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\begin{document}

\begin{center}
  {\Large\bfseries Problem Set 2}\\[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.  Computation may be
used where requested, but the purpose is to expose the mechanical structure, not just to obtain a
numerical answer.  If vector or matrix differentiation feels rusty, see the last question. 

\bigskip

% =====================================================================
% FLATTENED QUESTION: QABank/Q03_01.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q03_01
% Title: Triangle vectors and geometric structure
% Status: ready
% Introduced: 3
% BestFit: Newtonian mechanics; vectors
% Topics: vector spaces; inner products; cross products; wedge products; triangle geometry
% Type: guided conceptual derivation
% Difficulty: introductory / medium
% Computational: no
% Source: adapted from Physics 351 PSet 1
% Tags: vectors; inner product; law of cosines; area; bivector; geometric algebra
% Notes: 
% ============================================================

\begin{question}
Let \(\vect{u}\) and \(\vect{v}\) be two nonparallel vectors with a common tail, and
let \(\vect{w}\) point from the head of \(\vect{u}\) to the head of \(\vect{v}\), as
shown schematically below.

\begin{center}
\begin{tikzpicture}[scale=1.0,>=Stealth]
  \coordinate (O) at (0,0);
  \coordinate (U) at (3.4,1.0);
  \coordinate (V) at (0.8,3.0);
  \draw[->,line width=1.2pt] (O) -- (U)
    node[midway,below right] {$\vect{u}$};
  \draw[->,line width=1.2pt] (O) -- (V)
    node[midway,left] {$\vect{v}$};
  \draw[->,line width=1.2pt] (U) -- (V)
    node[midway,above right] {$\vect{w}$};
  \fill (O) circle (1.5pt);
\end{tikzpicture}
\end{center}

This problem deliberately builds the geometry in stages.  At each stage, pay attention
to what additional mathematical structure is being used.

\begin{QA}

\item Use only vector addition and scalar multiplication to express \(\vect{w}\) in
terms of \(\vect{u}\) and \(\vect{v}\).  What mathematical structure is required for
this statement?



\item Now suppose the vector space is equipped with an inner product.  Define
\[
  u^2=\vect{u}\cdot\vect{u},\qquad
  v^2=\vect{v}\cdot\vect{v},\qquad
  w^2=\vect{w}\cdot\vect{w}.
\]
Use part (a) to derive a relation among \(u^2\), \(v^2\), \(w^2\), and
\(\vect{u}\cdot\vect{v}\).  Then use
\[
  \vect{u}\cdot\vect{v}=uv\cos\theta
\]
to recover the Law of Cosines, where \(\theta\) is the angle between
\(\vect{u}\) and \(\vect{v}\).  What new structure has been introduced compared
with part (a)?



\item Now specialize to three-dimensional Euclidean space.  Show that the area
\(\mathcal A\) of the triangle is
\[
  \mathcal A=\frac12\lvert\vect{u}\times\vect{v}\rvert.
\]
Then use \(\vect{w}=\vect{v}-\vect{u}\) to prove algebraically that
\[
  \vect{u}\times\vect{w}=\vect{u}\times\vect{v},
  \qquad
  \vect{w}\times\vect{v}=-\vect{u}\times\vect{v}.
\]
Explain why all three cross products give the same (unsigned) triangle area even
though one of them has the opposite orientation.



\item A more geometric way to represent an oriented plane element is with the wedge
product \(\vect{u}\wedge\vect{v}\).  Use \(\vect{w}=\vect{v}-\vect{u}\) to show
\[
  \vect{u}\wedge\vect{w}=\vect{u}\wedge\vect{v},
  \qquad
  \vect{w}\wedge\vect{v}=-\vect{u}\wedge\vect{v}.
\]
What geometric information does the bivector \(\vect{u}\wedge\vect{v}\) represent?
Why might it be regarded as more fundamental than the cross product for describing
oriented area?



\end{QA}
\end{question}


% =====================================================================
% FLATTENED QUESTION: QABank/Q03_03.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q03_03
% Title: Prescribed motion and inverse dynamics
% Status: ready
% Introduced: 3
% BestFit: Newtonian mechanics
% Topics: prescribed trajectories; state; kinematics; inverse dynamics; work-energy; constraints
% Type: guided conceptual / calculation
% Difficulty: medium
% Computational: optional
% Source: adapted from Physics 351 PSet 1
% Tags: trajectory; state; velocity; acceleration; power; ideal constraint; roller coaster
% Notes: 
% ============================================================

\begin{question}
A roller coaster of mass \(M\) is prescribed to follow the trajectory
\[
  \vect{x}(t)
  =
  (1\,\mathrm m)
  \left[
    \tau\,\hat{\mathbf x}
    +\frac{\tau^2(\tau-10)(\tau-18)}{1260}\,\hat{\mathbf z}
  \right],
  \qquad
  \tau\equiv\frac{t}{1\,\mathrm s},
\]
for \(-5\leq \tau\leq 20\).  The \(x\)-axis is horizontal and the \(z\)-axis
points upward.  Take the gravitational acceleration to be
\[
  g_0=9.8\,\mathrm{m/s^2}.
\]

The word \emph{prescribed} is important: the function \(\vect{x}(t)\) gives a
particular history of the system.  In this problem you will ask what can, and what
cannot, be inferred about the dynamics that produced that history.

Define
\[
  f(\tau)=\frac{\tau^2(\tau-10)(\tau-18)}{1260},
\]
so that the height is \(h(t)=(1\,\mathrm m)f(\tau)\).

\begin{QA}

\item Is the specified function \(\vect{x}(t)\) itself a dynamical law?  What does
it determine about the net force along this particular history, and what does it
\emph{not} determine about the force law for arbitrary states or initial
conditions?



\item Find the velocity \(\vect{v}(t)\) and acceleration \(\vect{a}(t)\).  Write the
corresponding Newtonian state \(S(t)\).



\item Find all stationary points of the height \(h(t)\) in the interior of the
time interval and classify them as local maxima or minima.  Then determine the
global maximum and minimum heights on the full interval \(-5\leq\tau\leq20\).
Be careful about the endpoints.



\item At each interior stationary height, is the coaster instantaneously at rest?
Must its acceleration vanish there?  What is different about the global maximum at
\(t=-5\,\mathrm s\)?



\item Define the mechanical energy associated with kinetic energy and gravitational
potential energy by
\[
  E(t)=\frac12 M\,\vect{v}^{\,2}(t)+Mg_0h(t).
\]
Is \(E\) conserved along the prescribed trajectory?  Give a simple demonstration
without needing to analyze the entire function \(E(t)\).



\item Suppose the Newtonian equation of motion is written as
\[
  M\vect{a}=-Mg_0\hat{\mathbf z}+\vect{F}_{\rm other},
\]
where \(\vect{F}_{\rm other}\) represents everything other than gravity.  Show that
\[
  \frac{dE}{dt}=\vect{F}_{\rm other}\cdot\vect{v}.
\]
For the prescribed trajectory, express \(dE/dt\) in terms of \(h\), and verify at
\(t=3\,\mathrm s\) that it is nonzero.  Interpret the result physically.



\item Could gravity plus the normal force from an ideal, fixed, frictionless track
produce this prescribed motion by themselves?  Explain.  What kinds of additional
physics could account for the prescribed trajectory?



\end{QA}
\end{question}


% =====================================================================
% FLATTENED QUESTION: QABank/Q03_05.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q03_05
% Title: Two-body spring: forces and center-of-mass coordinates
% Status: ready
% Introduced: 3
% BestFit: Newtonian mechanics; two-body motion
% Topics: pair potentials; Newton third law; center of mass; relative coordinates; symmetry
% Type: guided conceptual derivation
% Difficulty: medium
% Computational: no
% Source: adapted from Physics 351 PSet 2
% Tags: two-body; spring; translation invariance; rotational invariance; central force; center of mass
% Notes: Q03_S01 provides optional background for the vector differentiation used here.
% ============================================================

\begin{question}
Two particles of masses $m_1$ and $m_2$ move in the $x$--$y$ plane.  Their
positions are $\mathbf{x}_1(t)$ and $\mathbf{x}_2(t)$.  They interact through a
spring whose potential energy depends only on the distance between the
particles,
\[
  V(l)=\frac12 K(l-l_0)^2,
\]
where
\[
  \mathbf l=\mathbf x_2-\mathbf x_1,
  \qquad
  l=\lVert\mathbf l\rVert,
  \qquad
  \hat{\mathbf l}=\frac{\mathbf l}{l}.
\]
Assume $l\neq0$ throughout the problem and that there are no external forces.

\begin{QA}

\item Consider a common translation of both particles,
\[
  \mathbf x_1\longmapsto \mathbf x_1+\mathbf a,
  \qquad
  \mathbf x_2\longmapsto \mathbf x_2+\mathbf a,
\]
where $\mathbf a$ is constant.  Show that $\mathbf l$, $l$, and $V$ are
unchanged.  What physical feature of the interaction does this make explicit?



\item Using
\[
  \nabla_{\mathbf x_1}l=-\hat{\mathbf l},
  \qquad
  \nabla_{\mathbf x_2}l=+\hat{\mathbf l},
\]
(which you may derive directly or take from the supplemental background
problem), find the force on each particle from
\[
  \mathbf F_i=-\nabla_{\mathbf x_i}V.
\]
Check the directions of the forces for both a stretched spring ($l>l_0$) and a
compressed spring ($l<l_0$).



\item Analyze Newton's third law for this interaction.

\begin{QA}
\item Show that the two forces obey the equal-and-opposite form of the third
law.



\item Show that the forces also obey the strong form of the third law: they
act along the line joining the particles.  Use this to show that the total
internal torque about any origin vanishes.



\item Which assumption is responsible for the equal-and-opposite property,
and which stronger assumption is responsible for the line-of-centers
property?


\end{QA}

\item Define the total mass and center-of-mass position by
\[
  M=m_1+m_2,
  \qquad
  \mathbf X=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{M}.
\]
Use Newton's second law and the result above to find $\ddot{\mathbf X}$.  What
does the result say about the motion of the center of mass?



\item Replace the two particle coordinates $(\mathbf x_1,\mathbf x_2)$ by the
center-of-mass and relative coordinates
\[
  \mathbf X=\frac{m_1\mathbf x_1+m_2\mathbf x_2}{M},
  \qquad
  \mathbf l=\mathbf x_2-\mathbf x_1.
\]
Find the inverse transformation: express $\mathbf x_1$ and $\mathbf x_2$ in
terms of $\mathbf X$ and $\mathbf l$.



\item Rewrite the potential $V$ in the new coordinates $(\mathbf X,\mathbf l)$.
Why is the absence of $\mathbf X$ important?  Explain, without doing the full
kinetic-energy calculation yet, why these coordinates are promising for
separating the overall motion of the system from its internal motion.



\end{QA}
\end{question}


% =====================================================================
% FLATTENED QUESTION: QABank/Q03_09.tex
% =====================================================================
\BeginQuestion[label=Oral Presentation]
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q03_09
% Title: Work, exactness, and conservative forces
% Status: ready
% Introduced: 3
% BestFit: Newtonian mechanics; work and energy
% Topics: work-energy theorem; exact differentials; conservative forces; potential energy; path independence
% Type: guided conceptual derivation
% Difficulty: medium
% Computational: no
% Source: original
% Tags: power; work; exactness; potential; gradient; conservative force; circulation
% Notes: Develops potential energy from exactness of the work form rather than positing F=-grad V.
% ============================================================

\begin{question}
Let a particle of constant mass $m$ move in the plane with position $\mathbf{x}(t)$,
velocity $\mathbf{v}=\dot{\mathbf{x}}$, and force $\mathbf{F}(\mathbf{x})$.  In this
problem we will treat potential energy as something to be \emph{discovered} from the
work done by the force.

\begin{QA}
\item Starting from Newton's law
\[
  m\dot{\mathbf{v}}=\mathbf{F},
\]
take the dot product with $\mathbf{v}$ and show that
\[
  \frac{d}{dt}\left(\frac12 m\,\mathbf{v}\cdot\mathbf{v}\right)
  =\mathbf{F}\cdot\mathbf{v}.
\]
Explain why the right-hand side is naturally interpreted as instantaneous power.



\item Along an infinitesimal displacement $d\mathbf{x}$ the work done by the force is
\[
  \delta W=\mathbf{F}\cdot d\mathbf{x}.
\]
Suppose there exists a scalar function $V(\mathbf{x})$ for which
\[
  \mathbf{F}\cdot d\mathbf{x}=-dV
\]
for every infinitesimal displacement.  Use
\[
  dV=(\nabla V)^T d\mathbf{x}
\]
to show that
\[
  \mathbf{F}=-\nabla V.
\]
Then show directly that $E=T+V$ is conserved.



\item Consider the force field
\[
  \mathbf{F}(x,y)=-(2x+y)\,\hat{\mathbf{x}}-(x+2y)\,\hat{\mathbf{y}}.
\]
Find a potential $V(x,y)$ such that $\mathbf{F}=-\nabla V$.  Check that the mixed
partial-derivative condition expected for a conservative field is satisfied.



\item Now consider
\[
  \mathbf{G}(x,y)=-y\,\hat{\mathbf{x}}+x\,\hat{\mathbf{y}}.
\]
Show that it cannot be written globally as $-\nabla V$ in the plane.  Then calculate the
work around the unit circle, oriented counterclockwise, using
\[
 x=\cos\theta,\qquad y=\sin\theta,\qquad 0\leq\theta\leq2\pi.
\]
How does this result distinguish $\mathbf G$ from the field in part (c)?



\item Summarize the logical chain connecting Newton's law, exact work, and energy conservation.
In particular, distinguish the statements ``$\mathbf F\cdot d\mathbf x$ is exact'' and
``$\mathbf F$ does no work.''



\end{QA}
\end{question}


% =====================================================================
% FLATTENED QUESTION: QABank/Q04_01.tex
% =====================================================================
\BeginQuestion
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q04_01
% Title: Discrete Newtonian updates: two step--turn conventions, clock, and mass
% Status: ready
% Introduced: 4
% BestFit: discrete Newtonian dynamics
% Topics: ordered configurations; first differences; second differences; state update; alternative update rules; clock; acceleration; mass; spring map; phase space; discretization error
% Type: guided derivation / computational
% Difficulty: medium
% Computational: yes
% Source: original
% Tags: step; turn; state map; Newton first law; constant acceleration; spring; phase space; MATLAB; symplectic Euler; Planck time; discretization comparison
% Notes: First half reproduces the Lecture 4 update. Second half compares the complementary turn-then-move convention. K is acceleration-like, not force-like.
% ============================================================

\begin{question}
Consider an ordered sequence of configurations $x_0,x_1,x_2,\ldots$ in one dimension.
Before assigning a physical clock, define the \emph{step}
\[
  s_n=x_{n+1}-x_n
\]
and the \emph{turn}
\[
  \kappa_n=s_n-s_{n-1},\qquad n\geq1.
\]
The word ``turn'' means any change in the step: it may change its magnitude, its direction,
or both.

\begin{QA}
\item Show that
\[
  s_n=s_0+\sum_{j=1}^{n}\kappa_j
\]
and then show that
\[
  x_n=x_0+n s_0+\sum_{j=1}^{n-1}(n-j)\kappa_j.
\]
Write the second expression first as a double sum and then reduce it to the single weighted sum.
Explain the analogy with integrating acceleration once to obtain velocity and twice to obtain position.



\item In more than one dimension, decompose a turn $\boldsymbol\kappa$ into a component
parallel to the current step $\mathbf s$ and a component perpendicular to it.  What qualitative
effect does each component have?  What special cases correspond to a turn angle of $0$ or $180^\circ$?



\item Let the state be $S_n=(x_n,s_n)$ and consider the semi-explicit update
\[
\boxed{
\begin{aligned}
 x_{n+1}&=x_n+s_n,\\
 s_{n+1}&=s_n+K(x_{n+1},s_n,t_n).
\end{aligned}}
\]
For $K=0$, derive $x_n$.  For constant $K=K_0$, derive both $s_n$ and $x_n$.



\item Now assign equally spaced clock times $t_n=n\Delta$.  Define
\[
  v_n=\frac{s_n}{\Delta},
  \qquad
  a_n\approx\frac{\kappa_n}{\Delta^2}.
\]
Show that Newton's law gives the discrete turn rule
\[
  \kappa\approx \Delta^2 M^{-1}F.
\]
For one particle, apply this result to:
\begin{QA}
  \item uniform gravity, $F=mg$;
  \item a spring, $F=-kx$.
\end{QA}
In the spring case identify the dimensionless parameter $c$ in $K=-cx$ and express it in terms of
$\omega=\sqrt{k/m}$ and $\Delta$.



\item For the spring-like rule $K(x_{n+1})=-c x_{n+1}$, show that
\[
\begin{pmatrix}x_{n+1}\\s_{n+1}\end{pmatrix}
=
\begin{pmatrix}
1&1\\
-c&1-c
\end{pmatrix}
\begin{pmatrix}x_n\\s_n\end{pmatrix}.
\]
Calculate the determinant.  What does the determinant say about area in the $(x,s)$ plane?
Then eliminate $s_n$ and obtain a second-order recurrence for $x_n$.



\item Implement the update in MATLAB for $N=40$ steps.  Make two plots for each case:
$x_n$ versus $n$, and $s_n$ versus $x_n$.
\begin{QA}
  \item $K=0$, with $x_0=0$, $s_0=1$;
  \item $K=-1$, with $x_0=0$, $s_0=6$;
  \item $K=-c x_{n+1}$, with $x_0=1$, $s_0=0$, $c=0.25$.
\end{QA}
Briefly describe what structure you see in each case.



\item Now explore a second update convention.  In the rule used above, $s_n$ is the
\emph{outgoing} step from $x_n$ to $x_{n+1}$: first move with $s_n$, then compute the turn
at the new point.  For this part only, shift the step index by one and reinterpret $s_n$ as the \emph{incoming} step that brought the
system to $x_n$,
\[
  s_n=x_n-x_{n-1}.
\]
At $x_n$, first apply the turn and then take the new step:
\[
\boxed{
\begin{aligned}
 s_{n+1}&=s_n+K(x_n,s_n,t_n),\\
 x_{n+1}&=x_n+s_{n+1}.
\end{aligned}}
\]
Draw a vector picture of one update for each convention.  In each picture label $x_n$,
$x_{n+1}$, the old step, the turn $K$, and the new step.  Explain in words where the turn is
applied in the two rules.



\item For the alternative rule, derive $s_n$ and $x_n$ for:
\begin{QA}
  \item $K=0$;
  \item constant $K=K_0$.
\end{QA}
Compare the constant-turn result with the original rule.  Specialize to $K_0=-1$.



\item Apply the alternative rule to the spring, now using
\[
 K(x_n)=-c x_n.
\]
Show that
\[
\begin{pmatrix}x_{n+1}\\s_{n+1}\end{pmatrix}
=
\begin{pmatrix}
1-c&1\\
-c&1
\end{pmatrix}
\begin{pmatrix}x_n\\s_n\end{pmatrix}.
\]
Compute the determinant and eliminate $s_n$ to obtain a second-order recurrence for $x_n$.
Compare the recurrence with the one from the original update rule.

Then modify your MATLAB code so that it implements \emph{both} update rules for the same
initial values.  For $c=0.25$, $x_0=1$, and $s_0=0$, plot both trajectories in the $(x,s)$
plane on the same axes.  What changes, and what does not?

\textbf{Optional analytic check.}  Verify directly that the original spring map preserves
\[
 I_A=s^2+cxs+cx^2,
\]
while the alternative map preserves
\[
 I_B=s^2-cxs+cx^2.
\]
Use these expressions to explain the opposite tilts of the phase-space curves.



\item Finally, use the two constant-acceleration updates as a scale estimate.  Let
\[
 K=a\Delta^2,\qquad s_0=v_0\Delta,\qquad T=n\Delta.
\]
Show that at the same elapsed time $T$ the two schemes predict
\[
 x_A(T)=x_0+v_0T+\frac12a\bigl(T^2-T\Delta\bigr),
\]
and
\[
 x_B(T)=x_0+v_0T+\frac12a\bigl(T^2+T\Delta\bigr).
\]
Hence find their separation $|x_B-x_A|$.  How does each compare with the exact continuum
constant-acceleration result?

As a thought experiment, take $\Delta$ to be the Planck time,
\[
 t_P\approx5.39\times10^{-44}\ \mathrm{s},
\]
and use $a=g=9.81\ \mathrm{m/s^2}$ for $T=1\ \mathrm{s}$.  Estimate the difference between
the two discrete predictions.  For scale, the Planck length is approximately
$\ell_P=1.62\times10^{-35}\ \mathrm{m}$.  What fraction of a Planck length is the difference?

\emph{Interpret this only as a discretization scale estimate, not as evidence that physical time
is discrete or that either update rule is fundamental.}



\end{QA}
\end{question}


% =====================================================================
% FLATTENED QUESTION: QABank/Q04_02.tex
% =====================================================================
\BeginQuestion[label=Challenge]
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q04_02
% Title: From stationary points to stationary paths
% Status: ready
% Introduced: 4
% BestFit: variation; motivation for the action principle
% Topics: stationary points; variations; gradients; saddles; functionals; shortest paths; Fermat principle
% Type: guided conceptual derivation
% Difficulty: medium / advanced
% Computational: no
% Source: original
% Tags: stationary; variation; saddle; functional; shortest path; Fermat; calculus of variations
% Notes: Stops before introducing a Lagrangian or deriving the Euler-Lagrange equation.
% ============================================================

\begin{question}
Before varying an entire trajectory, it is useful to remember what ``stationary'' means for an
ordinary function.  A stationary point need not be a minimum.

\begin{QA}
\item Consider
\[
  V(x)=x^4-2x^2.
\]
Find all stationary points and classify them.  Why would it be misleading to say that the condition
$dV/dx=0$ always finds a minimum?



\item Now consider the two-variable function
\[
  V(x,y)=x^2-y^2.
\]
Let $\mathbf q=(x,y)^T$ and $\delta\mathbf q=(\delta x,\delta y)^T$.
Show that
\[
  \delta V=(\nabla V)^T\delta\mathbf q.
\]
Show that the origin is stationary under every infinitesimal variation but is neither a maximum nor
a minimum.  Explain how the two coordinate directions reveal the saddle.



\item The length of a plane curve written as $x=x(y)$ between fixed endpoint values
$(x_1,y_1)$ and $(x_2,y_2)$ is
\[
  \ell[x]=\int_{y_1}^{y_2}
  \sqrt{1+\left(\frac{dx}{dy}\right)^2}\,dy.
\]
What is being varied in this problem: a number, a point, or a whole function?  If
$x(y)\mapsto x(y)+\epsilon\eta(y)$, what endpoint conditions should $\eta(y)$ satisfy when the
endpoints are fixed?  Why is the notation $\delta\ell=0$ the natural analog of $dV/dx=0$?



\item Suppose the local speed of light in a medium is $u(x,y)$.  The travel time for the same
curve can be written
\[
  T[x]=\int_{y_1}^{y_2}
  \frac{\sqrt{1+(dx/dy)^2}}{u(x,y)}\,dy.
\]
Identify the function $f(x,x',y)$ in the generic functional
\[
  \mathcal J[x]=\int f(x,x',y)\,dy,
  \qquad x'=\frac{dx}{dy}.
\]
What happens when $u$ is constant?  Why is ``stationary travel time'' more precise than saying
that light always follows a minimum of $T[x]$?



\item Compare the three levels
\[
  \frac{dV}{dx}=0,
  \qquad
  \delta V=(\nabla V)^T\delta\mathbf q=0,
  \qquad
  \delta\mathcal J[x]=0.
\]
What is being varied in each case, and why does the last condition suggest that a global-looking
integral can generate a local differential equation along a path?



\end{QA}
\end{question}



\noindent\rule{\textwidth}{0.4pt}
\medskip
If vector or matrix differentiation feels rusty, this question may be useful.  You do not need to turn in background questions.
% =====================================================================
% FLATTENED QUESTION: QABank/Q03_S01.tex
% =====================================================================
\BeginQuestion[label=Background]
% ============================================================
% QABank Metadata
% Schema: 1
% ID: Q03_S01
% Title: Background check: vector and matrix differentiation
% Status: ready
% Introduced: 3
% BestFit: mathematical background
% Topics: gradients; vector differentiation; quadratic forms; Hessians; composed quadratic energies
% Type: supplemental / background check
% Difficulty: introductory / medium
% Computational: no
% Source: adapted from Physics 351 PSet 2
% Tags: gradient; matrix calculus; quadratic form; Hessian; ATCA; norm derivative
% Notes: Supplemental background for force-from-potential and matrix-energy calculations.
% ============================================================

\begin{question}
Throughout this problem, vectors are column vectors.  For a differentiable
scalar function $f(\mathbf{x})$, define the gradient by
\[
  df=(\nabla_{\mathbf{x}}f)^T d\mathbf{x}.
\]
Thus the gradient is the vector that represents the first-order change in
$f$ under an arbitrary displacement $d\mathbf{x}$.

This problem is a background check on vector and matrix differentiation.  The
results will be used repeatedly in mechanics, especially when forces are
obtained from potentials.

\begin{QA}

\item Let $\mathbf{a}\in\mathbb{R}^n$ be constant and
\[
  f(\mathbf{x})=\mathbf{a}^T\mathbf{x}.
\]
Find $\nabla_{\mathbf{x}}f$.



\item Let
\[
  f(\mathbf{x})=\mathbf{x}^T\mathbf{x}=\lVert\mathbf{x}\rVert^2.
\]
Show that
\[
  \nabla_{\mathbf{x}}f=2\mathbf{x}.
\]



\item Let $B$ be a constant $n\times n$ matrix and
\[
  f(\mathbf{x})=\frac12\mathbf{x}^T B\mathbf{x}.
\]
Do not assume that $B$ is symmetric.

\begin{QA}
\item Show that
\[
  \nabla_{\mathbf{x}}f
  =\frac12(B+B^T)\mathbf{x}.
\]



\item What does the result become when $B=B^T$?  What is the Hessian
$H_{ij}=\partial^2 f/(\partial x_i\partial x_j)$ in that case?


\end{QA}

\item Let $A$ be an $m\times n$ matrix, $\mathbf{b}\in\mathbb{R}^m$, and let
$C=C^T$ be an $m\times m$ matrix.  Define
\[
  f(\mathbf{x})
  =\frac12(A\mathbf{x}-\mathbf{b})^T
  C(A\mathbf{x}-\mathbf{b}).
\]
Show that
\[
  \boxed{\nabla_{\mathbf{x}}f
  =A^T C(A\mathbf{x}-\mathbf{b}).}
\]



\item Apply the previous result to the discrete spring energy
\[
  U(\mathbf{u})=\frac12(A\mathbf{u})^T C(A\mathbf{u}),
\]
where $A$ is an incidence/difference matrix and $C=C^T$ is the constitutive
matrix.

\begin{QA}
\item Find $\nabla_{\mathbf{u}}U$ and the internal restoring force
$\mathbf{F}_{\mathrm{int}}$.



\item What is the Hessian of $U$?


\end{QA}

\item Finally, let
\[
  \mathbf{l}=\mathbf{x}_2-\mathbf{x}_1,
  \qquad
  l=\lVert\mathbf{l}\rVert,
  \qquad l\neq0.
\]
Use the differential definition of the gradient to show that
\[
  \boxed{\nabla_{\mathbf{x}_1}l=-\hat{\mathbf l},
  \qquad
  \nabla_{\mathbf{x}_2}l=+\hat{\mathbf l},}
\]
where $\hat{\mathbf l}=\mathbf l/l$.



\end{QA}
\end{question}



\end{document}