Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 56 insertions, 56 deletions

diff --git a/source/know/concept/harmonic-oscillator/index.md b/source/know/concept/harmonic-oscillator/index.md
index 6729dc2..6d9eac0 100644
--- a/source/know/concept/harmonic-oscillator/index.md
+++ b/source/know/concept/harmonic-oscillator/index.md
@@ -11,34 +11,34 @@ layout: "concept"
 A **harmonic oscillator** obeys
 the simple 1D version of [Hooke's law](/know/concept/hookes-law/):
 to displace the system away from its equilibrium,
-the needed force $F_d(x)$ scales linearly with the displacement $x(t)$:
+the needed force $$F_d(x)$$ scales linearly with the displacement $$x(t)$$:
 
 $$\begin{aligned}
     F_d(x) = k x
 \end{aligned}$$
 
-Where $k$ is a system-specific proportionality constant,
+Where $$k$$ is a system-specific proportionality constant,
 called the **spring constant**,
 since a spring is a good example of a harmonic oscillator,
 at least for small displacements.
 Hooke's law is also often stated for
-the restoring force $F_r(x)$ instead:
+the restoring force $$F_r(x)$$ instead:
 
 $$\begin{aligned}
     F_r(x) = - k x
 \end{aligned}$$
 
-Let a mass $m$ be attached to the end of the spring.
-After displacing it, we let it go $F_d = 0$,
-so Newton's second law for the restoring force $F_r$ demands that:
+Let a mass $$m$$ be attached to the end of the spring.
+After displacing it, we let it go $$F_d = 0$$,
+so Newton's second law for the restoring force $$F_r$$ demands that:
 
 $$\begin{aligned}
     F_r = m x''
 \end{aligned}$$
 
-But $F_r = - k x$,
-meaning $m x'' = - k x$,
-leading to the following equation for $x(t)$:
+But $$F_r = - k x$$,
+meaning $$m x'' = - k x$$,
+leading to the following equation for $$x(t)$$:
 
 $$\begin{aligned}
     \boxed{
@@ -46,7 +46,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $\omega_0 \equiv \sqrt{k / m}$ is the **natural frequency** of the system.
+Where $$\omega_0 \equiv \sqrt{k / m}$$ is the **natural frequency** of the system.
 This differential equation has the following general solution:
 
 $$\begin{aligned}
@@ -56,8 +56,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $C_1$ and $C_2$ are constants determined by the initial conditions.
-For example, for $x(0) = 1$ and $x'(0) = 0$, the solution becomes:
+Where $$C_1$$ and $$C_2$$ are constants determined by the initial conditions.
+For example, for $$x(0) = 1$$ and $$x'(0) = 0$$, the solution becomes:
 
 $$\begin{aligned}
     x(t) = \cos(\omega_0 t)
@@ -65,8 +65,8 @@ $$\begin{aligned}
 
 When using [Lagrangian](/know/concept/lagrangian-mechanics/)
 or Hamiltonian mechanics,
-we need to know the potential energy $V(x)$
-added to the system by a displacement to $x$.
+we need to know the potential energy $$V(x)$$
+added to the system by a displacement to $$x$$.
 This equals the work done by the displacement,
 and is therefore given by:
 
@@ -77,16 +77,16 @@ $$\begin{aligned}
 
 ## Damped oscillation
 
-If there is a **friction force** $F_f$ affecting the system,
+If there is a **friction force** $$F_f$$ affecting the system,
 then the oscillation amplitude will decrease,
 or it might not oscillate at all.
-We define $F_f$ using a **viscous damping coefficient** $c$:
+We define $$F_f$$ using a **viscous damping coefficient** $$c$$:
 
 $$\begin{aligned}
     F_f = - c x'
 \end{aligned}$$
 
-Both $F_r$ and $F_f$ are acting on the system,
+Both $$F_r$$ and $$F_f$$ are acting on the system,
 so Newton's second law states that:
 
 $$\begin{aligned}
@@ -94,7 +94,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This can be rewritten in the following conventional form
-by defining the **damping coefficient** $\zeta \equiv c / (2 \sqrt{m k})$,
+by defining the **damping coefficient** $$\zeta \equiv c / (2 \sqrt{m k})$$,
 which determines the expected behaviour of the system:
 
 $$\begin{aligned}
@@ -103,20 +103,20 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The general solution is found from the roots $u$ of the auxiliary quadratic equation:
+The general solution is found from the roots $$u$$ of the auxiliary quadratic equation:
 
 $$\begin{aligned}
     u^2 + 2 \zeta \omega_0 u + \omega_0^2 = 0
 \end{aligned}$$
 
-The discriminant $D = 4 \zeta^2 \omega_0^2 - 4 \omega_0^2$
+The discriminant $$D = 4 \zeta^2 \omega_0^2 - 4 \omega_0^2$$
 tells us that the behaviour changes substantially
-depending on the damping coefficient $\zeta$,
-with three possibilities: $\zeta < 1$ or $\zeta = 1$ or $\zeta > 1$.
+depending on the damping coefficient $$\zeta$$,
+with three possibilities: $$\zeta < 1$$ or $$\zeta = 1$$ or $$\zeta > 1$$.
 
-If $\zeta < 1$, there is **underdamping**:
+If $$\zeta < 1$$, there is **underdamping**:
 the system oscillates with exponentially decaying
-amplitude and reduced frequency $\omega_1 \equiv \omega_0 \sqrt{1 - \zeta^2}$.
+amplitude and reduced frequency $$\omega_1 \equiv \omega_0 \sqrt{1 - \zeta^2}$$.
 The general solution is:
 
 $$\begin{aligned}
@@ -126,7 +126,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-If $\zeta = 1$, there is **critical damping**:
+If $$\zeta = 1$$, there is **critical damping**:
 the system returns to its equilibrium point in minimum time.
 The general solution is given by:
 
@@ -137,10 +137,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-If $\zeta > 1$, there is **overdamping**:
+If $$\zeta > 1$$, there is **overdamping**:
 the system returns to equilibrium slowly.
 The general solution is as follows,
-where $\omega_1 \equiv \omega_0 \sqrt{\zeta^2 - 1}$:
+where $$\omega_1 \equiv \omega_0 \sqrt{\zeta^2 - 1}$$:
 
 $$\begin{aligned}
     \boxed{
@@ -161,9 +161,9 @@ $$\begin{aligned}
     x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = f(t)
 \end{aligned}$$
 
-Obviously, there exist infinitely many $f(t)$ to choose from,
+Obviously, there exist infinitely many $$f(t)$$ to choose from,
 and each needs a separate analysis.
-However, there is one type of $f(t)$ that deserves special mention,
+However, there is one type of $$f(t)$$ that deserves special mention,
 namely sinusoids:
 
 $$\begin{aligned}
@@ -172,25 +172,25 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $F$ is a constant force, $\chi$ is an arbitrary phase,
-and the frequency $\omega$ is not necessarily $\omega_0$.
-We solve this case for $x(t)$ in detail.
+Where $$F$$ is a constant force, $$\chi$$ is an arbitrary phase,
+and the frequency $$\omega$$ is not necessarily $$\omega_0$$.
+We solve this case for $$x(t)$$ in detail.
 Consider the complex version of the equation:
 
 $$\begin{aligned}
     X'' + 2 \zeta \omega_0 X' + \omega_0^2 X = \frac{F}{m} \exp\!\big(i (\omega t + \chi)\big)
 \end{aligned}$$
 
-Then $x(t) = \Real\{X(t)\}$.
-Inserting the ansatz $X(t) = C \exp(i \omega t)$,
-for some constant $C$:
+Then $$x(t) = \Real\{X(t)\}$$.
+Inserting the ansatz $$X(t) = C \exp(i \omega t)$$,
+for some constant $$C$$:
 
 $$\begin{aligned}
     - C \omega^2 + C 2 i \zeta \omega_0 \omega + C \omega_0^2 = \frac{F}{m} \exp(i \chi)
 \end{aligned}$$
 
-Where $\exp(i \omega t)$ has already been divided out.
-We isolate this equation for $C$:
+Where $$\exp(i \omega t)$$ has already been divided out.
+We isolate this equation for $$C$$:
 
 $$\begin{aligned}
     C
@@ -200,7 +200,7 @@ $$\begin{aligned}
     \exp(i \chi)
 \end{aligned}$$
 
-We would like to rewrite this in polar form $C = r \exp(i \theta)$,
+We would like to rewrite this in polar form $$C = r \exp(i \theta)$$,
 which turns out to be as follows:
 
 $$\begin{aligned}
@@ -209,8 +209,8 @@ $$\begin{aligned}
     \exp\!\bigg(i \chi - i \arctan\!\Big(\frac{2 \zeta \omega_0 \omega}{\omega_0^2 - \omega^2}\Big)\bigg)
 \end{aligned}$$
 
-For brevity, let us define the **impedance** $Z$
-and the **phase shift** $\phi$
+For brevity, let us define the **impedance** $$Z$$
+and the **phase shift** $$\phi$$
 in the following way:
 
 $$\begin{aligned}
@@ -221,8 +221,8 @@ $$\begin{aligned}
     \equiv \arctan\!\Big(\frac{2 \zeta \omega_0 \omega}{\omega_0^2 - \omega^2}\Big)
 \end{aligned}$$
 
-Returning to the original ansatz $X(t) = C \exp(i \omega t)$,
-we take its real part to find $x(t)$:
+Returning to the original ansatz $$X(t) = C \exp(i \omega t)$$,
+we take its real part to find $$x(t)$$:
 
 $$\begin{aligned}
     \boxed{
@@ -232,18 +232,18 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Two things are noteworthy here.
-Firstly, $f(t)$ and $x(t)$ are out of phase by $\phi$; there is some lag.
-This is caused by damping, because if $\zeta = 0$, it disappears $\phi = 0$.
+Firstly, $$f(t)$$ and $$x(t)$$ are out of phase by $$\phi$$; there is some lag.
+This is caused by damping, because if $$\zeta = 0$$, it disappears $$\phi = 0$$.
 
-Secondly, the amplitude of $x(t)$ depends on $\omega$ and $\omega_0$.
+Secondly, the amplitude of $$x(t)$$ depends on $$\omega$$ and $$\omega_0$$.
 This brings us to **resonance**,
 where the amplitude can become extremely large.
 Actually, resonance has two subtly different definitions,
-depending on which one of $\omega$ and $\omega_0$ is a free parameter,
+depending on which one of $$\omega$$ and $$\omega_0$$ is a free parameter,
 and which one is fixed.
 
-If the natural $\omega_0$ is fixed and the driving $\omega$ is variable,
-we find for which $\omega$ resonance occurs by minimizing the amplitude denominator $\omega Z$.
+If the natural $$\omega_0$$ is fixed and the driving $$\omega$$ is variable,
+we find for which $$\omega$$ resonance occurs by minimizing the amplitude denominator $$\omega Z$$.
 We thus find:
 
 $$\begin{aligned}
@@ -256,13 +256,13 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Meaning the resonant $\omega$ is lower than $\omega_0$,
-and resonance can only occur if $\zeta < 1 / \sqrt{2}$.
+Meaning the resonant $$\omega$$ is lower than $$\omega_0$$,
+and resonance can only occur if $$\zeta < 1 / \sqrt{2}$$.
 
-However, if the driving $\omega$ is fixed and the natural is $\omega_0$ is variable,
+However, if the driving $$\omega$$ is fixed and the natural is $$\omega_0$$ is variable,
 the problem is bit more subtle:
-the damping coefficient $\zeta = c / (2 m \omega_0)$
-depends on $\omega_0$.
+the damping coefficient $$\zeta = c / (2 m \omega_0)$$
+depends on $$\omega_0$$.
 This leads us to:
 
 $$\begin{aligned}
@@ -275,10 +275,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Surprisingly, the damping does not affect $\omega_0$, if $\omega$ is given.
+Surprisingly, the damping does not affect $$\omega_0$$, if $$\omega$$ is given.
 However, in both cases, the damping *does* matter for the eventual amplitude:
-$c \to 0$ leads to $x \to \infty$,
-and resonance disappears or becomes negligible for $c \to \infty$.
+$$c \to 0$$ leads to $$x \to \infty$$,
+and resonance disappears or becomes negligible for $$c \to \infty$$.