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Sander, Oliver
skript-numerik
Commits
44cef932
Commit
44cef932
authored
Jun 23, 2020
by
Sander, Oliver
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Detailverbesserungen symplektische Verfahren
parent
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#4177
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hamilton-systeme.tex
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hamilton-systeme.tex
View file @
44cef932
...
...
@@ -788,18 +788,27 @@ Für quadratische Hamilton-Funktionen ist das anders.
P(
\tau
J
^{
-1
}
C)
^
T JP(
\tau
J
^{
-1
}
C) = Q(
\tau
J
^{
-1
}
C)
^
T J Q(
\tau
J
^{
-1
}
C).
\end{equation}
\todoannot
{
1.5
\baselineskip
}{
Die folgende Rechnung muss nochmal im Detail überprüft werden!
}
\item
Betrachte das Produkt
\glqq
Polynom in
$
J
^{
-
1
}
C
$
\grqq\
mit
$
J
$
.
\item
Für jedes Monom
$
(
J
^{
-
1
}
C
)
^
k
$
gilt (
$
C
$
ist symmetrisch)
\begin{equation*}
\item
Für jedes Monom
$
(
J
^{
-
1
}
C
)
^
k
$
,
$
k
\in
\N
$
gilt
(
$
C
$
ist symmetrisch, und
$
J
^
T
=
-
J
$
)
\begin{align*}
((J
^{
-1
}
C)
^
k)
^
T J
=
(C
^
T J
^{
-T
}
)
^
k J
=
J(-(J
^{
-1
}
C)
^
k)
\quad
\forall
k=0,1,2,
\hdots
\end{equation*}
&
=
(C
^
T J
^{
-T
}
)
^
k J
\\
&
=
\underbrace
{
C
^
TJ
^{
-T
}
\dots
C
^
TJ
^{
-T
}}_{
\text
{$
k
$
mal
}}
J
\\
&
=
-C
^
T
\underbrace
{
J
^{
-T
}
C
^
T
\dots
J
^{
-T
}
C
^
T
}_{
\text
{$
k
-
1
$
mal
}}
\\
&
=
-C
\underbrace
{
J
^{
-T
}
C
\dots
J
^{
-T
}
C
}_{
\text
{$
k
-
1
$
mal
}}
\\
&
=
-J
^
T J
^{
-T
}
C
\underbrace
{
J
^{
-T
}
C
\dots
J
^{
-T
}
C
}_{
\text
{$
k
-
1
$
mal
}}
\\
&
=
-J
^
T (J
^{
-T
}
C)
^
k
\\
&
=
J(-J
^{
-1
}
C)
^
k.
\end{align*}
\item
Also folgt aus
\eqref
{
eq:beweis
_
symplektisch
_
symmetrisch
}
\begin{equation*}
P(-
\tau
J
^{
-1
}
C)
\cdot
P(
\tau
J
^{
-1
}
C) = Q(-
\tau
J
^{
-1
}
C)
\cdot
Q(
\tau
J
^{
-1
}
C)
...
...
@@ -808,7 +817,7 @@ Für quadratische Hamilton-Funktionen ist das anders.
\begin{equation*}
R(-
\tau
J
^{
-1
}
C)
\cdot
R(
\tau
J
^{
-1
}
C) = I.
\end{equation*}
\item
Das ist gerade die
Symmetrie
des Verfahrens.
\item
Das ist gerade die
Reversibilität
des Verfahrens.
\qedhere
\end{itemize}
\end{proof}
...
...
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