From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/lagrangian-mechanics/index.md | 42 +++++++++++------------
 1 file changed, 21 insertions(+), 21 deletions(-)

(limited to 'source/know/concept/lagrangian-mechanics')

diff --git a/source/know/concept/lagrangian-mechanics/index.md b/source/know/concept/lagrangian-mechanics/index.md
index 7b520a2..0a7066a 100644
--- a/source/know/concept/lagrangian-mechanics/index.md
+++ b/source/know/concept/lagrangian-mechanics/index.md
@@ -17,8 +17,8 @@ and hence it is built on the **principle of least action**,
 which states that the path taken by a system
 will be a minimum of the **action** (i.e. energy cost) of that path.
 
-For a moving object with position $x(t)$ and velocity $\dot{x}(t)$,
-we define the Lagrangian $L$ as the difference
+For a moving object with position $$x(t)$$ and velocity $$\dot{x}(t)$$,
+we define the Lagrangian $$L$$ as the difference
 between its kinetic and potential energies:
 
 $$\begin{aligned}
@@ -38,21 +38,21 @@ $$\begin{aligned}
 
 But compared to Newtonian mechanics,
 Lagrangian mechanics scales better for large systems.
-For example, to describe the dynamics of $N$ objects $x_1(t), ..., x_N(t)$,
-we only need a single $L$
+For example, to describe the dynamics of $$N$$ objects $$x_1(t), ..., x_N(t)$$,
+we only need a single $$L$$
 from which the equations of motion can easily be derived.
 Getting these equations directly from Newton's laws could get messy.
 
 At no point have we assumed Cartesian coordinates:
 the Euler-Lagrange equations keep their form
-for any independent coordinates $q_1(t), ..., q_N(t)$:
+for any independent coordinates $$q_1(t), ..., q_N(t)$$:
 
 $$\begin{aligned}
     \dv{}{t}\Big( \pdv{L}{\dot{q}_n} \Big) = \pdv{L}{q_n}
 \end{aligned}$$
 
-We define the **canonical momentum conjugate** $p_n(t)$
-and the **generalized force conjugate** $F_n(t)$ as follows,
+We define the **canonical momentum conjugate** $$p_n(t)$$
+and the **generalized force conjugate** $$F_n(t)$$ as follows,
 such that we can always get Newton's second law:
 
 $$\begin{aligned}
@@ -64,15 +64,15 @@ $$\begin{aligned}
 \end{aligned}$$
 
 But this is actually a bit misleading,
-since $p_n$ need not be a momentum, nor $F_n$ a force,
+since $$p_n$$ need not be a momentum, nor $$F_n$$ a force,
 although often they are.
-For example, $p_n$ could be angular momentum, and $F_n$ torque.
+For example, $$p_n$$ could be angular momentum, and $$F_n$$ torque.
 
 Another advantage of Lagrangian mechanics is that
-the conserved quantities can be extracted from $L$ using Noether's theorem.
-In the simplest case, if $L$ does not depend on $q_n$
+the conserved quantities can be extracted from $$L$$ using Noether's theorem.
+In the simplest case, if $$L$$ does not depend on $$q_n$$
 (then known as a **cyclic coordinate**),
-then we know that the "momentum" $p_n$ is a conserved quantity:
+then we know that the "momentum" $$p_n$$ is a conserved quantity:
 
 $$\begin{aligned}
     F_n = \pdv{L}{q_n} = 0
@@ -80,23 +80,23 @@ $$\begin{aligned}
     \dv{p_n}{t} = 0
 \end{aligned}$$
 
-Now, as the number of particles $N$ increases to infinity,
+Now, as the number of particles $$N$$ increases to infinity,
 variational calculus will give infinitely many coupled equations,
 which is obviously impractical.
 
-Such a system can be regarded as continuous, so the $N$ functions $q_n$
-can be replaced by a single density function $u(x,t)$.
+Such a system can be regarded as continuous, so the $$N$$ functions $$q_n$$
+can be replaced by a single density function $$u(x,t)$$.
 This approach can also be used for continuous fields,
-in which case the complex conjugate $u^*$ is often included.
-The Lagrangian $L$ then becomes:
+in which case the complex conjugate $$u^*$$ is often included.
+The Lagrangian $$L$$ then becomes:
 
 $$\begin{aligned}
     L(u, u^*, u_x, u_x^*, u_t, u_t^*, x, t)
     = \int_{-\infty}^\infty \mathcal{L}(u, u^*, u_x, u_x^*, u_t, u_t^*, x, t) \dd{x}
 \end{aligned}$$
 
-Where $\mathcal{L}$ is known as the **Lagrangian density**.
-By inserting this into the functional $J$
+Where $$\mathcal{L}$$ is known as the **Lagrangian density**.
+By inserting this into the functional $$J$$
 used for the derivation of the Euler-Lagrange equations, we get:
 
 $$\begin{aligned}
@@ -114,11 +114,11 @@ $$\begin{aligned}
     0 &= \pdv{\mathcal{L}}{u^*} - \pdv{}{x}\Big( \pdv{\mathcal{L}}{u_x^*} \Big) - \pdv{}{t}\Big( \pdv{\mathcal{L}}{u_t^*} \Big)
 \end{aligned}$$
 
-If $\mathcal{L}$ is real,
+If $$\mathcal{L}$$ is real,
 then these two Euler-Lagrange equations will in fact be identical.
 
 Finally, note that for abstract fields,
-the Lagrangian density $\mathcal{L}$ rarely has
+the Lagrangian density $$\mathcal{L}$$ rarely has
 a physical interpretation, and is not unique.
 Instead, it must be reverse-engineered from a relevant equation.
 
-- 
cgit v1.2.3