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Diffstat (limited to 'source/know/concept/hamiltonian-mechanics/index.md')
| -rw-r--r-- | source/know/concept/hamiltonian-mechanics/index.md | 33 |
1 files changed, 12 insertions, 21 deletions
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 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-canoneq"/> -<label for="proof-canoneq">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-canoneq">Proof.</label> + +{% 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" %} -</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 $$p_n$$: @@ -175,11 +173,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-diff-t"/> -<label for="proof-diff-t">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-diff-t">Proof.</label> + +{% 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" %} -</div> -</div> Assuming that $$H$$ does not explicitly depend on $$t$$, the above property naturally leads us to an alternative @@ -247,11 +241,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-cantrans"/> -<label for="proof-cantrans">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-cantrans">Proof.</label> + +{% 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. $$\{A, B\} = - \{B, A\}$$. -</div> -</div> +{% include proof/end.html id="proof-transformation" %} + If you have experience with quantum mechanics, the latter equation should look suspiciously similar |