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\begin{document}

\begin{center}
  {\Large\bfseries Problem Set 4}\\[2mm]
  {\large PHYS V1100: Analytical Dynamics}\\
  Fall 2026\\[1mm]
  Professor Mark Shattuck
\end{center}

\medskip
\noindent
Show enough reasoning that another person can follow your argument.  This set emphasizes the
transition from generalized coordinates to continuous symmetries and conservation laws, and then to
coupled oscillators viewed as a discrete field.  Look for structure before doing algebra: identify the
symmetry, boundary condition, or local interaction rule that controls the calculation.  Computation may
be used for checking and visualization, but it should support the mechanical argument rather than replace it.
Parts explicitly marked \textbf{Optional} or problems labeled \textbf{Challenge} are not required for submission.
In the oral-presentation problem, all students complete the written portion; only the designated presenter
prepares the oral extension.

\bigskip

% Bridge from generalized coordinates to symmetry.
% =====================================================================
% FLATTENED QUESTION: QABank/Q06_05.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q06_05
% Title: Block on a movable wedge: cyclic coordinate and coupled acceleration
% Status: ready
% Introduced: 6
% BestFit: generalized coordinates; cyclic coordinates; symmetry and conservation
% Topics: movable wedge; generalized coordinates; coordinate map; non-diagonal kinetic energy; cyclic coordinate; conserved momentum; energy conservation
% Type: guided derivation / structural interpretation
% Difficulty: medium / advanced
% Computational: no
% Source: adapted from Physics 351 Exam 1
% Tags: wedge; generalized coordinates; cyclic coordinate; horizontal momentum; energy; coupled motion
% Notes: Distinct from the fixed-incline Q06_04 problem because the support itself moves. The legacy sign convention for the block height is corrected, and the symbol mu is avoided because it is reserved elsewhere for reduced mass.
% ============================================================

\begin{question}
\textbf{Block on a movable wedge: cyclic coordinate and coupled acceleration.}
A block of mass $m$ slides without friction on a wedge of mass $M$ and fixed angle $\theta$.  The wedge itself slides without friction on a horizontal table.  Gravity acts vertically downward.

Let $X$ be the horizontal displacement of the wedge to the right, and let $s$ measure the displacement of the block \emph{down the incline}.  Take the vertical coordinate to increase upward.  Up to an irrelevant fixed vertical offset $H_0$, the block position in the laboratory frame is
\[
  \boxed{
  \mathbf r_m(X,s)
  =
  \begin{pmatrix}
  X+s\cos\theta\\
  H_0-s\sin\theta
  \end{pmatrix}.}
\]

\begin{center}
  \begin{tikzpicture}
    \draw[rotate=-40,transform shape] (-3,-1/2) rectangle (-2,1/2) node[pos=.5] {$m$};
    \draw[rotate=-40,transform shape] (-5,-1/2)--(1,-1/2);
    \draw (-5,-1)--(1.5,-1);
    \draw (-4.1,-1)--(-4.1,2.8);
    \node (angle) at (-1,-.5) {$\theta$};
    \draw[line width=3pt,-{Stealth[scale=1]},rotate=-40,transform shape](-4.95,-.5)--(-3,-.5) node[midway,above]{$s$};
    \draw[line width=3pt,-{Stealth[scale=1]}](-7,-1)--(-4.1,-1) node[midway,below]{$X$};
    \draw[line width=3pt,-{Stealth[scale=1]}](-4.1,-1)--(-4.1,2.8) node[midway,left]{$H_0$};
    \draw[->,thick] (1,1.8)--(1,0.6) node[midway,right] {$g$};    
    \node at (-3,0.25) {$M$};
  \end{tikzpicture}      
\end{center}

\begin{QA}

\item Differentiate the coordinate map and show that the block speed satisfies
\[
  \boxed{
  \dot{\mathbf r}_m^{\,2}
  =\dot X^2+2\dot X\dot s\cos\theta+\dot s^2.}
\]
Then write the full Lagrangian in the coordinates $(X,s)$.



\item Identify the cyclic coordinate.  Find its conjugate momentum and interpret the conserved quantity physically.



\item Derive the two Euler--Lagrange equations and solve them for $\ddot s$ and $\ddot X$.  Show that
\[
  \boxed{
  \ddot s
  =g\,\frac{(M+m)\sin\theta}{M+m\sin^2\theta}},
\]
\[
  \boxed{
  \ddot X
  =-g\,\frac{m\sin\theta\cos\theta}{M+m\sin^2\theta}.}
\]



\item Show that the mechanical energy is conserved and write it explicitly.  Why does this conservation law coexist with the changing gravitational potential energy of the block?



\item Check the limiting cases $M\to\infty$, $\theta\to0$, and $\theta\to\pi/2$.  Interpret each result.



\end{QA}
\end{question}



% Time-translation symmetry and the Lagrangian energy function.
% =====================================================================
% FLATTENED QUESTION: QABank/Q07_01.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q07_01
% Title: Autonomous Lagrangians and the conserved energy function
% Status: ready
% Introduced: 7
% BestFit: time translation; Lagrangian energy; Noether review
% Topics: Lagrangian energy; explicit time dependence; total derivative; Euler-Lagrange equations; time-translation symmetry; conservation of energy
% Type: guided derivation / conceptual interpretation
% Difficulty: medium
% Computational: no
% Source: adapted from Physics 351 PSet 5
% Tags: energy; autonomous Lagrangian; time translation; generalized momentum; Euler-Lagrange; Noether
% Notes: Uses E_L rather than calling the quantity the Hamiltonian, since Hamiltonian formalism has not yet been introduced. Replaces the legacy global iff wording by the precise identity dE_L/dt=-partial L/partial t along Euler-Lagrange trajectories.
% ============================================================

\begin{question}
\textbf{Autonomous Lagrangians and the conserved energy function.}
Consider a system with generalized coordinates
\[
  q_1(t),\ldots,q_N(t)
\]
and Lagrangian
\[
  L=L(\mathbf q,\dot{\mathbf q},t).
\]
Define the generalized momenta
\[
  p_j=\frac{\partial L}{\partial \dot q_j}
\]
and the Lagrangian energy function
\[
  \boxed{E_L=\sum_{j=1}^N p_j\dot q_j-L.}
\]
Assume the motion satisfies the Euler--Lagrange equations.

\begin{QA}

\item Write the total time derivative of the Lagrangian along a trajectory.  Be explicit about the distinction between the total derivative $dL/dt$ and the explicit partial derivative $\partial L/\partial t$.



\item Differentiate $E_L$ with respect to time and use the Euler--Lagrange equations to show that
\[
  \boxed{
  \frac{dE_L}{dt}=-\frac{\partial L}{\partial t}.}
\]



\item Suppose the Lagrangian has no explicit time dependence,
\[
  \frac{\partial L}{\partial t}=0.
\]
What follows?  Explain why this condition expresses time-translation symmetry of the dynamical description.  Does conservation of $E_L$ along one special trajectory by itself prove that $L$ is independent of time everywhere in configuration-velocity space?



\item For an ordinary mechanical Lagrangian
\[
  L=T-V,
\]
assume that $V=V(\mathbf q)$ is velocity independent and that the kinetic energy is quadratic in the generalized velocities,
\[
  T=\frac12\dot{\mathbf q}^{\,T}\mathsf M(\mathbf q)\dot{\mathbf q},
\]
with symmetric $\mathsf M$.  Show that
\[
  \boxed{E_L=T+V.}
\]



\end{QA}
\end{question}



% Translation and rotation as direct Noether examples.
% =====================================================================
% FLATTENED QUESTION: QABank/Q07_02.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q07_02
% Title: Translation and rotation symmetry of two interacting particles
% Status: ready
% Introduced: 7
% BestFit: symmetry; Noether theorem; momentum and angular momentum
% Topics: Noether theorem; continuous symmetry; translation invariance; rotation invariance; linear momentum; angular momentum; external symmetry breaking
% Type: guided derivation / structural interpretation
% Difficulty: medium
% Computational: no
% Source: original
% Tags: Noether; translation; rotation; two particles; momentum; angular momentum; symmetry breaking
% Notes: Uses only the symmetry structure of a pair potential U(|r_2-r_1|); it does not require the later central-force orbit analysis.
% ============================================================

\begin{question}
\textbf{Translation and rotation symmetry of two interacting particles.}
Two particles of masses $m_1$ and $m_2$ move in a plane.  Their interaction depends only on their separation,
\[
  L
  =\frac12m_1\dot{\mathbf r}_1^{\,2}
   +\frac12m_2\dot{\mathbf r}_2^{\,2}
   -U\!\left(\left\lVert\mathbf r_2-\mathbf r_1\right\rVert\right).
\]
Define
\[
  \mathbf p_i=\frac{\partial L}{\partial\dot{\mathbf r}_i}
  =m_i\dot{\mathbf r}_i.
\]

\begin{QA}

\item Let $\mathbf a$ be an arbitrary constant vector and consider the infinitesimal common translation
\[
  \delta_{\rm sym}\mathbf r_1=\epsilon\mathbf a,
  \qquad
  \delta_{\rm sym}\mathbf r_2=\epsilon\mathbf a.
\]
Show directly that $\delta_{\rm sym}L=0$.  Use Noether's theorem to find the conserved quantity associated with this one-parameter transformation.



\item Why does the arbitrariness of $\mathbf a$ imply conservation of the full vector
\[
  \boxed{\mathbf P=\mathbf p_1+\mathbf p_2}
\]
rather than only one scalar projection of it?



\item Introduce the $90^\circ$ rotation matrix
\[
  \mathsf J=
  \begin{pmatrix}
    0&-1\\
    1&0
  \end{pmatrix}
\]
and consider the infinitesimal common rotation
\[
  \delta_{\rm sym}\mathbf r_i
  =\epsilon\mathsf J\mathbf r_i,
  \qquad i=1,2.
\]
Show that the kinetic energy and the pair separation are unchanged to first order.  Then use Noether's theorem to obtain the conserved scalar angular momentum about the origin.



\item Now add a centered isotropic external potential
\[
  V_{\rm ext}
  =\frac12\kappa_1\lVert\mathbf r_1\rVert^2
   +\frac12\kappa_2\lVert\mathbf r_2\rVert^2,
  \qquad \kappa_1,\kappa_2>0.
\]
Which of the two continuous symmetries above survives?  Which associated conservation law survives?  Explain without deriving the equations of motion.



\end{QA}
\end{question}



% Boundary conditions, spectral transforms, zero modes, and discrete-field structure.
% =====================================================================
% FLATTENED QUESTION: QABank/Q08_05.tex
% =====================================================================
\BeginQuestion[label=Written + Oral Presentation]
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q08_05
% Title: Same local operator, different boundaries: sine and Fourier modes
% Status: ready
% Introduced: 8
% BestFit: coupled oscillators; normal modes; discrete fields; boundary conditions
% Topics: discrete field; local operator; fixed boundaries; periodic boundaries; normal modes; discrete sine transform; discrete Fourier transform; zero mode; symmetry; topology
% Type: guided derivation / structural interpretation / oral extension
% Difficulty: medium / advanced
% Computational: optional
% Source: original
% Tags: coupled oscillators; fixed chain; ring; DST; DFT; Fourier modes; zero mode; boundary conditions; locality; topology
% Requires: Q02_01
% Notes: Complements Q08_04 by emphasizing that fixed and periodic chains share the same local second-difference eigenvalue relation while the boundary condition selects different allowed wave numbers and transforms. The final part connects the vector q and incidence structure to the discrete-field viewpoint of Lecture 8.
% ============================================================

\begin{question}
\textbf{Same local operator, different boundaries: sine and Fourier modes.}
Consider $N$ identical masses $m$ with nearest-neighbor springs of stiffness $\kappa$ and lattice spacing $a$.  Away from boundaries, the dimensionless stiffness operator acts as
\[
  (K\mathbf q)_n
  =2q_n-q_{n-1}-q_{n+1}.
\]
The equations of motion are
\[
  m\ddot{\mathbf q}=-\kappa K\mathbf q.
\]
We will compare two ways of closing the same local nearest-neighbor rule: a fixed--fixed chain and a periodic ring.

\begin{QA}

\item Let $A$ map nodal displacements to signed nearest-neighbor spring extensions, so that
\[
  \mathbf e=A\mathbf q,
  \qquad
  V=\frac{\kappa}{2}\mathbf e^T\mathbf e
  =\frac{\kappa}{2}\mathbf q^TA^TA\mathbf q.
\]
For the fixed--fixed chain, take $q_0=q_{N+1}=0$ and include the two boundary springs.  For the ring, identify $q_N$ periodically with $q_0$ and include only the $N$ cyclic nearest-neighbor edges.  Explain why
\[
  \boxed{K=A^TA}
\]
has the same interior action
\[
  (K\mathbf q)_n=2q_n-q_{n-1}-q_{n+1}
\]
in both systems, while the global matrices differ at the boundary.  What structural information is carried by $A$ that is not contained in the list of values $\mathbf q$ alone?



\item Try the local complex pattern
\[
  q_n\propto e^{ikna}.
\]
Show that wherever the nearest-neighbor stencil applies,
\[
  \boxed{
  \lambda(k)
  =2-2\cos(ka)
  =4\sin^2\!\left(\frac{ka}{2}\right).}
\]
Explain why this is a statement about the \emph{local operator}, before any boundary condition has been imposed.



\item For fixed endpoints, take moving sites $n=1,\ldots,N$ and impose fictitious endpoint values
\[
  q_0=q_{N+1}=0.
\]
Show that the allowed wave numbers are
\[
  \boxed{k_j a=\frac{j\pi}{N+1},\qquad j=1,\ldots,N,}
\]
and that an orthonormal real eigenbasis is
\[
  \boxed{
  Q_{nj}
  =\sqrt{\frac{2}{N+1}}
   \sin\!\left(\frac{nj\pi}{N+1}\right).}
\]
Explain why
\[
  \mathbf a=Q^T\mathbf q
\]
is a discrete sine transform of the site values.



\item For the ring, label sites by $n=0,\ldots,N-1$ and impose
\[
  q_{n+N}=q_n.
\]
Show that
\[
  \boxed{k_r a=\frac{2\pi r}{N},\qquad r=0,\ldots,N-1.}
\]
Using the normalized complex basis
\[
  \boxed{
  U_{nr}=\frac1{\sqrt N}e^{i2\pi rn/N},}
\]
explain why the modal transform
\[
  \widetilde{\mathbf q}=U^\dagger\mathbf q
\]
is a discrete Fourier transform.  Why can a real calculation replace conjugate $+k$ and $-k$ modes by real sine/cosine combinations?



\item Compare the lowest mode of the fixed chain with the $r=0$ mode of the ring.  Show that the ring has
\[
  \boxed{K\mathbf 1=0,}
\]
while the fixed--fixed chain does not.  Explain the difference using continuous symmetry rather than matrix algebra alone.



\item For a fixed--fixed chain with $N=3$, suppose
\[
  \mathbf q(0)=d
  \begin{pmatrix}1\\0\\0\end{pmatrix},
  \qquad
  \dot{\mathbf q}(0)=0.
\]
Using the normalized sine basis, show that the initial modal amplitudes are
\[
  \boxed{
  \mathbf a(0)
  =d
  \begin{pmatrix}
    1/2\\[1mm]
    1/\sqrt2\\[1mm]
    1/2
  \end{pmatrix}.}
\]
Write the resulting motion as a sum of the three normal modes.  You do not need to expand the final answer back into three separate site-coordinate formulas.



\item \textbf{Structural interpretation / oral extension.}
Suppose someone hands you only the column of numbers
\[
  \mathbf q=(q_1,\ldots,q_N)^T.
\]
Explain why this alone does not tell you whether the values live on a chain, a ring, or part of a higher-dimensional discrete manifold.  What extra information is carried by the incidence/interaction structure?  In a higher-dimensional model, why might one introduce geometric ``virtual edges'' or higher simplices that are not additional physical springs?



\end{QA}
\end{question}



% Quasi-symmetry / boundary-term Noether theorem.
% =====================================================================
% FLATTENED QUESTION: QABank/Q07_03.tex
% =====================================================================
\BeginQuestion[label=Challenge]
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q07_03
% Title: Galilean boosts as a Noether quasi-symmetry
% Status: ready
% Introduced: 7
% BestFit: Noether theorem; quasi-symmetry; boundary terms
% Topics: Noether theorem; quasi-symmetry; total derivative; Galilean boost; free particle; center-of-mass motion
% Type: challenge / guided derivation
% Difficulty: advanced
% Computational: no
% Source: original
% Tags: Noether; quasi-symmetry; Galilean boost; boundary term; center of mass
% Notes: Optional challenge illustrating the Lecture 7 extension delta_sym L = epsilon dB/dt. The transformation is treated only to first order in the infinitesimal parameter.
% ============================================================

\begin{question}
\textbf{Galilean boosts as a Noether quasi-symmetry.}
This problem uses the quasi-symmetry form of Noether's theorem.  If an infinitesimal transformation
\[
  \delta_{\rm sym}q=\epsilon h
\]
changes the Lagrangian by a total derivative,
\[
  \delta_{\rm sym}L
  =\epsilon\frac{dB}{dt},
\]
then along an Euler--Lagrange trajectory
\[
  \boxed{p\,h-B=\text{constant}.}
\]

\begin{QA}

\item For a free particle in one dimension,
\[
  L=\frac12m\dot x^2,
\]
consider the infinitesimal Galilean boost
\[
  x\longrightarrow x'=x+\epsilon t.
\]
Find $\delta_{\rm sym}x$ and $\delta_{\rm sym}\dot x$.  Show, to first order in $\epsilon$, that
\[
  \delta_{\rm sym}L
  =\epsilon\frac{d}{dt}(mx).
\]



\item Identify $h$ and $B$.  Use the quasi-symmetry form of Noether's theorem to show that
\[
  \boxed{t p-mx=\text{constant}.}
\]
Verify the result directly using the free-particle equation of motion.



\item Interpret this conservation law.  Show that it is equivalent to uniform straight-line motion
\[
  x(t)=x_0+vt.
\]
Why is this conservation law not the same statement as conservation of momentum, even though both hold for the free particle?



\item \textbf{Optional extension.}
For $N$ particles in one dimension with
\[
  L=\sum_{i=1}^N\frac12m_i\dot x_i^2
  -V(\{x_i-x_j\}),
\]
apply the common boost
\[
  \delta_{\rm sym}x_i=\epsilon t.
\]
Show that the corresponding Noether charge can be written
\[
  \boxed{tP-MX_{\rm cm}=\text{constant},}
\]
where
\[
  P=\sum_i m_i\dot x_i,
  \qquad
  M=\sum_i m_i,
  \qquad
  X_{\rm cm}=\frac1M\sum_i m_i x_i.
\]
What physical statement does this encode?



\end{QA}
\end{question}



\noindent\rule{\textwidth}{0.4pt}
\medskip
\noindent
\textbf{Additional practice, not required.}
The following bank problem works through the exact sine spectrum of a fixed-boundary chain and includes
an optional numerical check and continuum-limit interpretation.
% =====================================================================
% FLATTENED QUESTION: QABank/Q08_04.tex
% =====================================================================
\BeginQuestion[label=Optional Practice]
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q08_04
% Title: Fixed-boundary chain: sine modes and the discrete spectrum
% Status: ready
% Introduced: 8
% BestFit: coupled oscillators; discrete fields; normal modes
% Topics: fixed-boundary chain; stiffness matrix; discrete Laplacian; normal modes; sine eigenvectors; mode spectrum; continuum limit
% Type: guided derivation / computational verification
% Difficulty: medium
% Computational: optional
% Source: adapted from Physics 34100 Applied Differential Equations PSet 8
% Tags: coupled oscillators; fixed boundaries; sine modes; eigenvalues; discrete field; normal modes; continuum limit; MATLAB
% Requires: Q02_01
% Notes: Extends the fixed-boundary stiffness assembly of Q02_01 from statics to dynamics. The legacy ODE exercise supplied the sine eigenvectors and eigenvalue formula for K_N; the mechanics version restores masses and spring constants, derives the normal frequencies, and adds an optional continuum-limit interpretation.
% ============================================================

\begin{question}
\textbf{Fixed-boundary chain: sine modes and the discrete spectrum.}
Consider $N$ identical point masses $m$ moving along a line between two fixed walls.  Consecutive nodes are joined by identical springs of stiffness $k$.  Let $u_j(t)$ be the displacement of mass $j$, with fixed endpoint values
\[
  u_0(t)=u_{N+1}(t)=0.
\]
For the $N$ free masses, write
\[
  \mathbf u=(u_1,\ldots,u_N)^T.
\]
For unit spring stiffness, the fixed-boundary stiffness matrix is
\[
  \mathsf K_N=
  \begin{pmatrix}
    2&-1&0&\cdots&0\\
    -1&2&-1&\ddots&\vdots\\
    0&\ddots&\ddots&\ddots&0\\
    \vdots&\ddots&-1&2&-1\\
    0&\cdots&0&-1&2
  \end{pmatrix}.
\]
The equations of motion are therefore
\[
  \boxed{m\ddot{\mathbf u}+k\mathsf K_N\mathbf u=0.}
\]

\begin{QA}

\item For
\[
  r=1,\ldots,N,
  \qquad
  \theta_r=\frac{r\pi}{N+1},
\]
consider the vector $\mathbf y^{(r)}$ with components
\[
  \boxed{y_j^{(r)}=\sin(j\theta_r),\qquad j=1,\ldots,N.}
\]
Explain how the fixed endpoint conditions are already encoded by this choice.



\item Apply an interior row of $\mathsf K_N$ to $\mathbf y^{(r)}$.  Using
\[
  \sin((j+1)\theta)+\sin((j-1)\theta)
  =2\sin(j\theta)\cos\theta,
\]
show that
\[
  \boxed{\mathsf K_N\mathbf y^{(r)}
  =\lambda_r\mathbf y^{(r)}},
  \qquad
  \boxed{\lambda_r
  =2\left(1-\cos\theta_r\right)
  =4\sin^2\!\left(\frac{\theta_r}{2}\right).}
\]



\item Seek a normal-mode motion
\[
  \mathbf u(t)=a_r(t)\mathbf y^{(r)}.
\]
Show that $a_r$ is a simple harmonic oscillator and derive the normal frequency
\[
  \boxed{
  \omega_r
  =2\sqrt{\frac{k}{m}}
  \sin\!\left(\frac{r\pi}{2(N+1)}\right).}
\]
What is the physical meaning of increasing $r$?



\item For $N=5$, calculate the five analytic eigenvalues $\lambda_r$.  Use MATLAB or Python to compute the eigenvalues and eigenvectors of $\mathsf K_5$, and compare them with the analytic formulas.  Plot the five mode shapes $y_j^{(r)}$ against the node index $j$.



\item \textbf{Optional continuum-limit interpretation.}
Let the node spacing be $a$ and the total fixed-end length be
\[
  \ell=(N+1)a.
\]
Show that for low mode number $r\ll N$,
\[
  \omega_r
  \approx
  \sqrt{\frac{k}{m}}\frac{r\pi a}{\ell}.
\]
If the discrete chain approximates a string with mass density $\rho$ and tension $T$, so that $m=\rho a$ and $k=T/a$, recover the fixed-end string spectrum.



\end{QA}
\end{question}



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