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Difference between revisions of "Kustaanheimo-Stiefel transformation"

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A particular case of the surjective application <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102101.png" /> defined by
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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{{TEX|done}}
  
subject to the constraint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102103.png" /> with
+
A particular case of the surjective application  $  \mathbf R  ^ {4} \rightarrow \mathbf R  ^ {3} : ( u _ {0} ,u _ {1} ,u _ {2} ,u _ {3} ) \mapsto ( x _ {0} ,x _ {2} ,x _ {3} ) $
 +
defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a1 }
 +
\left .
  
where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102106.png" /> can take the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102107.png" />. The transformation introduced by P. Kustaanheimo and E. Stiefel [[#References|[a1]]] corresponds to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102108.png" />, while the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102109.png" /> [[#References|[a2]]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021010.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021011.png" />) [[#References|[a3]]] correspond to two other (inequivalent) transformations.
+
\begin{array}{c}
 +
x _ {0} = u _ {0}  ^ {2} - p u _ {1}  ^ {2} + q u _ {2}  ^ {2} - pq u _ {3}  ^ {2} ,  \\
 +
x _ {2} = 2 ( u _ {0} u _ {2} + p u _ {1} u _ {3} ) , \\
 +
x _ {3} = 2 ( u _ {0} u _ {3} + u _ {1} u _ {2} ) ,  \\
 +
\end{array}
  
There are several ways of introducing these three transformations. For instance, the Kustaanheimo–Stiefel mapping is inherent to the Cartan spinor theory, since (a1), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021012.png" />, can be rewritten as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021015.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021016.png" /> (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021017.png" />) are the [[Pauli matrices|Pauli matrices]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021018.png" /> is a spinor of components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021020.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021021.png" /> stands for the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021022.png" />). Furthermore, the Kustaanheimo–Stiefel transformation, along with the two other transformations, can be introduced as a particular Hurwitz transformation (see also [[Hurwitz transformation|Hurwitz transformation]]) in the framework of the [[Cayley–Dickson algebra|Cayley–Dickson algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021023.png" />. (The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021024.png" /> turns out to be the algebra of ordinary (or elliptic) quaternions when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021025.png" /> and the algebra of hyperbolic quaternions when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021026.png" />.) The main properties of the transformations (a1)–(a2) are as follows.
+
\right \}
 +
$$
 +
 
 +
subject to the constraint  $  \omega = 0 $
 +
with
 +
 
 +
$$ \tag{a2 }
 +
\omega = 2 ( u _ {1}  du _ {0} - u _ {0}  du _ {1} + q u _ {3}  du _ {2} - q u _ {2}  du _ {3} ) ,
 +
$$
 +
 
 +
where the parameters  $  p $
 +
and  $  q $
 +
can take the values  $  \pm  1 $.
 +
The transformation introduced by P. Kustaanheimo and E. Stiefel [[#References|[a1]]] corresponds to  $  p = q = - 1 $,
 +
while the cases  $  p = - q = - 1 $[[#References|[a2]]] and  $  p = - q = 1 $(
 +
or  $  p = q = 1 $)
 +
[[#References|[a3]]] correspond to two other (inequivalent) transformations.
 +
 
 +
There are several ways of introducing these three transformations. For instance, the Kustaanheimo–Stiefel mapping is inherent to the Cartan spinor theory, since (a1), with $  p = q = - 1 $,  
 +
can be rewritten as $  x _ {0} = \xi  ^  \dag  \sigma _ {3} \xi $,  
 +
$  x _ {2} = \xi  ^  \dag  \sigma _ {1} \xi $
 +
and $  x _ {3} = \xi  ^  \dag  \sigma _ {2} \xi $,  
 +
where the $  \sigma _ {i} $(
 +
with $  i = 1,2,3 $)  
 +
are the [[Pauli matrices|Pauli matrices]] and $  \xi $
 +
is a spinor of components $  u _ {0} - i u _ {1} $
 +
and $  u _ {2} + i u _ {3} $(
 +
$  \xi  ^  \dag  $
 +
stands for the adjoint of $  \xi $).  
 +
Furthermore, the Kustaanheimo–Stiefel transformation, along with the two other transformations, can be introduced as a particular Hurwitz transformation (see also [[Hurwitz transformation|Hurwitz transformation]]) in the framework of the [[Cayley–Dickson algebra|Cayley–Dickson algebra]] $  A ( p,q ) $.  
 +
(The algebra $  A ( p,q ) $
 +
turns out to be the algebra of ordinary (or elliptic) quaternions when $  p + q = - 2 $
 +
and the algebra of hyperbolic quaternions when $  p + q \neq - 2 $.)  
 +
The main properties of the transformations (a1)–(a2) are as follows.
  
 
1) These transformations satisfy
 
1) These transformations satisfy
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021027.png" /></td> </tr></table>
+
$$
 +
dx _ {0}  ^ {2} - q  dx _ {2}  ^ {2} + pq  dx _ {3}  ^ {2} = 4 r ( {} ^ {\textrm{ t } } d \mathbf u  \eta  d \mathbf u )
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
$$ \tag{a3 }
 +
x _ {0}  ^ {2} - q x _ {2}  ^ {2} + pq x _ {3}  ^ {2} = r  ^ {2} ,
 +
$$
  
where the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021029.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021030.png" />, so that
+
where the metric $  \eta $
 +
is $  \eta = { \mathop{\rm diag} } ( 1, - p, - q, pq ) $,  
 +
so that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
$$ \tag{a4 }
 +
r = ( {} ^ {\textrm{ t } } \mathbf u \eta \mathbf u ) = u _ {0}  ^ {2} - p u _ {1}  ^ {2} - q u _ {2}  ^ {2} + pq u _ {3}  ^ {2}
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021032.png" /> denotes a column vector, the transposed of which is the row vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021033.png" />).
+
( $  \mathbf u $
 +
denotes a column vector, the transposed of which is the row vector $  {} ^ {\textrm{ t } } \mathbf u = ( u _ {0} , u _ {1} , u _ {2} , u _ {3} ) $).
  
2) Equations (a3)–(a4) show that the various mappings (a1) correspond to three types of fibrations (cf. [[Fibration|Fibration]]): the celebrated [[Hopf fibration|Hopf fibration]] (on spheres) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021034.png" /> of compact fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021035.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021036.png" /> and two other fibrations (on hyperboloids), namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021037.png" /> of compact fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021038.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021040.png" /> of non-compact fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021041.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021043.png" />.
+
2) Equations (a3)–(a4) show that the various mappings (a1) correspond to three types of fibrations (cf. [[Fibration|Fibration]]): the celebrated [[Hopf fibration|Hopf fibration]] (on spheres) $  \mathbf S  ^ {3} \rightarrow \mathbf S  ^ {2} $
 +
of compact fibre $  \mathbf S  ^ {1} $
 +
for $  p = q = - 1 $
 +
and two other fibrations (on hyperboloids), namely, $  \mathbf R  ^ {2} \times \mathbf S  ^ {1} \rightarrow \mathbf R  ^ {2} $
 +
of compact fibre $  \mathbf S  ^ {1} $
 +
for $  p = - q = - 1 $
 +
and $  \mathbf R  ^ {2} \times \mathbf S  ^ {1} \rightarrow \mathbf R \times \mathbf S  ^ {1} $
 +
of non-compact fibre $  \mathbf R $
 +
for $  p = - q = 1 $
 +
and $  p = q = 1 $.
  
3) The preceding geometrical property can be made precise by transformation properties of (first- and) second-order elliptic and hyperbolic differential operators. For example, this yields (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021044.png" />)
+
3) The preceding geometrical property can be made precise by transformation properties of (first- and) second-order elliptic and hyperbolic differential operators. For example, this yields (for $  f \in C  ^ {2} ( \mathbf R  ^ {3} ) $)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021045.png" /></td> </tr></table>
+
$$
 +
\Delta _ {x} f = {
 +
\frac{1}{4r }
 +
} \Delta _ {u} f,  Xf = 0,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021046.png" /></td> </tr></table>
+
$$
 +
\Delta _ {x} = \partial  _ {x _ {0}  x _ {0} } - q \partial  _ {x _ {2}  x _ {2} } + pq \partial  _ {x _ {3}  x _ {3} } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021047.png" /></td> </tr></table>
+
$$
 +
\Delta _ {u} = \partial  _ {u _ {0}  u _ {0} } - p \partial  _ {u _ {1}  u _ {1} } - q \partial  _ {u _ {2}  u _ {2} } + pq \partial  _ {u _ {3}  u _ {3} } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021048.png" /></td> </tr></table>
+
$$
 +
X = - p u _ {1} \partial  _ {u _ {0}  } - u _ {0} \partial  _ {u _ {1}  } + p u _ {3} \partial  _ {u _ {2}  } + u _ {2} \partial  _ {u _ {3}  } .
 +
$$
  
4) The [[Vector field|vector field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021049.png" /> associated to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021050.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021051.png" /> (with an action satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021052.png" />) is an element of the symplectic Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021054.png" /> which generates either the subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021055.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021056.png" />, according to whether the fibre is compact or non-compact. The introduction of the (Dirac primary) constraint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021058.png" /> produces a Lie algebra under constraint (i.e., the centralizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021059.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021060.png" />), which is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021061.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021062.png" />, depending on the compactness or non-compactness of the fibre, respectively (see [[#References|[a3]]] and the references therein).
+
4) The [[Vector field|vector field]] $  X $
 +
associated to the $  1 $-
 +
form $  \omega $(
 +
with an action satisfying $  \omega [ {1 / {( 4r ) } } X ] = 1 $)  
 +
is an element of the symplectic Lie algebra $  { \mathop{\rm sp} } ( 8, \mathbf R ) $
 +
which generates either the subalgebra $  { \mathop{\rm so} } ( 2 ) $
 +
or $  { \mathop{\rm so} } ( 1,1 ) $,  
 +
according to whether the fibre is compact or non-compact. The introduction of the (Dirac primary) constraint $  X = 0 $
 +
in $  { \mathop{\rm sp} } ( 8, \mathbf R ) $
 +
produces a Lie algebra under constraint (i.e., the centralizer of $  X $
 +
in $  { \mathop{\rm sp} } ( 8, \mathbf R ) $),  
 +
which is isomorphic to $  { \mathop{\rm so} } ( 4,2 ) $
 +
or $  { \mathop{\rm so} } ( 3,3 ) $,  
 +
depending on the compactness or non-compactness of the fibre, respectively (see [[#References|[a3]]] and the references therein).
  
The applications of the transformations (a1)–(a2) range from number theory to physics (classical and quantum mechanics, gauge theories). By way of illustration, the (generalized) Hurwitz matrix corresponding to (a1)–(a2) is connected to the (generalized) problem of factoring the sum of four squared numbers, addressed by A. Hurwitz (1898) in the specific case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021063.png" />. In addition, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021065.png" /> fixed, the solutions of the generalized Pythagorean equation (a3) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021066.png" /> are parametrized by (a1) and (a4) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021067.png" />. In classical mechanics, the Kustaanheimo–Stiefel transformation is used for the regularization of the Kepler problem. In quantum mechanics (cf. also [[MIC-Kepler problem|MIC-Kepler problem]]), the latter transformation makes it possible to transform the [[Schrödinger equation|Schrödinger equation]] for the three-dimensional hydrogen atom (in an electromagnetic field) into a Schrödinger equation for a four-dimensional isotropic harmonic oscillator (with quartic and sextic anharmonic terms) subject to a constraint [[#References|[a4]]], [[#References|[a5]]]. Conversely, the mappings (a1)–(a2) may be used in some dimensional reduction process for converting a dynamical system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021068.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021069.png" /> into a dynamical system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021070.png" />.
+
The applications of the transformations (a1)–(a2) range from number theory to physics (classical and quantum mechanics, gauge theories). By way of illustration, the (generalized) Hurwitz matrix corresponding to (a1)–(a2) is connected to the (generalized) problem of factoring the sum of four squared numbers, addressed by A. Hurwitz (1898) in the specific case $  p = q = - 1 $.  
 +
In addition, for $  p $
 +
and $  q $
 +
fixed, the solutions of the generalized Pythagorean equation (a3) with $  ( x _ {0} , x _ {1} , x _ {2} , r ) \in \mathbf Z  ^ {4} $
 +
are parametrized by (a1) and (a4) with $  ( u _ {0} ,u _ {1} , u _ {2} , u _ {3} ) \in \mathbf Z  ^ {4} $.  
 +
In classical mechanics, the Kustaanheimo–Stiefel transformation is used for the regularization of the Kepler problem. In quantum mechanics (cf. also [[MIC-Kepler problem|MIC-Kepler problem]]), the latter transformation makes it possible to transform the [[Schrödinger equation|Schrödinger equation]] for the three-dimensional hydrogen atom (in an electromagnetic field) into a Schrödinger equation for a four-dimensional isotropic harmonic oscillator (with quartic and sextic anharmonic terms) subject to a constraint [[#References|[a4]]], [[#References|[a5]]]. Conversely, the mappings (a1)–(a2) may be used in some dimensional reduction process for converting a dynamical system in $  \mathbf R  ^ {4} $
 +
or $  \mathbf R  ^ {2} \times \mathbf R  ^ {2} $
 +
into a dynamical system in $  \mathbf R  ^ {3} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Kustaanheimo,  E. Stiefel,  "Perturbation theory of Kepler motion based on spinor regularization"  ''J. Reine Angew. Math.'' , '''218'''  (1965)  pp. 204</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Iwai,  "On reduction of two degrees of freedom Hamiltonian system by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021071.png" /> action, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021072.png" /> as a dynamical group"  ''J. Math. Phys.'' , '''26'''  (1985)  pp. 885</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Lambert,  M. Kibler,  "An algebraic and geometric approach to non-bijective quadratic transformations"  ''J. Phys. A: Math. Gen.'' , '''21'''  (1988)  pp. 307</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Boiteux,  "Theory of nonbijective canonical transformations in mechanics: Application to the Coulomb problem"  ''J. Math. Phys.'' , '''23'''  (1982)  pp. 1311</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Kibler,  T. Négadi,  "Hydrogen atom in a uniform electromagnetic field as an anharmonic oscillator"  ''Lett. Nuovo Cim.'' , '''39'''  (1984)  pp. 319</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Kustaanheimo,  E. Stiefel,  "Perturbation theory of Kepler motion based on spinor regularization"  ''J. Reine Angew. Math.'' , '''218'''  (1965)  pp. 204</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Iwai,  "On reduction of two degrees of freedom Hamiltonian system by an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021071.png" /> action, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k11021072.png" /> as a dynamical group"  ''J. Math. Phys.'' , '''26'''  (1985)  pp. 885</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Lambert,  M. Kibler,  "An algebraic and geometric approach to non-bijective quadratic transformations"  ''J. Phys. A: Math. Gen.'' , '''21'''  (1988)  pp. 307</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Boiteux,  "Theory of nonbijective canonical transformations in mechanics: Application to the Coulomb problem"  ''J. Math. Phys.'' , '''23'''  (1982)  pp. 1311</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Kibler,  T. Négadi,  "Hydrogen atom in a uniform electromagnetic field as an anharmonic oscillator"  ''Lett. Nuovo Cim.'' , '''39'''  (1984)  pp. 319</TD></TR></table>

Latest revision as of 22:15, 5 June 2020


A particular case of the surjective application $ \mathbf R ^ {4} \rightarrow \mathbf R ^ {3} : ( u _ {0} ,u _ {1} ,u _ {2} ,u _ {3} ) \mapsto ( x _ {0} ,x _ {2} ,x _ {3} ) $ defined by

$$ \tag{a1 } \left . \begin{array}{c} x _ {0} = u _ {0} ^ {2} - p u _ {1} ^ {2} + q u _ {2} ^ {2} - pq u _ {3} ^ {2} , \\ x _ {2} = 2 ( u _ {0} u _ {2} + p u _ {1} u _ {3} ) , \\ x _ {3} = 2 ( u _ {0} u _ {3} + u _ {1} u _ {2} ) , \\ \end{array} \right \} $$

subject to the constraint $ \omega = 0 $ with

$$ \tag{a2 } \omega = 2 ( u _ {1} du _ {0} - u _ {0} du _ {1} + q u _ {3} du _ {2} - q u _ {2} du _ {3} ) , $$

where the parameters $ p $ and $ q $ can take the values $ \pm 1 $. The transformation introduced by P. Kustaanheimo and E. Stiefel [a1] corresponds to $ p = q = - 1 $, while the cases $ p = - q = - 1 $[a2] and $ p = - q = 1 $( or $ p = q = 1 $) [a3] correspond to two other (inequivalent) transformations.

There are several ways of introducing these three transformations. For instance, the Kustaanheimo–Stiefel mapping is inherent to the Cartan spinor theory, since (a1), with $ p = q = - 1 $, can be rewritten as $ x _ {0} = \xi ^ \dag \sigma _ {3} \xi $, $ x _ {2} = \xi ^ \dag \sigma _ {1} \xi $ and $ x _ {3} = \xi ^ \dag \sigma _ {2} \xi $, where the $ \sigma _ {i} $( with $ i = 1,2,3 $) are the Pauli matrices and $ \xi $ is a spinor of components $ u _ {0} - i u _ {1} $ and $ u _ {2} + i u _ {3} $( $ \xi ^ \dag $ stands for the adjoint of $ \xi $). Furthermore, the Kustaanheimo–Stiefel transformation, along with the two other transformations, can be introduced as a particular Hurwitz transformation (see also Hurwitz transformation) in the framework of the Cayley–Dickson algebra $ A ( p,q ) $. (The algebra $ A ( p,q ) $ turns out to be the algebra of ordinary (or elliptic) quaternions when $ p + q = - 2 $ and the algebra of hyperbolic quaternions when $ p + q \neq - 2 $.) The main properties of the transformations (a1)–(a2) are as follows.

1) These transformations satisfy

$$ dx _ {0} ^ {2} - q dx _ {2} ^ {2} + pq dx _ {3} ^ {2} = 4 r ( {} ^ {\textrm{ t } } d \mathbf u \eta d \mathbf u ) $$

and

$$ \tag{a3 } x _ {0} ^ {2} - q x _ {2} ^ {2} + pq x _ {3} ^ {2} = r ^ {2} , $$

where the metric $ \eta $ is $ \eta = { \mathop{\rm diag} } ( 1, - p, - q, pq ) $, so that

$$ \tag{a4 } r = ( {} ^ {\textrm{ t } } \mathbf u \eta \mathbf u ) = u _ {0} ^ {2} - p u _ {1} ^ {2} - q u _ {2} ^ {2} + pq u _ {3} ^ {2} $$

( $ \mathbf u $ denotes a column vector, the transposed of which is the row vector $ {} ^ {\textrm{ t } } \mathbf u = ( u _ {0} , u _ {1} , u _ {2} , u _ {3} ) $).

2) Equations (a3)–(a4) show that the various mappings (a1) correspond to three types of fibrations (cf. Fibration): the celebrated Hopf fibration (on spheres) $ \mathbf S ^ {3} \rightarrow \mathbf S ^ {2} $ of compact fibre $ \mathbf S ^ {1} $ for $ p = q = - 1 $ and two other fibrations (on hyperboloids), namely, $ \mathbf R ^ {2} \times \mathbf S ^ {1} \rightarrow \mathbf R ^ {2} $ of compact fibre $ \mathbf S ^ {1} $ for $ p = - q = - 1 $ and $ \mathbf R ^ {2} \times \mathbf S ^ {1} \rightarrow \mathbf R \times \mathbf S ^ {1} $ of non-compact fibre $ \mathbf R $ for $ p = - q = 1 $ and $ p = q = 1 $.

3) The preceding geometrical property can be made precise by transformation properties of (first- and) second-order elliptic and hyperbolic differential operators. For example, this yields (for $ f \in C ^ {2} ( \mathbf R ^ {3} ) $)

$$ \Delta _ {x} f = { \frac{1}{4r } } \Delta _ {u} f, Xf = 0, $$

where

$$ \Delta _ {x} = \partial _ {x _ {0} x _ {0} } - q \partial _ {x _ {2} x _ {2} } + pq \partial _ {x _ {3} x _ {3} } , $$

$$ \Delta _ {u} = \partial _ {u _ {0} u _ {0} } - p \partial _ {u _ {1} u _ {1} } - q \partial _ {u _ {2} u _ {2} } + pq \partial _ {u _ {3} u _ {3} } , $$

$$ X = - p u _ {1} \partial _ {u _ {0} } - u _ {0} \partial _ {u _ {1} } + p u _ {3} \partial _ {u _ {2} } + u _ {2} \partial _ {u _ {3} } . $$

4) The vector field $ X $ associated to the $ 1 $- form $ \omega $( with an action satisfying $ \omega [ {1 / {( 4r ) } } X ] = 1 $) is an element of the symplectic Lie algebra $ { \mathop{\rm sp} } ( 8, \mathbf R ) $ which generates either the subalgebra $ { \mathop{\rm so} } ( 2 ) $ or $ { \mathop{\rm so} } ( 1,1 ) $, according to whether the fibre is compact or non-compact. The introduction of the (Dirac primary) constraint $ X = 0 $ in $ { \mathop{\rm sp} } ( 8, \mathbf R ) $ produces a Lie algebra under constraint (i.e., the centralizer of $ X $ in $ { \mathop{\rm sp} } ( 8, \mathbf R ) $), which is isomorphic to $ { \mathop{\rm so} } ( 4,2 ) $ or $ { \mathop{\rm so} } ( 3,3 ) $, depending on the compactness or non-compactness of the fibre, respectively (see [a3] and the references therein).

The applications of the transformations (a1)–(a2) range from number theory to physics (classical and quantum mechanics, gauge theories). By way of illustration, the (generalized) Hurwitz matrix corresponding to (a1)–(a2) is connected to the (generalized) problem of factoring the sum of four squared numbers, addressed by A. Hurwitz (1898) in the specific case $ p = q = - 1 $. In addition, for $ p $ and $ q $ fixed, the solutions of the generalized Pythagorean equation (a3) with $ ( x _ {0} , x _ {1} , x _ {2} , r ) \in \mathbf Z ^ {4} $ are parametrized by (a1) and (a4) with $ ( u _ {0} ,u _ {1} , u _ {2} , u _ {3} ) \in \mathbf Z ^ {4} $. In classical mechanics, the Kustaanheimo–Stiefel transformation is used for the regularization of the Kepler problem. In quantum mechanics (cf. also MIC-Kepler problem), the latter transformation makes it possible to transform the Schrödinger equation for the three-dimensional hydrogen atom (in an electromagnetic field) into a Schrödinger equation for a four-dimensional isotropic harmonic oscillator (with quartic and sextic anharmonic terms) subject to a constraint [a4], [a5]. Conversely, the mappings (a1)–(a2) may be used in some dimensional reduction process for converting a dynamical system in $ \mathbf R ^ {4} $ or $ \mathbf R ^ {2} \times \mathbf R ^ {2} $ into a dynamical system in $ \mathbf R ^ {3} $.

References

[a1] P. Kustaanheimo, E. Stiefel, "Perturbation theory of Kepler motion based on spinor regularization" J. Reine Angew. Math. , 218 (1965) pp. 204
[a2] T. Iwai, "On reduction of two degrees of freedom Hamiltonian system by an action, and as a dynamical group" J. Math. Phys. , 26 (1985) pp. 885
[a3] D. Lambert, M. Kibler, "An algebraic and geometric approach to non-bijective quadratic transformations" J. Phys. A: Math. Gen. , 21 (1988) pp. 307
[a4] M. Boiteux, "Theory of nonbijective canonical transformations in mechanics: Application to the Coulomb problem" J. Math. Phys. , 23 (1982) pp. 1311
[a5] M. Kibler, T. Négadi, "Hydrogen atom in a uniform electromagnetic field as an anharmonic oscillator" Lett. Nuovo Cim. , 39 (1984) pp. 319
How to Cite This Entry:
Kustaanheimo-Stiefel transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kustaanheimo-Stiefel_transformation&oldid=22693
This article was adapted from an original article by M. Kibler (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article