From 6e70f28ccbd5afc1506f71f013278a9d157ef03a Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 27 Oct 2022 20:40:09 +0200
Subject: Optimize last images, add proof template, improve CSS
---
source/know/concept/hamiltonian-mechanics/index.md | 33 ++++++++--------------
1 file changed, 12 insertions(+), 21 deletions(-)
(limited to 'source/know/concept/hamiltonian-mechanics')
diff --git a/source/know/concept/hamiltonian-mechanics/index.md b/source/know/concept/hamiltonian-mechanics/index.md
index 19e55b0..03ff2dd 100644
--- a/source/know/concept/hamiltonian-mechanics/index.md
+++ b/source/know/concept/hamiltonian-mechanics/index.md
@@ -15,6 +15,7 @@ It is built on the shoulders of [Lagrangian mechanics](/know/concept/lagrangian-
which is in turn built on [variational calculus](/know/concept/calculus-of-variations/).
+
## Definitions
In Lagrangian mechanics, use a Lagrangian $$L$$,
@@ -90,6 +91,7 @@ $$\begin{aligned}
\end{aligned}$$
+
## Canonical equations
Lagrangian mechanics has a single Euler-Lagrange equation per object,
@@ -105,11 +107,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-canonical" -%}
For the first equation,
we differentiate $$H$$ with respect to $$q_n$$,
and use the chain rule:
@@ -148,9 +147,8 @@ $$\begin{aligned}
- 0 \pdv{L}{q_j} - p_j \pdv{\dot{q}_j}{p_n} \Big)
= \dot{q}_n
\end{aligned}$$
+{% include proof/end.html id="proof-canonical" %}
-
-
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 $$p_n$$:
@@ -175,11 +173,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-dv-t" -%}
We differentiate via the multivariate chain rule,
insert the canonical equations,
and eventually recognize the PB definition:
@@ -192,9 +187,8 @@ $$\begin{aligned}
\\
&= \sum_{n} \Big( \pdv{A}{q_n} \pdv{H}{p_n} - \pdv{A}{p_n} \pdv{H}{q_n} \Big) + \pdv{A}{t}
\end{aligned}$$
+{% include proof/end.html id="proof-dv-t" %}
-
-
Assuming that $$H$$ does not explicitly depend on $$t$$,
the above property naturally leads us to an alternative
@@ -247,11 +241,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-
-
-
-
-
+
+{% include proof/start.html id="proof-transformation" -%}
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:
@@ -296,8 +287,8 @@ if and only if $$\{P_n, P_j\} = 0$$,
and $$\{Q_n, P_j\} = - \delta_{nj}$$.
The PB is anticommutative,
i.e. $$\{A, B\} = - \{B, A\}$$.
-
-
+{% include proof/end.html id="proof-transformation" %}
+
If you have experience with quantum mechanics,
the latter equation should look suspiciously similar
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