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Diffstat (limited to 'source/know/concept/hamiltonian-mechanics')
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1 files changed, 53 insertions, 51 deletions
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$$. |