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diff --git a/source/know/concept/hamiltonian-mechanics/index.md b/source/know/concept/hamiltonian-mechanics/index.md
index 466a78f..19e55b0 100644
--- a/source/know/concept/hamiltonian-mechanics/index.md
+++ b/source/know/concept/hamiltonian-mechanics/index.md
@@ -17,12 +17,12 @@ which is in turn built on [variational calculus](/know/concept/calculus-of-varia
 
 ## Definitions
 
-In Lagrangian mechanics, use a Lagrangian $L$,
-which depends on position $q(t)$ and velocity $\dot{q}(t)$,
-to define the momentum $p(t)$ as a derived quantity.
-Hamiltonian mechanics switches the roles of $\dot{q}$ and $p$:
-the **Hamiltonian** $H$ is a function of $q$ and $p$,
-and the velocity $\dot{q}$ is derived from it:
+In Lagrangian mechanics, use a Lagrangian $$L$$,
+which depends on position $$q(t)$$ and velocity $$\dot{q}(t)$$,
+to define the momentum $$p(t)$$ as a derived quantity.
+Hamiltonian mechanics switches the roles of $$\dot{q}$$ and $$p$$:
+the **Hamiltonian** $$H$$ is a function of $$q$$ and $$p$$,
+and the velocity $$\dot{q}$$ is derived from it:
 
 $$\begin{aligned}
     \pdv{L(q, \dot{q})}{\dot{q}} = p
@@ -32,9 +32,9 @@ $$\begin{aligned}
 
 Conveniently, this switch turns out to be
 [Legendre transformation](/know/concept/legendre-transform/):
-$H$ is the Legendre transform of $L$,
-with $p = \partial L / \partial \dot{q}$ taken as
-the coordinate to replace $\dot{q}$.
+$$H$$ is the Legendre transform of $$L$$,
+with $$p = \partial L / \partial \dot{q}$$ taken as
+the coordinate to replace $$\dot{q}$$.
 Therefore:
 
 $$\begin{aligned}
@@ -44,15 +44,15 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This almost always works,
-because $L$ is usually a second-order polynomial of $\dot{q}$,
+because $$L$$ is usually a second-order polynomial of $$\dot{q}$$,
 and thus convex as required for Legendre transformation.
 In the above expression,
-$\dot{q}$ must be rewritten in terms of $p$ and $q$,
-which is trivial, since $p$ is proportional to $\dot{q}$ by definition.
+$$\dot{q}$$ must be rewritten in terms of $$p$$ and $$q$$,
+which is trivial, since $$p$$ is proportional to $$\dot{q}$$ by definition.
 
-The Hamiltonian $H$ also has a direct physical meaning:
-for a mass $m$, and for $L = T - V$,
-it is straightforward to show that $H$ represents the total energy $T + V$:
+The Hamiltonian $$H$$ also has a direct physical meaning:
+for a mass $$m$$, and for $$L = T - V$$,
+it is straightforward to show that $$H$$ represents the total energy $$T + V$$:
 
 $$\begin{aligned}
     H
@@ -64,8 +64,8 @@ $$\begin{aligned}
 
 Just as Lagrangian mechanics,
 Hamiltonian mechanics scales well for large systems.
-Its definition is generalized as follows to $N$ objects,
-where $p$ is shorthand for $p_1, ..., p_N$:
+Its definition is generalized as follows to $$N$$ objects,
+where $$p$$ is shorthand for $$p_1, ..., p_N$$:
 
 $$\begin{aligned}
     \boxed{
@@ -74,13 +74,13 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The positions and momenta $(q, p)$ form a phase space,
+The positions and momenta $$(q, p)$$ form a phase space,
 i.e. they fully describe the state.
 
 An extremely useful concept in Hamiltonian mechanics
 is the **Poisson bracket** (PB),
-which is a binary operation on two quantities $A(q, p)$ and $B(q, p)$,
-denoted by $\{A, B\}$:
+which is a binary operation on two quantities $$A(q, p)$$ and $$B(q, p)$$,
+denoted by $$\{A, B\}$$:
 
 $$\begin{aligned}
     \boxed{
@@ -93,8 +93,8 @@ $$\begin{aligned}
 ## Canonical equations
 
 Lagrangian mechanics has a single Euler-Lagrange equation per object,
-yielding $N$ second-order equations of motion in total.
-In contrast, Hamiltonian mechanics has $2 N$ first-order equations of motion,
+yielding $$N$$ second-order equations of motion in total.
+In contrast, Hamiltonian mechanics has $$2 N$$ first-order equations of motion,
 known as **Hamilton's canonical equations**:
 
 $$\begin{aligned}
@@ -111,7 +111,7 @@ $$\begin{aligned}
 <div class="hidden" markdown="1">
 <label for="proof-canoneq">Proof.</label>
 For the first equation,
-we differentiate $H$ with respect to $q_n$,
+we differentiate $$H$$ with respect to $$q_n$$,
 and use the chain rule:
 
 $$\begin{aligned}
@@ -133,7 +133,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The second equation is somewhat trivial,
-since $H$ is defined to satisfy it in the first place.
+since $$H$$ is defined to satisfy it in the first place.
 Nevertheless, we can prove it by brute force,
 using the same approach as above:
 
@@ -148,11 +148,12 @@ $$\begin{aligned}
     - 0 \pdv{L}{q_j} - p_j \pdv{\dot{q}_j}{p_n} \Big)
     = \dot{q}_n
 \end{aligned}$$
+
 </div>
 </div>
 
-Just like in Lagrangian mechanics, if $H$ does not explicitly contain $q_n$,
-then $q_n$ is called a **cyclic coordinate**, and leads to the conservation of $p_n$:
+Just like in Lagrangian mechanics, if $$H$$ does not explicitly contain $$q_n$$,
+then $$q_n$$ is called a **cyclic coordinate**, and leads to the conservation of $$p_n$$:
 
 $$\begin{aligned}
     \dot{p}_n = - \pdv{H}{q_n} = 0
@@ -161,11 +162,11 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Of course, there may be other conserved quantities.
-Generally speaking, the $t$-derivative of an arbitrary quantity $A(q, p, t)$ is as follows,
-where $\ipdv{}{t}$ is a "soft" derivative
-(only affects explicit occurrences of $t$),
-and $\idv{}{t}$ is a "hard" derivative
-(also affects implicit $t$ inside $q$ and $p$):
+Generally speaking, the $$t$$-derivative of an arbitrary quantity $$A(q, p, t)$$ is as follows,
+where $$\ipdv{}{t}$$ is a "soft" derivative
+(only affects explicit occurrences of $$t$$),
+and $$\idv{}{t}$$ is a "hard" derivative
+(also affects implicit $$t$$ inside $$q$$ and $$p$$):
 
 $$\begin{aligned}
     \boxed{
@@ -191,10 +192,11 @@ $$\begin{aligned}
     \\
     &= \sum_{n} \Big( \pdv{A}{q_n} \pdv{H}{p_n} - \pdv{A}{p_n} \pdv{H}{q_n} \Big) + \pdv{A}{t}
 \end{aligned}$$
+
 </div>
 </div>
 
-Assuming that $H$ does not explicitly depend on $t$,
+Assuming that $$H$$ does not explicitly depend on $$t$$,
 the above property naturally leads us to an alternative
 way of writing Hamilton's canonical equations:
 
@@ -208,7 +210,7 @@ $$\begin{aligned}
 
 ## Canonical coordinates
 
-So far, we have assumed that the phase space coordinates $(q, p)$
+So far, we have assumed that the phase space coordinates $$(q, p)$$
 are the *positions* and *canonical momenta*, respectively,
 and that led us to Hamilton's canonical equations.
 
@@ -220,11 +222,11 @@ $$\begin{aligned}
     p \to P(q, p)
 \end{aligned}$$
 
-However, most choices of $(Q, P)$ would not preserve Hamilton's equations.
-Any $(Q, P)$ that do keep this form
+However, most choices of $$(Q, P)$$ would not preserve Hamilton's equations.
+Any $$(Q, P)$$ that do keep this form
 are known as **canonical coordinates**,
 and the corresponding transformation is a **canonical transformation**.
-That is, any $(Q, P)$ that satisfy:
+That is, any $$(Q, P)$$ that satisfy:
 
 $$\begin{aligned}
     - \pdv{H}{Q_n} = \dot{P}_n
@@ -232,10 +234,10 @@ $$\begin{aligned}
     \pdv{H}{P_n} = \dot{Q}_n
 \end{aligned}$$
 
-Then we might as well write $H(q, p)$ as $H(Q, P)$.
-So, which $(Q, P)$ fulfill this?
-It turns out that the following must be satisfied for all $n, j$,
-where $\delta_{nj}$ is the Kronecker delta:
+Then we might as well write $$H(q, p)$$ as $$H(Q, P)$$.
+So, which $$(Q, P)$$ fulfill this?
+It turns out that the following must be satisfied for all $$n, j$$,
+where $$\delta_{nj}$$ is the Kronecker delta:
 
 $$\begin{aligned}
     \boxed{
@@ -250,8 +252,8 @@ $$\begin{aligned}
 <label for="proof-cantrans">Proof</label>
 <div class="hidden" markdown="1">
 <label for="proof-cantrans">Proof.</label>
-Assuming that $Q_n$, $P_n$ and $H$ do not explicitly depend on $t$,
-we use our expression for the $t$-derivative of an arbitrary quantity,
+Assuming that $$Q_n$$, $$P_n$$ and $$H$$ do not explicitly depend on $$t$$,
+we use our expression for the $$t$$-derivative of an arbitrary quantity,
 and apply the multivariate chain rule to it:
 
 $$\begin{aligned}
@@ -268,11 +270,11 @@ $$\begin{aligned}
     &= \sum_{j} \bigg( \pdv{H}{Q_j} \{Q_n, Q_j\} + \pdv{H}{P_j} \{Q_n, P_j\} \bigg)
 \end{aligned}$$
 
-This is equivalent to Hamilton's equation $\dot{Q}_n = \ipdv{H}{P_n}$
-if and only if $\{Q_n, Q_j\} = 0$ for all $n$ and $j$,
-and if $\{Q_n, P_j\} = \delta_{nj}$.
+This is equivalent to Hamilton's equation $$\dot{Q}_n = \ipdv{H}{P_n}$$
+if and only if $$\{Q_n, Q_j\} = 0$$ for all $$n$$ and $$j$$,
+and if $$\{Q_n, P_j\} = \delta_{nj}$$.
 
-Next, we do the exact same thing with $P_n$ instead of $Q_n$,
+Next, we do the exact same thing with $$P_n$$ instead of $$Q_n$$,
 giving an analogous result:
 
 $$\begin{aligned}
@@ -289,17 +291,17 @@ $$\begin{aligned}
     &= \sum_{j} \bigg( \pdv{H}{Q_j} \{P_n, Q_j\} + \pdv{H}{P_j} \{P_n, P_j\} \bigg)
 \end{aligned}$$
 
-Which is equivalent to Hamilton's equation $\dot{P}_n = -\ipdv{H}{Q_n}$
-if and only if $\{P_n, P_j\} = 0$,
-and $\{Q_n, P_j\} = - \delta_{nj}$.
+Which is equivalent to Hamilton's equation $$\dot{P}_n = -\ipdv{H}{Q_n}$$
+if and only if $$\{P_n, P_j\} = 0$$,
+and $$\{Q_n, P_j\} = - \delta_{nj}$$.
 The PB is anticommutative,
-i.e. $\{A, B\} = - \{B, A\}$.
+i.e. $$\{A, B\} = - \{B, A\}$$.
 </div>
 </div>
 
 If you have experience with quantum mechanics,
 the latter equation should look suspiciously similar
-to the *canonical commutation relation* $[\hat{Q}, \hat{P}] = i \hbar$.
+to the *canonical commutation relation* $$[\hat{Q}, \hat{P}] = i \hbar$$.