View source for Birkhoff normal form

''Birkhoff–Gustavson normal form''

Usually, a formal normal form (cf. [[Normal form|Normal form]] of a system of differential equations) for a time-independent [[Hamiltonian system|Hamiltonian system]] in the neighbourhood of a stationary point (cf. [[Normal form|Normal form]] in a neighbourhood of a fixed point) for which the linearized system at the stationary point has only purely imaginary eigenvalues.

Consider a [[Hamiltonian system|Hamiltonian system]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b1105701.png" /> with [[Hamiltonian|Hamiltonian]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b1105702.png" />, i.e.

<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/b/b110/b110570/b1105703.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b1105704.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b1105705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b1105706.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/b/b110/b110570/b1105707.png" /></td> </tr></table>

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b1105708.png" />. The origin is a stationary point and the Hamiltonian evaluated at the origin is

<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/b/b110/b110570/b1105709.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057010.png" /> denotes the homogeneous terms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057011.png" />. Furthermore, suppose that the matrix of the linearized system, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057012.png" />, is diagonalizable (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057013.png" />) with purely imaginary eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057016.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057017.png" />. The eigenvalues are called resonant if they are rationally dependent, i.e. if there is an integer-valued vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057018.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057020.png" /> is the standard [[Inner product|inner product]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057021.png" />. The eigenvalues are non-resonant if there is no such relation.

On <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057022.png" />, define [[Poisson brackets|Poisson brackets]] by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057023.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057024.png" />. Considering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057025.png" /> with the [[Symplectic structure|symplectic structure]] given by the standard symplectic form (see [[#References|[a1]]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057026.png" /> is the Hamiltonian vector field generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057027.png" />.

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057028.png" /> is said to be in normal form up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057029.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057030.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057032.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057033.png" /> can be transformed into normal form using transformations of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057034.png" />. These transformations can be considered as the time-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057035.png" /> flow of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057036.png" />, and therefore as symplectic diffeomorphisms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057037.png" />. They can also be considered as differential operators acting on the space of homogeneous polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057038.png" />. These two points of view are related by the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057039.png" />. Applying a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057040.png" /> with generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057041.png" /> homogeneous of degree three gives

<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/b/b110/b110570/b11057042.png" /></td> </tr></table>

The terms of degree three are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057043.png" />. Consequently all terms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057044.png" /> that are in the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057045.png" /> can be removed by making the appropriate choice for the generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057046.png" />. After having made a choice for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057047.png" /> one gets

<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/b/b110/b110570/b11057048.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057049.png" /> in some complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057050.png" />.

Next, consider a transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057051.png" /> now taking a generating function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057052.png" /> homogeneous of degree four. This gives

<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/b/b110/b110570/b11057053.png" /></td> </tr></table>

Thus, now one can remove all terms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057054.png" /> that are in the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057055.png" />. Repeating this process means that up to arbitrary degree one can remove all terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057056.png" /> that are in the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057057.png" />. This leads to the following idea of normal form: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057058.png" /> is in normal form up to degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057059.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057060.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057062.png" />, are in some complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057063.png" />. When the linearized system is diagonalizable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057064.png" /> can be chosen as a complement to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057065.png" />, giving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057067.png" />. Letting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057068.png" />, one gets a formal normal form.

More on the basic ideas sketched above can be found in [[#References|[a10]]], [[#References|[a1]]], [[#References|[a2]]]. The above idea was first used, although in an implicit way, by G.D. Birkhoff [[#References|[a3]]] for deriving a normal form in the non-resonant case. Attention was again drawn to normal forms by F.G. Gustavson's paper [[#References|[a7]]], where he obtained a formal normal form for the resonant cases. A similar normal form was obtained earlier by J. Moser [[#References|[a9]]]. Gustavson emphasized that by normalizing up to infinite order extra formal integrals are obtained. The normal form has more symmetry than the original system.

The above ideas have been extended to the case where the linearized system has purely imaginary eigenvalues but is not diagonalizable [[#References|[a8]]]. A normal form theory for non–Hamiltonian vector fields has also been developed along the above lines by using the [[Lie bracket|Lie bracket]] of vector fields rather than [[Poisson brackets|Poisson brackets]] of functions [[#References|[a4]]], [[#References|[a6]]]. The most general context to formulate the theory is that of reductively filtered Lie algebras [[#References|[a11]]].

Normal forms are of importance in the [[Qualitative theory of differential equations|qualitative theory of differential equations]]. In particular, they play a role in [[Bifurcation|bifurcation]] theory. Using Lyapunov–Schmidt reduction and the theory of [[Singularities of differentiable mappings|singularities of differentiable mappings]] one can determine which number of terms of the normal form is sufficient to describe the bifurcation of stationary points and periodic solutions up to topological equivalence [[#References|[a5]]], [[#References|[a6]]], [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Abraham,   J.E. Marsden,   "Foundations of mechanics" , Benjamin/Cummings  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.I. Arnol'd,   V.V. Kozlov,   A.I. Neishstadt,   "Mathematical aspects of classical and celestial mechanics" , ''Dynamical systems III'' , Springer  (1988)  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.D. Birkhoff,   "Dynamical systems" , ''Amer. Math. Soc. Colloqium Publ.'' , '''IX''' , Amer. Math. Soc.  (1927)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.H. Cushman,   J.A. Sanders,   "Nilpotent normal forms and representation theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110570/b11057069.png" />"  M. Golubitsky (ed.)  J. Guckenheimer (ed.) , ''Multiparameter Bifurcation Theory'' , ''Contemp. Math.'' , '''56''' , Amer. Math. Soc.  (1986)  pp. 31–51</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Golubitsky,   D.G. Schaeffer,   "Singularities and groups in bifurcation theory I" , ''Appl. Math. Sci.'' , '''51''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Golubitsky,   I. Stewart,   D.G. Schaeffer,   "Singularities and groups in bifurcation theory II" , ''Appl. Math. Sci.'' , '''69''' , Springer  (1988)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  F.G. Gustavson,   "On constructing formal integrals of a Hamiltonian system near an equilibrium point"  ''Astron. J.'' , '''71'''  (1966)  pp. 670–686</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.C. van der Meer,   "The Hamiltonian Hopf bifurcation" , ''Lecture Notes in Mathematics'' , '''1160''' , Springer  (1985)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  J. Moser,   "New aspects in the theory of Hamiltonian systems"  ''Comm. Pure Appl. Math.'' , '''9'''  (1958)  pp. 81–114</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J. Moser,   "Lectures on Hamiltonian systems" , ''Memoirs'' , '''81''' , Amer. Math. Soc.  (1968)  pp. 1–60</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  J.A. Sanders,   "Versal normal form computations and representation theory"  E. Tournier (ed.) , ''Computer Algebra and Differential Equations'' , ''London Math. Soc. Lecture Notes'' , '''193''' , Cambridge Univ. Press  (1994)  pp. 185–210</TD></TR></table>