# Birkhoff normal form

*Birkhoff–Gustavson normal form*

Usually, a formal normal form (cf. Normal form of a system of differential equations) for a time-independent Hamiltonian system in the neighbourhood of a stationary point (cf. 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 on $ \mathbf R ^ {2n } $ with Hamiltonian $ H \in C ^ \infty ( \mathbf R ^ {2n } ) $, i.e.

$$ {\dot{z} } = J dH ( z ) = \left ( \begin{array}{c} ( { {\partial H } / {\partial y } } ) ( x,y ) \\ ( - { {\partial H } / {\partial x } } ) ( x,y ) \\ \end{array} \right ) $$

with $ x \in \mathbf R ^ {n} $, $ y \in \mathbf R ^ {n} $, $ z = ( x,y ) $,

$$ J = \left ( \begin{array}{cc} 0 &I _ {n} \\ - I _ {n} & 0 \\ \end{array} \right ) . $$

Suppose that $ H ( 0 ) = dH ( 0 ) = 0 $. The origin is a stationary point and the Hamiltonian evaluated at the origin is

$$ H ( z ) = H _ {2} ( z ) + H _ {3} ( z ) + \dots + H _ {k} ( z ) + \dots , $$

where $ H _ {k} ( z ) $ denotes the homogeneous terms of degree $ k $. Furthermore, suppose that the matrix of the linearized system, $ D ( J dH ) ( 0 ) $, is diagonalizable (over $ \mathbf C $) with purely imaginary eigenvalues $ i \omega _ {k} $, $ - i \omega _ {k} $, $ k = 1 \dots n $. Let $ \Omega = ( \omega _ {1} \dots \omega _ {n} ) $. The eigenvalues are called resonant if they are rationally dependent, i.e. if there is an integer-valued vector $ v $ such that $ \langle {\Omega,v } \rangle = 0 $, where $ \langle {\ , } \rangle $ is the standard inner product on $ \mathbf R ^ {2n } $. The eigenvalues are non-resonant if there is no such relation.

On $ C ^ \infty ( \mathbf R ^ {2n } ) $, define Poisson brackets by $ \{ G,F \} = \langle {dF,J dG } \rangle $, where $ G,F \in C ^ \infty ( \mathbf R ^ {2n } ) $. Considering $ \mathbf R ^ {2n } $ with the symplectic structure given by the standard symplectic form (see [a1]), $ X _ {H} = \{ \ ,H \} = J dH $ is the Hamiltonian vector field generated by $ H $.

$ H $ is said to be in normal form up to order $ k $ with respect to $ H _ {2} $ if $ \{ H _ {m} ,H _ {2} \} = 0 $ for $ m = 2 \dots k $. $ H $ can be transformed into normal form using transformations of the type $ { \mathop{\rm exp} } ( X _ {F} ) $. These transformations can be considered as the time- $ 1 $ flow of the vector field $ X _ {F} ( z ) $, and therefore as symplectic diffeomorphisms on $ \mathbf R ^ {2n } $. They can also be considered as differential operators acting on the space of homogeneous polynomials of degree $ k $. These two points of view are related by the fact that $ H _ {k} \circ { \mathop{\rm exp} } ( X _ {F} ) = { \mathop{\rm exp} } ( X _ {F} ) ( H _ {k} ) $. Applying a transformation $ { \mathop{\rm exp} } ( X _ {F _ {3} } ) $ with generating function $ F _ {3} $ homogeneous of degree three gives

$$ { \mathop{\rm exp} } ( X _ {F _ {3} } ) ( H ) = H _ {2} + X _ {F _ {3} } H _ {2} + H _ {3} + \textrm{ h.o.t. } . $$

The terms of degree three are $ \{ H _ {2} ,F _ {3} \} + H _ {3} $. Consequently all terms in $ H _ {3} $ that are in the image of $ X _ {H _ {2} } $ can be removed by making the appropriate choice for the generating function $ F _ {3} $. After having made a choice for $ F _ {3} $ one gets

$$ { \mathop{\rm exp} } ( X _ {F _ {3} } ) ( H ) = {\widehat{H} } = H _ {2} + {\widehat{H} } _ {3} + {\widehat{H} } _ {4} + \textrm{ h.o.t. } , $$

with $ {\widehat{H} } _ {3} $ in some complement of $ { \mathop{\rm im} } X _ {H _ {2} } $.

Next, consider a transformation $ { \mathop{\rm exp} } ( X _ {F _ {4} } ) $ now taking a generating function $ F _ {4} $ homogeneous of degree four. This gives

$$ { \mathop{\rm exp} } ( X _ {F _ {4} } ) ( H ) = H _ {2} + {\widehat{H} } _ {3} + X _ {F _ {4} } H _ {2} + {\widehat{H} } _ {4} + \textrm{ h.o.t. } . $$

Thus, now one can remove all terms in $ H _ {4} $ that are in the image of $ X _ {H _ {2} } $. Repeating this process means that up to arbitrary degree one can remove all terms of $ H $ that are in the image of $ X _ {H _ {2} } $. This leads to the following idea of normal form: $ H = H _ {2} + H _ {3} + \dots $ is in normal form up to degree $ k $ with respect to $ H _ {2} $ if $ H _ {m} $, $ m = 3 \dots k $, are in some complement of $ { \mathop{\rm im} } ( X _ {H _ {2} } ) $. When the linearized system is diagonalizable, $ { \mathop{\rm ker} } ( X _ {H _ {2} } ) $ can be chosen as a complement to $ { \mathop{\rm im} } ( X _ {H _ {2} } ) $, giving $ \{ H _ {2} , H _ {m} \} = 0 $, $ m = 2 \dots k $. Letting $ k \rightarrow \infty $, one gets a formal normal form.

More on the basic ideas sketched above can be found in [a10], [a1], [a2]. The above idea was first used, although in an implicit way, by G.D. Birkhoff [a3] for deriving a normal form in the non-resonant case. Attention was again drawn to normal forms by F.G. Gustavson's paper [a7], where he obtained a formal normal form for the resonant cases. A similar normal form was obtained earlier by J. Moser [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 [a8]. A normal form theory for non–Hamiltonian vector fields has also been developed along the above lines by using the Lie bracket of vector fields rather than Poisson brackets of functions [a4], [a6]. The most general context to formulate the theory is that of reductively filtered Lie algebras [a11].

Normal forms are of importance in the qualitative theory of differential equations. In particular, they play a role in bifurcation theory. Using Lyapunov–Schmidt reduction and the theory of 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 [a5], [a6], [a8].

#### References

[a1] | R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978) MR0515141 Zbl 0393.70001 |

[a2] | V.I. Arnol'd, V.V. Kozlov, A.I. Neishstadt, "Mathematical aspects of classical and celestial mechanics" , Dynamical systems III , Springer (1988) (In Russian) Zbl 0785.00010 Zbl 0674.70003 Zbl 0612.70002 |

[a3] | G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. Colloqium Publ. , IX , Amer. Math. Soc. (1927) MR1555257 Zbl 53.0733.03 Zbl 53.0732.01 |

[a4] | R.H. Cushman, J.A. Sanders, "Nilpotent normal forms and representation theory of $sl(2,\RR)$" M. Golubitsky (ed.) J. Guckenheimer (ed.) , Multiparameter Bifurcation Theory , Contemp. Math. , 56 , Amer. Math. Soc. (1986) pp. 31–51 MR0855083 Zbl 0604.58005 |

[a5] | M. Golubitsky, D.G. Schaeffer, "Singularities and groups in bifurcation theory I" , Appl. Math. Sci. , 51 , Springer (1985) MR771477 Zbl 0607.35004 |

[a6] | M. Golubitsky, I. Stewart, D.G. Schaeffer, "Singularities and groups in bifurcation theory II" , Appl. Math. Sci. , 69 , Springer (1988) MR950168 Zbl 0691.58003 |

[a7] | F.G. Gustavson, "On constructing formal integrals of a Hamiltonian system near an equilibrium point" Astron. J. , 71 (1966) pp. 670–686 |

[a8] | J.C. van der Meer, "The Hamiltonian Hopf bifurcation" , Lecture Notes in Mathematics , 1160 , Springer (1985) Zbl 0585.58019 |

[a9] | J. Moser, "New aspects in the theory of Hamiltonian systems" Comm. Pure Appl. Math. , 9 (1958) pp. 81–114 Zbl 0082.40801 |

[a10] | J. Moser, "Lectures on Hamiltonian systems" , Memoirs , 81 , Amer. Math. Soc. (1968) pp. 1–60 MR0230498 Zbl 0172.11401 |

[a11] | 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 MR1278060 Zbl 0804.17018 |

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Birkhoff normal form.

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