Due Friday, April 22.
In the 18.06 ODE lecture, we found that the position ($x$) and velocity ($v$) of a mass bouncing on a spring without friction are described by the equations:
$$ \frac{d}{dt} \underbrace{\begin{pmatrix} x \\ v \end{pmatrix}}_\vec{x} = \underbrace{\begin{pmatrix} 0 & 1 \\ -k/m & 0 \end{pmatrix}}_A \vec{x} $$for (real) spring constant $k > 0$ and mass $m > 0$.
(a) Show that by simply rescaling the variables $\vec{x} = (x,v)$ to new variables $\vec{y} = (a,b) = D \vec{x}$ for some diagonal matrix $D$, then you can get equations: $$ \frac{d\vec{y}}{dt} = B \vec{y} $$ where $B$ is (real) anti-Hermitian (also called skew-Hermitian): $B^H = -B$. (What is the formula relating $B$ and $A$ via $D$? $B$ and $A$ are ______ to one another.)
(b) For any anti-Hermitian $B = -B^H$, the matrix $iB$ is a ________ matrix because $(iB)^H = \_\_\_$. It follows that the eigenvalues of $B$ are purely ________ and hence the solutions $\vec{y}(t)$ of $d\vec{y}/dt = B \vec{y}$ are ________ (oscillating/growing/decaying).
(c) For any anti-Hermitian $B = -B^H$, the matrix $Q = e^{Bt}$ is unitary (for any real $t$) because ________. (Note: from the series definition of $e^A$, you can immediately see that $(e^A)^H = e^{A^H}$.)
(d) Recall from class that $\Vert Q \vec{y} \Vert = \Vert \vec{y} \Vert$ for any unitary $Q$ (or more generally any $Q$ with orthonormal columns). From this, derive that the solution $$\vec{y}(t) = e^{Bt} \vec{y}(0)$$ for your $B$ from part (a) conserves energy: $\frac{1}{2} mv^2 + \frac{1}{2} kx^2$ (kinetic + potential energy) is a constant as a function of time $t$.
(Based on Strang, section 6.3, problem 18.)
(a) Write five terms of the infinite series for $e^{At}$. Take the derivative $d/dt$, and show that you have four terms of $Ae^{At}$.
Conclusion: $\frac{d}{dt} e^{At} = Ae^{At}$ as claimed in class, and $x(t) = e^{At}x(0)$ solves $dx/dt = Ax$.
(b) Using the same five terms of the infinite series for $f(A) = e^A$ (no $t$), write down four terms of $f(A+dA) - f(A)$ as a linear operator $f'(A)[dA]$ acting on $dA$ as in Lecture 10 (i.e. drop all terms higher than first order in the "infinitesimal" change $dA$).
$f'(A)[dA]$ is not simply $e^A dA$ (as it would be for scalars in 18.01) except for very special perturbations $dA$ that ______________.
(a) Write down two familiar functions that solve the equation $d^2 y /dt^2 = -9y$. What is a solution $y(t)$ that starts with $y(0)=3$ and $y'(0)=0$?
(b) Find (by hand) the eigenvalues $\lambda_1$ and $\lambda_2$ and corresponding eigenvectors $x_1$ and $x_2$ of $$A = \begin{pmatrix} 1 & -5\\ 10 & -14\\ \end{pmatrix} .$$
(c) Consider the system of ODEs $\frac{d^2x}{dt^2} = Ax$ (note 2nd derivative!) with the $A$ from part (b). If $x(t) = c_1(t) x_1 + c_2(t) x_2$, what equations do $c_1(t)$ and $c_2(t)$ satisfy, and hence what solutions $c_1(t)$ and $c_2(t)$ are possible? (See part (a).)
(d) For the ODE of part (c), solve for $x(t)$ given the initial conditions $x(0) = (2,3)$ and $x'(0) = (1,4)$.
(e) Alternatively, if we let $y$ be the 4-component vector $y = (x, dx/dt)$, then $dy/dt = By$ for $B = \_\_\_\_$. Since this must ultimately give the same solutions $x(t)$ as part (c), what must be the eigenvalues of $B$?
In chemistry, the stoichiometry matrix is often used to describe a set of $m$ reactions among $n$ different chemical "species" (e.g. H₂O, C₈H₁₀N₄O₂, and so on).
For example, consider the following 3 (fictitious) chemical reactions among 4 species, labeled $x_1, x_2, x_3, x_4$:
$$ x_1 + 2x_2 \longleftrightarrow 3 x_2 + 2x_4, \;\;\\ 2x_2 + 4x_3 \longleftrightarrow x_1 + x_4, \;\;\\ x_1 + 4x_3 \longleftrightarrow 5x_4 $$which would be represented by the stoichiometry matrix
$$ S = \begin{pmatrix} -1 & 3-2 & 0 & 2 \\ 1 & -2 & -4 & 1 \\ -1 & 0 & -4 & 5 \end{pmatrix} $$whose rows are the reactions and whose columns are the species. (Some authors use the transpose of this matrix instead.)
If we use a vector $\vec{x} \in \mathbb{R}^4$ to represent the concentrations of each of these four species, and a vector $\vec{r} \in \mathbb{R}^3$ to represent the rates of each reaction, then the rate of change of the concentrations is given by the system of ordinary differential equations (ODEs):
$$ \frac{d\vec{x}}{dt} = S^T \vec{r} $$(where the rates $\vec{r}$ are not generally constant: they may depend on the concentrations $\vec{x}$ in a complicated way … so you can't solve this just by multiplying the right-hand side by $t$).
(a) Describe (find a basis for) all possible reaction rates $\vec{r}$ for which $\frac{d\vec{x}}{dt} = 0$ (the system is in steady state).
(b) Certain linear combinations of the concentrations are conserved: there are some (time-independent) vectors $\vec{v} \in \mathbb{R}^4$ for which $\frac{d(\vec{v}^T \vec{x})}{dt} = 0$ for all possible rate vectors $\vec{r}$. If $\vec{v}$ doesn't depend on $t$, then $\frac{d(\vec{v}^T \vec{x})}{dt}$ is ________ times $\frac{d\vec{x}}{dt}$. These vectors $\vec{v}$ all lie within the ___-dimensional ________ space of $S$. Why?