View source for Kustaanheimo-Stiefel transformation

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

<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>

subject to the constraint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110210/k1102103.png" /> with

<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>

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.

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.

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>

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>

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

<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>

(<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" />).

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" />.

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" />)

<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>

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>

<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>

<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>

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).

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" />.

====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>