by parts as needed.  Do not discard the spatial boundary contribution before interpreting it.

Derive both the field equation in the interior $0<x<\ell$ and the boundary condition at $x=\ell$.  State clearly why the left boundary contribution vanishes and why the variation at $x=\ell$ is not set to zero.



\item \textbf{Interpret and check.}
Use your boundary condition to recover the two limiting behaviors predicted in part (a).

Then answer briefly:
\begin{enumerate}[label=(\roman*)]
\item Is $\phi(x,t_0)$ by itself a complete dynamical state for this system?  If not, what additional initial data are required?
\item In deriving a model of this form from a discrete chain or string, distinguish the continuum/smoothness approximation from a small-slope or linearization approximation.  Why are they logically different assumptions?
\item Identify the decisive idea in this problem, one assumption your result uses, and one check that would expose an error.
\end{enumerate}



\end{QA}
\end{question}



% Reconstruct A, C, and A^T C A for the same physical system.
% =====================================================================
% FLATTENED QUESTION: QABank/Q09_03.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q09_03
% Title: Reconstruct the discrete mechanics of an elastic boundary
% Status: ready
% Introduced: 9
% BestFit: continuum-discrete synthesis; compatibility; force balance
% Topics: incidence matrix; compatibility; constitutive matrix; ATCA; boundary force balance; continuum limit; complete state
% Type: modeling / structural derivation / validation
% Difficulty: medium
% Computational: no
% Source: original
% Tags: discrete mechanics; incidence matrix; compatibility; constitutive response; force balance; elastic boundary; continuum limit; ATCA
% Requires: Q09_02
% Notes: Uses the same physical system as Q09_02. Students construct A rather than receiving it and then recover the continuum boundary condition independently from discrete nodal balance.
% ============================================================

\begin{question}
\textbf{Reconstruct the discrete mechanics of an elastic boundary.}
Consider a discrete elastic representation of the same spatial system.  Take nodes
\[
  x_j=ja,
  \qquad
  j=0,1,2,3,
  \qquad
  \ell=3a,
\]
with
\[
  q_0=0
\]
at the fixed left boundary.  The moving nodal values are
\[
  \boxed{
  \mathbf q=
  \begin{pmatrix}
    q_1\\q_2\\q_3
  \end{pmatrix}.}
\]
Each of the three bulk segments has effective stiffness
\[
  \kappa=\frac{\mathcal T}{a},
\]
and the final node $q_3$ is attached to the fixed endpoint spring of stiffness $k_b$.

\begin{QA}

\item \textbf{Construct the discrete structure.}
Choose an orientation for each elastic element and construct an extension vector
\[
  \mathbf e=A\mathbf q.
\]
Your model should contain the three bulk elastic segments and the endpoint spring.  Construct an appropriate constitutive matrix $C$ and write the elastic potential energy in the form
\[
  \boxed{V=\frac12\mathbf e^TC\mathbf e.}
\]
Different consistent choices of edge orientation are acceptable.



\item \textbf{From compatibility to force balance.}
Starting from your energy, derive the internal elastic force
\[
  \boxed{
  \mathbf F_{\rm int}
  =-A^TC\mathbf e
  =-A^TCA\mathbf q.}
\]
Compute $A^TCA$.  In your explanation, distinguish the roles of $A$, $C$, and $A^T$: what represents compatibility, what represents constitutive response, and what returns edge forces to nodal force balance?



\item \textbf{Recover the continuum boundary condition.}
For this part consider a static configuration, so the last nodal force must balance to zero.  Use the last component of $\mathbf F_{\rm int}$ to show that
\[
  \frac{\mathcal T}{a}(q_3-q_2)+k_bq_3=0.
\]
Then use
\[
  q_j\simeq\phi(x_j)
\]
to recover the continuum boundary condition from the previous problem.

Finally explain briefly:
\begin{enumerate}[label=(\roman*)]
\item What additional information would be required to turn the vector $\mathbf q$ into a complete dynamical state?
\item Does the existence of this discrete model imply that the underlying physical system must be fundamentally discrete?
\end{enumerate}



\end{QA}
\end{question}



% Complete the central-force unit with an inverse-square synthesis problem.
% =====================================================================
% FLATTENED QUESTION: QABank/Q11_03.tex
% =====================================================================
\BeginQuestion[label=Written + Oral Presentation]
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q11_03
% Title: From initial data to a Kepler orbit
% Status: ready
% Introduced: 11
% BestFit: central-force motion; Kepler problem; orbital geometry
% Topics: two-body reduction; angular momentum; energy; effective potential; circular orbit; escape speed; Binet equation; conic sections; eccentricity; validation
% Type: synthesis / modeling choice / derivation / interpretation / oral extension
% Difficulty: medium / advanced
% Computational: no
% Source: original, informed by Q11_02
% Tags: central force; inverse square; Kepler; effective potential; orbit classification; Binet equation; eccentricity; validation; oral presentation
% Requires: Q03_06
% Notes: The written portion is for all students. The optional oral extension tests when the total effective potential does or does not produce a small-r barrier for V(r) ~ -a/r^n. Uses J for angular momentum so L remains reserved for the Lagrangian.
% ============================================================

\begin{question}
\textbf{From initial data to a Kepler orbit.}
The center-of-mass motion of a two-body system has already been separated.  Begin with the exact relative-coordinate problem for a particle of reduced mass $\mu$ moving in the attractive central potential
\[
  \boxed{
  V(r)=-\frac{k}{r},
  \qquad
  k>0.}
\]
At $t=0$, suppose
\[
  r=r_0,
  \qquad
  \dot r=0,
\]
and the relative velocity is purely tangential with magnitude $v_0>0$.  Choose the orientation of $\theta$ so that the conserved scalar angular momentum is positive.

The written portion is for everyone.  The oral extension at the end is prepared only by the student assigned to present this problem in class.

\medskip
\noindent\textbf{Written portion --- all students submit.}

\begin{QA}

\item \textbf{Choose a useful reduced description and predict the motion.}
Choose the conserved quantities or reduced description that you think are most useful for deciding whether the motion is bound and whether the particles can approach $r=0$.  Explain your choice before calculating.

Using the initial data:
\begin{enumerate}[label=(\roman*)]
\item determine the conserved angular momentum $J$ and relative energy $E$;
\item construct the effective potential governing the radial motion;
\item explain whether a finite-energy trajectory with $J\ne0$ can reach $r=0$;
\item derive the tangential speed $v_c$ that gives a circular orbit at $r_0$ and the escape speed $v_{\rm esc}$ at $r_0$;
\item classify the motion qualitatively for $0<v_0<v_{\rm esc}$, $v_0=v_{\rm esc}$, and $v_0>v_{\rm esc}$.
\end{enumerate}
For a bound noncircular orbit, also determine whether the initial radius $r_0$ is periapsis or apoapsis.



\item \textbf{Recover the orbit shape.}
The radial reduction determines $r(t)$, but the actual orbit is a curve in the plane.  Eliminate $t$ in favor of $\theta$ and choose a variable that makes the orbit equation simple.  Derive an equation for the shape $r(\theta)$.



\medskip
\noindent\textit{Checkpoint for continuation.}
Whether or not you complete the reduction, you may use for the remaining work
\[
  \boxed{
  \frac{d^2u}{d\theta^2}+u=\frac{\mu k}{J^2},
  \qquad
  u(\theta)=\frac1{r(\theta)}.}
\]
Solve this equation and show that, after choosing the angular origin conveniently,
\[
  \boxed{
  r(\theta)=\frac{p}{1+e\cos\theta},
  \qquad
  p=\frac{J^2}{\mu k}.}
\]
Explain what information is carried by $p$ and $e$.



\item \textbf{Connect the geometry back to the dynamics.}
Using a method of your choice, derive or independently verify
\[
  \boxed{
  e^2=1+\frac{2EJ^2}{\mu k^2}.}
\]
Use this result to connect $E<0$, $E=0$, and $E>0$ with ellipse, parabola, and hyperbola.

Finally define
\[
  \boxed{
  \alpha=\frac{\mu r_0v_0^2}{k}=\frac{v_0^2}{v_c^2}.}
\]
Show that the initial conditions imply
\[
  \boxed{e=|\alpha-1|,}
\]
and use this as an independent check of your qualitative predictions in part (a).



\end{QA}

\medskip
\noindent\textbf{Oral extension --- designated presenter only.}
Suppose that near the origin a central potential behaves as
\[
  V(r)\sim-\frac{a}{r^n},
  \qquad
  a>0.
\]
For $J\ne0$,
\[
  V_{\rm eff}(r)
  \sim
  -\frac{a}{r^n}
  +\frac{J^2}{2\mu r^2}.
\]
Prepare a short board argument comparing the cases
\[
  n<2,
  \qquad
  n=2,
  \qquad
  n>2.
\]
When does the total effective potential necessarily rise to $+\infty$ as $r\to0^+$?  When can the attractive microscopic potential dominate?  What extra comparison is needed in the marginal case $n=2$?



\end{question}



\end{document}