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--- equations:hamiltons_equations [2018/04/08 14:36]
jakobadmin [Concrete]
+++ equations:hamiltons_equations [2019/02/12 15:21] (current)
129.13.36.189 [Why is it interesting?]
@@ Line 5: / Line 5: @@
 <tabbox Intuitive>
+<blockquote>Hamilton’s Equations show how the $q_i$'s and $p_i$'s undergo a ‘dance to the music of time’, a dance in which, as some $q_i$'s or $p_i$'s increase in value, others decrease in value, but always such as to keep the energy constant (in conservative systems), and always such as to keep the total action minimized, both instant by instant, and over the whole path between ‘surfaces-of-common-action’. This ‘dance’ is governed by one function, $H$, - that is to say, while $H$ is different for different systems (orbiting planets, a statistical ensemble, an electrical circuit, positrons orbiting an atomic antinucleus, a spinning top, juggling pins, a flowing river and so on) yet within any one system there is just one overaching function (there is no need for individual functions, $H_1$, $H_2$,...,$H_n)$.
+<cite>The Lazy Universe by Coopersmith</cite>
+</blockquote>
-<blockquote>
+<blockquote>
 Now, how are we to visualize Hamilton's equations in terms of [[basic_tools:phase_space|phase
 space]]? First, we should bear in mind what a single point Q of phase space
@@ Line 34: / Line 37: @@
 {{ :basic_notions:vectorfieldphasespacepenroseemprerornewmind.png?nolink&600 |}}
 <cite>page 174ff in "The Emperors new Mind" by R. Penrose</cite></blockquote>
+test
 <tabbox Concrete>
 Hamiltons equations are
@@ Line 67: / Line 69: @@
 **Symmetries of Hamiltons equations - Canonical Transformations**
-Hamilton's equations can be written more compactly when we introduce the 2n-dimensional vector $x\equiv ( q_1,q_2,\ldots, p_1,p_2,\ldots) and the $(2N \times 2N)$ matrix
+Hamilton's equations can be written more compactly when we introduce the 2n-dimensional vector $x\equiv ( q_1,q_2,\ldots, p_1,p_2,\ldots)$ and the $(2N \times 2N)$ matrix
 \[
@@ Line 107: / Line 109: @@
 We call a Jacobian $\mathcal{J}$ that fulfills this property symplectic. A transformation whose Jacobian is __[[advanced_tools:symplectic_structure|symplectic]]__ is called a __canonical transformation__.
+Take note that also [[advanced_notions:poisson_bracket|Poisson brackets]] are invariant under canonical transformations and in turn, all transformations that leave the Poisson bracket unchanged:
+$$
+\{ Q_i , Q_j\} =0 , \quad
+\{ P_i , P_j \} = 0  , \quad
+\{ Q_i , P_j \} = \delta_{ij}
+$$
+are called canonical transformations. For a proof, see page 104 [[http://www.damtp.cam.ac.uk/user/tong/dynamics/four.pdf|here]].
+-->Example of a canonical transformation#
+\begin{eqnarray*}
+q_1 &=&Q_1\cos Q_2 \\
+q_2 &=&Q_1\sin Q_2 \\
+p_1 &=&P_1\cos Q_2-\frac{P_2}{Q_1}\sin Q_2 \\
+p_2 &=&P_1\sin Q_2+\frac{P_2}{Q_1}\cos Q_2
+\end{eqnarray*}
+The associated Jacobian matrix is given by
+\[
+M=\left[
+\begin{array}{cc}
+A & B \\
+C & D
+\end{array}
+\right]
+\]
+where
+\begin{eqnarray*}
+A &=&\left[
+\begin{array}{cc}
+\cos Q_2 & -Q_1\sin Q_2 \\
+\sin Q_2 & Q_1\cos Q_2
+\end{array}
+\right] \\
+B &=&\left[
+\begin{array}{cc}
+& 0 \\
+& 0
+\end{array}
+\right] \\
+C &=&\left[
+\begin{array}{cc}
+\frac{P_1}{Q_1^2}\sin Q_2 & -P_1\sin Q_2-\frac{P_2}{Q_1}\cos Q_2 \\
+-\frac{P_2}{Q_1^2}\cos Q_2 & P_1\cos Q_2-\frac{P_2}{Q_1}\sin Q_2
+\end{array}
+\right] \\
+D &=&\left[
+\begin{array}{cc}
+\cos Q_2 & -\frac{\sin Q_2}{Q_1} \\
+\sin Q_1 & \frac{\cos Q_2}{Q_1}
+\end{array}
+\right]
+\end{eqnarray*}
+Using the fact that
+\[
+\left[
+\begin{array}{cc}
+A & B \\
+C & D
+\end{array}
+\right] ^T=\left[
+\begin{array}{cc}
+A^T & C^T \\
+B^T & D^T
+\end{array}
+\right]
+\]
+we check directly that $M^TJM=J$:
+\[
+\left[
+\begin{array}{cc}
+A^T & C^T \\
+B^T & D^T
+\end{array}
+\right] \left[
+\begin{array}{cc}
+&
+\begin{array}{cc}
+-1 & 0 \\
+& -1
+\end{array}
+\\
+\begin{array}{cc}
+& 0 \\
+& 1
+\end{array}
+& 0
+\end{array}
+\right] \left[
+\begin{array}{cc}
+A & B \\
+C & D
+\end{array}
+\right] =\left[
+\begin{array}{cc}
+& -I \\
+I & 0
+\end{array}
+\right]
+\]
+<--
+----
+**Generalized Hamilton's equations and infinitesimal canonical transformations**
+Next, we consider infinitesimal canonical transformations, which will help us to understand normal canonical transformations better.
+We can write an infinitesimal transformation as
+\begin{align}
+q_i \to Q_i &= q_i + \alpha F_i(q,p) \notag \\
+p_i \to P_i &= p_i + \alpha E_i(q,p) \notag ,
+\end{align}
+where $\alpha$ is infinitesimally small.
+The question we now need to answer is: which functions $F_i(q,p)$ and $E_i(q,p)$ are allowed such that these transformations are indeed canonical? As we found out above, canonical transformations are defined through their Jacobian. Thus we now calculate the Jacobian of our infinitesimal transformation
+\[
+\mathcal{J} = \left[
+\begin{array}{cc}
+\partial_{ij} + \alpha \partial F_i/\partial q_j  & \alpha \partial F_i / \partial p_j \\
+\alpha \partial E_i & \partial_{ij} + \alpha \partial E_i/\partial p_j
+\end{array}
+\right] .
+\]
+The defining condition for canonical transformations $\mathcal{J} J \mathcal{J}^T \stackrel{!}{=} J$ then tells us that
+$$ \frac{\partial F_i}{\partial q_j} \stackrel{!}{=} - \frac{\partial E_i}{\partial p_j} $$
+must hold.
+This equation is fulfilled if
+$$ F_i =   \frac{\partial G}{\partial p_i } \quad \text{ and } \quad  E_i = -  \frac{\partial G}{\partial q_i },$$
+since partial derivatices commute:
+$$ \frac{\partial F_i}{\partial q_j} = \frac{\partial \partial G}{\partial q_j\partial p_i  } \stackrel{!}{=}       \frac{\partial\partial G }{\partial p_j \partial q_i} = - \frac{\partial E_i}{\partial p_j}  \checkmark $$
+The functions $G(q,p)$ therefore __generate__ the infinitesimal transformations.
+Next we can consider families of transformations
+$$ q_i \to Q_i (q,p; \alpha) \quad \text{ and } \quad p_i \to P_i (q,p; \alpha) . $$
+For each value of $\alpha$ we have a different transformation and
+$$ Q_i (q,p; 0) = q_i \quad \text{ and } \quad  P_i (q,p; 0) = p_i . $$
+Starting in one state of the system $(q,p)$ we can use canonical transformations to get to another state of the system, i.e. a different point in phase space. As we vary $\alpha$ continuously we trace out a path in phase space.
+We can then put our generating function $G$ into the formulas for a general infinitesimal transformation that we started with
+\begin{align}
+q_i \to Q_i (q,p; \alpha) &= q_i + \alpha F_i(q,p) =  q_i + \alpha    \frac{\partial G}{\partial p_i }  \notag \\
+p_i \to P_i(q,p; \alpha) &= p_i + \alpha E_i(q,p) =   p_i - \alpha    \frac{\partial G}{\partial q_i }  \notag .
+\end{align}
+we rewrite this as
+\begin{align}
+ \frac{Q_i (q,p; \alpha) - q_i}{\alpha} &=      \frac{\partial G}{\partial p_i }  \notag \\
+ \frac{P_i(q,p; \alpha) - p_i}{\alpha}  &=   -  \frac{\partial G}{\partial q_i }  \notag .
+\end{align}
+Therefore, as we take $\alpha \to 0$ we get the derivative with respect to $\alpha$
+$$ \frac{ dq_i}{d\alpha} = \frac{\partial G}{\partial p_i}  \quad \text{ and } \quad  \frac{ dp_i}{d\alpha} = - \frac{\partial G}{\partial q_i}. $$
+($\alpha$ is the parameter that parametrizes our curve in phase space.)
+These look exactly like Hamilton's equations, but instead of the Hamiltonian we have the general generating function $G$ and instead of the time, we have the general parameter $\alpha$.
+We can now understand that Hamilton's equations as written above are just a special case where we consider time-translations as our transformation. The corresponding generating function is the Hamiltonian.
+Another example would be where $G= p_k$, i.e. the momentum along the $k$ axis. Then, our canonical infinitesimal transformation reads $q_i \to q_i + \alpha \delta _{ik}$ and $p_i \to p_i$ which is simply a translation. Therefore, we can say that translations are generated by the conjugate momentum.
 <tabbox Abstract>
@@ Line 129: / Line 310: @@
 This comes at the cost of doubling the size of the system.
+----
+<blockquote>[E]verybody loves Hamilton’s equations: there are just two, and they summarize the entire essence of classical mechanics.
+<cite>[[https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/|John Baez]]</cite></blockquote>
+<blockquote>You get equations like Hamilton's whenever a system *extremizes something subject to constraints*.   A moving particle minimizes action; a box of gas maximizes entropy.   <cite>[[https://twitter.com/johncarlosbaez/status/1065715514381557761|John Baez]]</cite></blockquote>
 <tabbox Origin>
-"Hamilton’s equations and the [[equations:maxwell_relations|Maxwell relations]]—are mathematically just the same. They both say simply that partial derivatives commute." See  https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/
+"Hamilton’s equations and the [[formulas:maxwell_relations|Maxwell relations]]—are mathematically just the same. They both say simply that partial derivatives commute." See  https://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/
 and  https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/

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