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 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
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 @@ $$\begin{aligned} } \end{aligned}$$ -
- - - -
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/end.html id="proof-transformation" %} + If you have experience with quantum mechanics, the latter equation should look suspiciously similar -- cgit v1.2.3