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\begin{document}

\begin{center}
  {\Large\bfseries Problem Set 6}\\[2mm]
  {\large PHYS V1100: Analytical Dynamics}\\
  Fall 2026\\[1mm]
  Professor Mark Shattuck
\end{center}

\medskip
\noindent
This set closes the central-force unit and develops the course transition from local quadratic dynamics to nonlinear phase space and rigid-body motion.  The emphasis is structural: choose representations that expose the mechanics, distinguish local predictions from global behavior, and use independent checks before trusting algebra or computation.

\noindent
Show enough reasoning that another person can follow your argument.  In the small-oscillation problem, keep the distinction between Euclidean and mass-weighted orthogonality explicit.  In the nonlinear problem, make the analytical phase-space prediction before computing.  In the rigid-body problems, distinguish the reference point matrix, the moving frame, and body- versus laboratory-frame component lists.  MATLAB is the primary computational language; Python/NumPy is also acceptable where computation is requested.  In the written-plus-oral problem, all students complete the written portion; only the designated presenter prepares the oral extension.

\bigskip

% Central-force bridge: a non-Kepler central potential.
% =====================================================================
% FLATTENED QUESTION: QABank/Q10_02.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q10_02
% Title: Harmonic central force: exact orbit geometry and comparison with Kepler
% Status: ready
% Introduced: 10
% BestFit: central-force motion; exact orbit geometry; synthesis
% Topics: isotropic harmonic potential; reduced mass; planarity; angular momentum; ellipse; degeneracy; Kepler comparison
% Type: synthesis / geometric interpretation
% Difficulty: medium
% Computational: no
% Source: adapted from Q10_01 and course central-force development
% Requires: Q03_06
% Tags: central force; harmonic potential; ellipse; angular momentum; planarity; Kepler comparison
% Notes: Short synthesis version for PSet6. Avoids repeating the inverse-square derivation already required in PSet5.
% ============================================================

\begin{question}
\textbf{A different central force: the isotropic harmonic potential.}
After the center-of-mass motion of a two-body system has been separated, let the relative coordinate \(\mathbf r\) have reduced mass \(\mu\) and potential
\[
  V(r)=\frac12kr^2,
  \qquad k>0,
  \qquad r=\lVert\mathbf r\rVert.
\]
The initial data are
\[
  \mathbf r(0)=\mathbf r_0,
  \qquad
  \dot{\mathbf r}(0)=\mathbf v_0.
\]
Do not rederive the center-of-mass transformation.

\begin{QA}

\item Derive the relative equation of motion and show that
\[
  \boxed{
  \mathbf r(t)=\mathbf r_0\cos(\omega t)
  +\frac{\mathbf v_0}{\omega}\sin(\omega t),
  \qquad
  \omega=\sqrt{\frac{k}{\mu}}.}
\]
Use this solution to explain why the motion is planar and identify the conserved angular-momentum vector \(\mathbf J\).



\item Choose Cartesian axes in the orbital plane and define
\[
  B=
  \begin{pmatrix}
    x_0 & u_0/\omega\\
    y_0 & v_0/\omega
  \end{pmatrix},
  \qquad
  \mathbf s(t)=
  \begin{pmatrix}
    \cos\omega t\\
    \sin\omega t
  \end{pmatrix},
\]
where \(\mathbf r_0=(x_0,y_0)^T\) and \(\mathbf v_0=(u_0,v_0)^T\). Show that \(\mathbf r=B\mathbf s\). If \(\det B\ne0\), eliminate time and show that the orbit is an ellipse centered on the force center.



\item Interpret the exceptional case \(\det B=0\). Show that it is precisely the zero-angular-momentum case and describe the orbit.



\item Both the attractive inverse-square force and the isotropic harmonic force can produce closed elliptical bound orbits. State one essential geometric difference between the two ellipses, and explain why that difference is physically meaningful.



\end{QA}
\end{question}



% Post-L12 small-oscillation synthesis.
% =====================================================================
% FLATTENED QUESTION: QABank/Q12_01.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q12_01
% Title: General small oscillations: mass weighting and normal coordinates
% Status: ready
% Introduced: 12
% BestFit: small oscillations; generalized eigenproblem; normal coordinates
% Topics: mass matrix; stiffness matrix; generalized eigenproblem; M-orthogonality; mass normalization; normal coordinates; initial-state projection
% Type: synthesis / structural derivation
% Difficulty: medium / advanced
% Computational: no
% Source: adapted from Q08_01 and Lectures 11--12
% Tags: small oscillations; mass matrix; stiffness matrix; M-orthogonality; normal coordinates
% Notes: Post-L12 synthesis version. Uses spring-extension coordinates to make K simple while M becomes off diagonal.
% ============================================================

\begin{question}
\textbf{General small oscillations: mass weighting and normal coordinates.}
Two blocks move on a horizontal frictionless line. A mass \(m\) is attached to a wall by a spring of stiffness \(3k\), and a mass \(2m/3\) is attached to the first mass by a spring of stiffness \(k\). Let \(q_1\) and \(q_2\) be the signed extensions of the two springs measured from equilibrium. Then the physical displacements of the masses are
\[
  y_1=q_1,
  \qquad
  y_2=q_1+q_2.
\]
Define \(\mathbf q=(q_1,q_2)^T\) and \(\omega_0^2=k/m\).

\begin{QA}

\item Starting from the particle kinetic energy and the spring energies, show that
\[
  L^{(2)}=\frac12\dot{\mathbf q}^{\,T}M\dot{\mathbf q}
  -\frac12\mathbf q^TK\mathbf q,
\]
with
\[
  \boxed{M=\frac{m}{3}
  \begin{pmatrix}5&2\\2&2\end{pmatrix},
  \qquad
  K=k\begin{pmatrix}3&0\\0&1\end{pmatrix}.}
\]
Explain what this coordinate choice has simplified and what it has made less simple.



\item Derive
\[
  \boxed{M\ddot{\mathbf q}+K\mathbf q=0}
\]
and seek a normal mode \(\mathbf q=\mathbf u\,a(t)\) with \(\ddot a+\omega^2a=0\). Solve
\[
  \boxed{K\mathbf u=\omega^2M\mathbf u}
\]
to find the two frequencies and one convenient unnormalized mode shape for each.



\item Verify that the two mode shapes are not Euclidean-orthogonal but are \(M\)-orthogonal:
\[
  \boxed{\mathbf u_1^TM\mathbf u_2=0.}
\]
Find normalization constants so that the normalized vectors \(\widehat{\mathbf u}_\alpha\) satisfy
\[
  \widehat{\mathbf u}_\alpha^TM\widehat{\mathbf u}_\beta
  =\delta_{\alpha\beta}.
\]



\item Let
\[
  U=\begin{pmatrix}\widehat{\mathbf u}_1&\widehat{\mathbf u}_2\end{pmatrix},
  \qquad
  \mathbf q=U\mathbf a.
\]
Show that
\[
  \boxed{U^TMU=I_2,\qquad
  U^TKU=\operatorname{diag}(\omega_1^2,\omega_2^2),}
\]
and hence write the quadratic Lagrangian in the normal coordinates \(a_1,a_2\).



\item Suppose initially the first mass is at equilibrium, the second is displaced a distance \(d\) to the right, and both are released from rest:
\[
  y_1(0)=0,
  \qquad y_2(0)=d,
  \qquad \dot y_1(0)=\dot y_2(0)=0.
\]
Determine \(\mathbf q(0)\). Use the mass-weighted projection
\[
  a_\alpha(0)=\widehat{\mathbf u}_\alpha^TM\mathbf q(0)
\]
to show that both normal modes are excited. Explain why the subsequent motion is generally not a single-frequency oscillation.



\end{QA}
\end{question}



% Nonlinear phase portraits, stability, and limits of linearization.
% =====================================================================
% FLATTENED QUESTION: QABank/Q14_02.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q14_02
% Title: Nonlinear phase portraits: pendulum separatrix and marginal stability
% Status: ready
% Introduced: 14
% BestFit: nonlinear oscillations; stability; phase-plane methods; limits of linearization
% Topics: pendulum; phase portrait; energy level sets; separatrix; nonlinear stability; quartic potential; marginal equilibrium; numerical validation
% Type: synthesis / computational validation
% Difficulty: medium / advanced
% Computational: yes
% Source: course Lecture 12 development; informed by Q14_01
% Tags: nonlinear oscillator; phase plane; stability; separatrix; quartic potential; linearization; MATLAB; Python
% Notes: Focuses on conservative phase-space structure and the limits of quadratic theory. Damping is left to Q14_01 as optional practice.
% ============================================================

\begin{question}
\textbf{Nonlinear phase portraits: what the quadratic approximation cannot see.}
Consider first the exact simple pendulum
\[
  \ddot\theta+\omega_0^2\sin\theta=0,
  \qquad
  \omega_0^2=\frac{g}{\ell}.
\]
Its energy is
\[
  E=\frac12m\ell^2\dot\theta^2+mg\ell(1-\cos\theta).
\]

\begin{QA}

\item Introduce the dimensionless velocity \(u=\dot\theta/\omega_0\) and dimensionless energy \(\varepsilon=E/(mg\ell)\). Show that the phase curves satisfy
\[
  \boxed{\frac12u^2+1-\cos\theta=\varepsilon.}
\]
Use this equation to classify the motion for
\[
  0<\varepsilon<2,
  \qquad \varepsilon=2,
  \qquad \varepsilon>2.
\]
Sketch the corresponding qualitative phase portrait and identify the separatrix.



\item Linearize the pendulum equation about the stable equilibrium \(\theta=0\). What local behavior does the linear theory predict correctly? Name at least two global features of the exact phase portrait that the linearized equation cannot reveal.



\item Now compare the one-dimensional potentials
\[
  V_+(x)=\frac{\lambda}{4}x^4,
  \qquad
  V_-(x)=-\frac{\lambda}{4}x^4,
  \qquad \lambda>0,
\]
with unit mass. Show that both have the same equilibrium and the same quadratic linearization at \(x=0\), yet one equilibrium is stable and the other unstable. Explain what this demonstrates about a zero eigenvalue of the Hessian.



\item \textbf{Focused computation.} Work in dimensionless time \(\tau=\omega_0t\), so
\[
  \frac{d^2\theta}{d\tau^2}+\sin\theta=0.
\]
Numerically integrate the two initial states
\[
  \text{A: }\theta(0)=1.5,\quad \frac{d\theta}{d\tau}(0)=0,
\]
\[
  \text{B: }\theta(0)=0,\quad \frac{d\theta}{d\tau}(0)=2.2.
\]
Before integrating, predict from the energy which trajectory librates and which rotates. Then make one phase-plane plot containing both trajectories. Along each numerical solution compute
\[
  \varepsilon(\tau)=\frac12\left(\frac{d\theta}{d\tau}\right)^2+1-\cos\theta
\]
and report a simple measure of its numerical variation. Explain whether the computation agrees with the analytical prediction.



\end{QA}
\end{question}



% Rigid-body kinematics in the point-matrix / moving-frame formulation.
% =====================================================================
% FLATTENED QUESTION: QABank/Q15_01.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q15_01
% Title: Rigid-body kinematics from a reference point matrix
% Status: ready
% Introduced: 15
% BestFit: rigid-body kinematics; moving frames; angular velocity
% Topics: point matrix; reference configuration; center of mass; rotation matrix; angular velocity; skew matrix; body frame; laboratory frame; point velocity
% Type: synthesis / structural derivation
% Difficulty: medium / advanced
% Computational: no
% Source: original, revised to match the point-matrix formulation of Lecture 13
% Tags: rigid body; point matrix; rotation matrix; angular velocity; body frame; laboratory frame; center of mass
% Notes: Exam-II study problem for Lecture 13. Euler-angle and rotor extensions removed from the required problem.
% ============================================================

\begin{question}
\textbf{Rigid-body kinematics from a reference point matrix.}
A rigid body consists of material points with masses $m_i$.  Arrange their positions as columns of
\[
X(t)=\begin{pmatrix}\bm x_1(t)&\cdots&\bm x_N(t)\end{pmatrix}\in\mathbb R^{3\times N},
\]
and let
\[
\mathsf M=\operatorname{diag}(m_1,\ldots,m_N),
\qquad
\bm 1=(1,\ldots,1)^T,
\qquad
m_T=\bm 1^T\mathsf M\bm 1.
\]
Choose body coordinates so that the reference configuration $X_0$ is centered at the center of mass,
\[
\boxed{X_0\mathsf M\bm 1=\bm 0.}
\]
A rigid configuration is written
\[
\boxed{X(t)=\bm R(t)\bm 1^T+Q(t)X_0,\qquad Q(t)^TQ(t)=I_3,\quad \det Q(t)=1.}
\]
The matrix $Q$ maps body components into laboratory components.

\begin{QA}

\item Show directly that this map preserves every pairwise distance. Then show that the center of mass of the current configuration is exactly $\bm R(t)$.



\item Define
\[
\boxed{W_s\equiv \dot Q Q^T.}
\]
Differentiate $QQ^T=I_3$ to show that $W_s$ is skew-symmetric.  Explain why in three dimensions there is a unique vector $\boldsymbol\omega_s$ such that
\[
\boxed{W_s\bm r=\boldsymbol\omega_s\times\bm r}
\]
for every vector $\bm r$.



\item Differentiate the point-matrix configuration and show that
\[
\boxed{
\dot X
=\dot{\bm R}\bm 1^T
+W_s\bigl(X-\bm R\bm 1^T\bigr).}
\]
Deduce the velocity of an individual material point,
\[
\boxed{
\dot{\bm x}_i
=\dot{\bm R}+\boldsymbol\omega_s\times(\bm x_i-\bm R).}
\]
Use skew symmetry to verify that the instantaneous velocity relative to the center of mass is perpendicular to the radius from the center of mass.



\item Consider the explicit orientation
\[
Q(t)=Q_z(\Omega t)Q_x(\beta),
\]
where $\Omega$ and $\beta$ are constants and
\[
Q_x(\beta)=
\begin{pmatrix}
1&0&0\\
0&\cos\beta&-\sin\beta\\
0&\sin\beta&\cos\beta
\end{pmatrix},
\qquad
Q_z(\alpha)=
\begin{pmatrix}
\cos\alpha&-\sin\alpha&0\\
\sin\alpha&\cos\alpha&0\\
0&0&1
\end{pmatrix}.
\]
Calculate $W_s=\dot QQ^T$ and identify $\boldsymbol\omega_s$. Then find the same angular velocity in body components,
\[
\boldsymbol\omega_b=Q^T\boldsymbol\omega_s.
\]
Explain why $\boldsymbol\omega_s$ and $\boldsymbol\omega_b$ are not two different physical angular velocities.



\item A vector $\bm b$ has space and body component columns related by
\[
\bm b_s=Q\bm b_b.
\]
Show that
\[
\boxed{
Q^T\dot{\bm b}_s
=\dot{\bm b}_b+\boldsymbol\omega_b\times\bm b_b.}
\]
Interpret the formula in the two special cases: (i) $\bm b$ is painted on the body, and (ii) $\bm b$ is fixed in the laboratory.



\end{QA}
\end{question}



% Inertia, Euler equations, symmetric top, torque-free stability.
% =====================================================================
% FLATTENED QUESTION: QABank/Q16_03.tex
% =====================================================================
\BeginQuestion[label=Written + Oral Presentation]
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q16_03
% Title: Rigid-body dynamics: inertia, Euler equations, precession, and intermediate-axis stability
% Status: ready
% Introduced: 16
% BestFit: rigid-body dynamics; Euler equations; torque-free motion; stability
% Topics: inertia tensor; principal axes; angular momentum; rotational kinetic energy; Euler equations; symmetric top; torque-free precession; intermediate-axis theorem
% Type: synthesis / structural derivation
% Difficulty: advanced
% Computational: no
% Source: original, informed by Q16_01 and course rigid-body arc
% Tags: rigid body; inertia tensor; principal axes; Euler equations; symmetric top; torque-free motion; stability; precession
% Notes: Main PSet6 rigid-body synthesis. Uses a simple four-mass body to keep inertia arithmetic secondary to dynamics.
% ============================================================

\begin{question}
\textbf{Rigid-body dynamics: from inertia to torque-free stability.}
Four equal point masses \(m\) form a rigid planar body with body-frame positions
\[
  (\pm a,0,0),
  \qquad
  (0,\pm b,0),
  \qquad
  a>b>0.
\]
The origin is the center of mass and the displayed axes are fixed in the body. For a point mass at \(\boldsymbol\rho\), its inertia contribution about the origin is
\[
  m\left[(\boldsymbol\rho\cdot\boldsymbol\rho)I_3
  -\boldsymbol\rho\boldsymbol\rho^T\right].
\]

\begin{QA}

\item Before carrying out the full sum, use the reflection symmetries of the mass distribution to predict which products of inertia vanish. Then calculate the inertia tensor and identify the principal axes and principal moments. Put the moments in increasing order.



\item For angular velocity
\[
  \boldsymbol\omega=\Omega(\mathbf e_1+\mathbf e_2),
\]
find the body-frame angular momentum \(\mathbf J\) and rotational kinetic energy. Are \(\mathbf J\) and \(\boldsymbol\omega\) parallel? State the condition under which they are parallel for a body with distinct principal moments.



\item The laboratory torque and angular momentum satisfy
\[
  \boldsymbol\tau_s=\left(\frac{d\mathbf J}{dt}\right)_s.
\]
For any vector, the laboratory and body derivatives are related by
\[
  \left(\frac{d\mathbf J}{dt}\right)_s
  =Q\left[\left(\frac{d\mathbf J}{dt}\right)_b
  +\boldsymbol\omega_b\times\mathbf J_b\right].
\]
Use \(\mathbf J_b=\mathsf I\boldsymbol\omega_b\) in principal axes to derive Euler's equations
\[
  \boxed{
  I_1\dot\omega_1+(I_3-I_2)\omega_2\omega_3=\tau_1,}
\]
with the other two equations obtained cyclically.



\item Set \(\boldsymbol\tau=0\) and suppose \(I_1<I_2<I_3\). Linearize Euler's equations about steady rotation with angular speed \(\Omega\) about each principal axis. Show that rotations about axes 1 and 3 are linearly stable, while rotation about axis 2 is unstable.



\item Now consider a torque-free symmetric top with
\[
  I_1=I_2=I_\perp,
  \qquad
  I_3=I_\parallel.
\]
Show that \(\omega_3\) is constant and that the transverse body components satisfy
\[
  \dot\omega_1=-\Omega_b\omega_2,
  \qquad
  \dot\omega_2=\Omega_b\omega_1,
\]
where
\[
  \boxed{\Omega_b=\frac{I_\parallel-I_\perp}{I_\perp}\,\omega_3.}
\]
Interpret the result in both the body frame and the laboratory frame.



\item \textbf{Oral extension for the designated presenter.} Demonstrate the intermediate-axis instability with a book or rectangular object. Identify which physical axes correspond to the smallest, intermediate, and largest principal moments, and connect the observed behavior to the signs in the linearized equations above. Explain why conservation of laboratory angular momentum does not imply that the body components \((J_1,J_2,J_3)\) remain constant.



\end{QA}
\end{question}



\end{document}