Hamiltonian normal forms

The essence of normalization (of which the method of averaging is an example) is to use near-identity coordinate transformations to simplify a Hamiltonian system. The simplified system is called a normal form and the transformations are symplectic which means that the Hamiltonian character of the system is preserved under transformation. Introductions to normalization can be found in Arnold (1983) and Verhulst (2000). The methods have been extensively discussed in Sanders et al. (2007). A fundamental aspect is, that for a given number of degrees of freedom \(n\) and a given resonance relation between the basic frequencies \(\omega_j, \,j=1, \cdots, n\ ,\) there exist a finite number of polynomial invariants from which the normal form can be constructed.

[edit] Normalization

The equilibria of a Hamiltonian vector field coincide with the critical points of the Hamiltonian. Suppose we have found such a critical point and (without loss of generality) take it to be the origin. To express the fact that we are expanding about the equilibrium, we introduce a small parameter \(\varepsilon\) and rescale the position and momentum variables \(q \rightarrow \varepsilon q,\, p \rightarrow \varepsilon p\ .\) Consequently, the energy deviates from the equilibrium value by order \(\varepsilon^2\ .\) If the Hamiltonian is in polynomial form and starts with quadratic terms, we usually divide by \(\varepsilon^2\ .\) This implies that the "size" of a Hamiltonian term is its degree minus two. In most (but not all) cases, putting \(\varepsilon =0\ ,\) the equations of motion will reduce to linear decoupled oscillators.

Since the value of the Hamiltonian at the critical point is not important, we take it to be zero, and we expand the Hamiltonian in the local coordinates in a Taylor-expansion: \[ {H} = {H}{}_{2} + \varepsilon {H}{}_{3} + \varepsilon^2 {H}{}_{4} + \cdots \ ,\] where \({H}{}_{k} \) is homogeneous of degree \(k\) in position and momentum \((q, p)\) and \(\varepsilon \) is the scaling factor.

We shall assume \({H}{}_{2} \) to be in the following standard form \[ H_2 = \frac{1}{2} {\sum}_{ j = 1 }^{ n } \omega_j ( q_j^2 + p_j^2 ) \] with the frequencies \(\omega_j >0\ ;\) this is the so-called semisimple case. Often, other coordinate systems play a part. Instead of the position-momentum variable, action-angle variables \(\tau_j,\, \phi_j\) can be useful.

[edit] The normalization process and its implications

The notion of Poisson bracket plays an important part in calculations for Hamiltonian systems. Consider the functions of \(2n\) variables \(F(q, p)\) and \(G(q, p)\ .\) The Poisson bracket \(\{., .\}\) is defined as \[\{F, G\} = \sum_{j = 1}^{ n }\left(\frac{\partial F}{\partial q_j}\frac{\partial G}{\partial p_j} - \frac{\partial F}{\partial p_j}\frac{\partial G}{\partial q_j}\right).\] Two functionally independent functions are in involution if \(\{F, G\}=0.\) Each step of the normalization process involves solving so-called normal form (homological) equations. The first step to solve the normal form equation is to remove nonresonant (cubic) terms from \(H_3\ ,\) producing the normalized \(\bar{H}_3\) with Poisson bracket \[ \{H_2, \bar{H}_3\}=0\ .\] Carrying on to normalize to degree \(k\ ,\) we have \[ \{H_2, \bar{H}_k\}=0,\,\,k\geq 3\ .\] In each step of the normalization procedure we remove terms which are not in involution with \({H}{}_{2}\ ;\) the consequence is that the Hamiltonian in normal form has \({H}{}_{2}\) as additional integral. The implication is that two-degree-of-freedom systems in normal form are integrable. More generally: Consider the \(n\) degree-of-freedom, time-independent Hamiltonian \[\tag{1} H(p,q) = \frac{1}{2}\sum_{j=1}^n\omega_j\left(p_j^2 + q_j^2 \right) + \varepsilon H_3 + \varepsilon^2 H_4 + \varepsilon^3 \cdots. \]

In practice we have to stop the normalization process at a certain degree \(m\ :\) \[\tag{2} \bar{H}= H_2 + \varepsilon \bar{H}_3 + \varepsilon^2 \bar{H}_4 + \cdots + \varepsilon^{m-2} \bar{H}_m. \]

Because of the construction we have the following results:

• \({H}_2\) and \(\bar{H}\) are (exact) integrals of the Hamiltonian system induced by \(\bar{H}\ .\) So the normal form has at least two integrals.
• \(\bar{H}\) is conserved for the original Hamiltonian system (1) with error \(O(\varepsilon^{m-1})\) for all time.
• \(H_2\) is conserved for the original Hamiltonian system (1) with error \(O(\varepsilon)\) for all time.
• If there are additional symmetries, then there may also be more integrals, see Tuwankotta and Verhulst (2000). Any additional integrals of the normal form have slightly weaker error estimates. Explicitly, suppose that \(F(p, q)\) is an independent integral of the truncated Hamiltonian system (2), we have for the solutions of the original Hamiltonian system (1), expressed in the original variables, the estimate \[ F(p, q) - F(p(0), q(0))= O(\varepsilon^{m-1}t)\ .\]

An important consequence is the following statement: if the flow induced by the truncated Hamiltonian (2) is completely integrable, the flow of the original Hamiltonian (1) is approximately integrable in the sense described above. In this case the original system is called formally integrable. This implies that the irregular, chaotic component in the flow of the original Hamiltonian is limited by the given error estimates and must be a small-scale phenomenon on a long timescale. For details see Sanders et al. (2007).

To determine whether a normal form of a Hamiltonian system with three or more degrees of freedom is integrable or not, is not easy. The earliest proofs are of a negative character, showing that integrals of a certain kind are not present. Nevertheless, this is still a useful approach, for instance showing that algebraic integrals to a certain degree do not exist.

[edit] Two degrees of freedom, examples

The resonance studied extensively in the literature is the case \(\omega_1: \omega_2 = 1:2\ .\) This is not surprising as in this resonance case we have only to normalize to \(H_3\) to obtain significant results.

Other prominent resonances are \(1:1\) and \(1:3\ .\) They involve normalization at least to \(H_4\ .\) Consider the case \(\omega_1: \omega_2 = k:l\) with \(k, l\) relative prime, \(k>l\ ,\) thus excluding the resonance \(1:1\ .\) In this nearly general case the normal form of the Hamiltonian is determined by four polynomial invariants. The normalized Hamiltonian becomes, in action-angle coordinates, \[ \bar{H} = {\omega}_1 {\tau}_1 + {\omega}_2 {\tau}_2 + {\varepsilon}^{ k + l - 2 } | D | \sqrt{ 2 \tau_1} ( 2 {\tau}_2 {)}^{ \frac{k}{2} } \cos ( l {\phi}_1 - k {\phi}_2 + \alpha ) \ :\]

\[ + \varepsilon^2 ( A \tau_1^2 + 2 B {\tau}_1 {\tau}_2 + C \tau_2^2 ) + \cdots \]

with constants \(|D|, \alpha, A, B, C\ .\) The first resonant term arrives from \(H_{k+l}\) at \(O({\varepsilon}^{ k + l - 2 })\ ,\) the dots represent terms of size smaller that \(O({\varepsilon}^{ 2 })\) and depend on \(\tau_1, \tau_2\) only.

As a classical example we consider the elastic pendulum, a pendulum where the suspending, inflexible string is replaced by a linear spring. In particular we will look at the higher order resonances defined by \(k+l \geq 5\ .\) It turns out there are two domains in phase-space where the dynamics is very different and is characterized by different timescales:

• The resonance domain \(D_I \ ,\) which is a neighborhood of the resonance manifold \(M \ .\) In terms of singular perturbations, this is the inner boundary layer. Introducing the distance \(d ( P , M ) \) for a point \(P \) on the energy manifold to the manifold \(M \) we have

\[ D_I = \{ P | d ( P , M ) = O( \varepsilon^{\frac{ k + l - 4 }{2} })\},\,\,\,k+l \geq 5. \]

• The remaining part of phase-space, outside the resonance domain, is \(D_o\ ,\) the outer domain. In the domain \(D_o \ ,\) there is, to a certain approximation, hardly any exchange of energy between the two degrees of freedom.

Following Tuwankotta and Verhulst (2000) we summarize some results in table .

The table presents the most prominent higher-order resonances of the elastic pendulum with lowest order resonant terms \(O(\varepsilon^{k+l-2})\ .\) The third column gives the size of the resonance domain in which the resonance manifold \(M\) is embedded, while in the fourth column we find the timescale of interaction in the resonance domain.

An example of a Poincare map for the \(1:6\)-resonance is shown in Figure 1.

Figure 1: The Poincare map for the \(1:6\)-resonance of the elastic pendulum (\(\varepsilon= 0.75\ ,\) large for illustration purposes). In the resonance domain, the saddles are connected by heteroclinic cycles and inside the cycles are centers see Tuwankotta and Verhulst (2000), courtesy SIAP.

[edit] Three degrees of freedom, an example

In the case of three degrees of freedom we still have two integrals of the normal form, \(\bar{H}\) and \(H_2\ ,\) but three are needed for the system to be integrable. To find a third integral is a nontrivial problem: in some cases it can be shown to exist, but there are also cases where it has been shown that a third analytic integral does not exist, see Duistermaat (1984), Van der Aa and Verhulst (1984) and Hoveijn and Verhulst (1990). This makes the global description of the phase-flow of the normalized system essentially more difficult in the case of three degrees of freedom. For a survey of results see Sanders et al. (2007).

As an example we discuss the \(1:2:1\)–resonance with general \(H_3\) and the case with certain symmetries. It turns out that by normalizing, the \(56\) constants (parameters) of the general \({H}_{3} \) are reduced to \(6\) real constants \(a_1, \cdots, a_6\ .\) In action-angle variables the normal form of \(H_2 + \varepsilon {H}_{3} \) is \[ \bar{H} = \tau_1 + 2 \tau_2 + \tau_3 + 2 \varepsilon \sqrt{ 2 \tau_2 } [ a_1 \tau_1 \cos ( 2 \phi_1 - \phi_2 - a_2 ) \ :\]

\[ + a_3 \sqrt{ \tau_1 \tau_3 } \cos ( \phi_1 - \phi_2 + \phi_3 - a_4 ) + a_5 \tau_3 \cos ( 2 {\phi}_3 - {\phi}_2 - a_6 ) ] \ .\]

Analyzing the critical points of the equation of motion we find in the general case 7 periodic orbits (for each value of the energy) of the following three types:

1. one unstable normal mode in the \(\tau_2 \)-direction;
2. two stable periodic solutions in the \(\tau_2 = 0 \) hyperplane;
3. two stable and two unstable periodic solutions in general position (i.e. \(\tau_1 \tau_2 \tau_3 > 0 \)).

The results are displayed in (Figure 2).

Figure 2: Action simplex for the the \( 1 : 2 : 1 \)–resonance; dots indicate periodic solutions, normal modes are at the vertices. The stability characteristics are denoted by \( E \) (conjugate pair of imaginary eigenvalues) and \( H \) (conjugate pair of real eigenvalues).

It is shown in Duistermaat (1984) that the normal form without additional assumptions on the six free constants is non-integrable.

[edit] Symmetry assumptions

In applications, assumptions arise which often induce certain symmetries in the Hamiltonian. Such symmetries cause special bifurcations and other phenomena which are of practical interest. Now consider some of the consequences of the assumption of discrete (mirror) symmetry.

We consider the case of discrete symmetry in \(p_1 , q_1 \) or \(p_3 , q_3 \) (or both). In the normal form this results in \(a_3 = 0 \ ,\) since the Hamiltonian has to be invariant under M, defined by \[ M {\phi}_i = {\phi}_i + \pi ,\quad i = 1 , 3 \ .\] Analysis of the critical points of the averaged equation shows that no periodic orbits in general position exist. There are still 7 periodic orbits, but the four in general position have moved into the \(\tau_1 = 0 \) and \(\tau_3 = 0 \) hyperplanes; see the action simplex in Figure 3.

Figure 3: Action simplex for the \( 1 : 2 : 1 \)–resonance, discrete symmetric in \(q_1, p_1\ ,\) \(q_3, p_3\) or both.

[edit] Many degrees of freedom

There are not many results for \(n\) degrees of freedom normal forms of Hamiltonian systems with \(n \geq 4\ .\)

A remarkable result is that the normal form of \(H_2+H_3\) of the \(1:2, \cdots, 2\)–resonance is integrable with \(n\) arbitrary, see Van der Aa and Verhulst (1984).

An important paper is Rink (2001) where it is shown that to a certain order the normal form of the Fermi-Pasta-Ulam chain is integrable. This finally explains rigorously a classical problem: the recurrence behavior of this chain at low energy levels.

[edit] References

V.I. Arnold, (1983), Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag.

J.J. Duistermaat, (1984), Non-integrability of the 1:1:2-resonance, Ergodic Theory and Dynamical Systems 4, pp. 553-568.

I. Hoveijn and F. Verhulst (1990), Chaos in the \(1:2: 3\) Hamiltonian normal form, Physica D, 44, pp. 397–406.

B. Rink, (2001) Symmetry and resonance in periodic FPU chains, Comm. Math. Phys., 218, pp. 665–685.

J.A. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems, 2d. ed., Applied Math. Sciences 59, Springer (2007).

J.M. Tuwankotta and F. Verhulst, (2000) Symmetry and resonance in Hamiltonian systems, SIAM J. Appl. Math., 61, pp. 1369–1385.

E. Van der Aa and F. Verhulst, (1984) Asymptotic integrability and periodic solutions of a Hamiltonian system in \(1:2:2\)-resonance, SIAM J. Math. Anal. 15, pp. 890–911.

Ferdinand Verhulst, (2000) Nonlinear Differential Equations and Dynamical Systems, Springer-Verlag.

Internal references

• Jan A. Sanders (2006) Averaging. Scholarpedia, 1(11):1760.
• John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
• James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.
• James Murdock (2006) Normal forms. Scholarpedia, 1(10):1902.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

[edit] External Links

• Ferdinand Verhulst's website

[edit] See Also

Averaging, Hamiltonian Systems, Normal Forms