aa r X i v : . [ m a t h . D S ] J u l Higher order normal modes
Giuseppe Gaeta ∗ Dipartimento di Matematica,Universit`a degli Studi di Milano,via Saldini 50, 20133 Milano (Italy); and
SMRI, 00058 Santa Marinella (Italy)
Sebastian Walcher † Mathematik A, RWTH Aachen, 52056 Aachen (Germany)
Normal modes are intimately related to the quadratic approximation of a potential at its hy-perbolic equilibria. Here we extend the notion to the case where the Taylor expansion for thepotential at a critical point starts with higher order terms, and show that such an extension sharessome of the properties of standard normal modes. Some symmetric examples are considered in detail.
Dedicated to James Montaldi on his 25+something anniversary
I. INTRODUCTION
The concept of normal modes is a fundamental one inthe study of Hamiltonian dynamical systems [1–4]; it isbased on the quadratic part of the Taylor expansion ofthe Hamiltonian around a non-degenerate (isolated andhyperbolic) stable equilibrium, and under certain fairlygeneral assumptions it can be conveniently employed alsoin considering higher order expansions of the Hamilto-nian. That is, under such assumptions (including a non-resonance condition) normal modes persist, at least lo-cally, when one considers also higher order terms – or fornonlinear Hamiltonian dynamics [5, 6] – as also studiedby James in a series of papers [7, 8].Normal modes span the dynamics of the quadraticHamiltonian, i.e. any motion for this can be describedas the superposition of normal modes; more precisely,normal modes define invariant lines in a neighborhood U ≃ R n of the equilibrium in the position space, andthese provide a basis for U .Here we want to discuss the (partial) generalization ofthis concept to the case where the equilibrium is stableand isolated, but not hyperbolic. This resonates withsome recent studies in a different field, i.e. liquid crystals .More precisely, we will find some connection with recentstudies of liquid crystals described by higher order tensororder parameter [9–12].We will show that normal modes – in the sense of in-variant lines – also exist for the dynamics of fully nonlin-ear homogeneous Hamiltonian systems (more precisely,we will confine ourselves to systems with a natural Hamil-tonian H = T + V and a potential V ( x ) homogeneousof degree k > ∗ [email protected] † [email protected] system) are met, it is not possible to describe a genericmotion as a (nonlinear) superposition of these higher or-der normal modes. II. NORMAL MODES
We will consider natural Hamiltonians H = T + V = p m + V ( q ) (1)with q ∈ R n ; the potential V is such that the origin isan isolated stable equilibrium point.Our discussion could be extended to encompass a gen-eral Hamiltonian on a symplectic manifold M of dimen-sion 2 n , and a non-degenerate equilibrium point p forthis; however our considerations will be local , so by Dar-boux’ theorem we can always consider a standard sym-plectic form.The physical interest of the natural Hamiltonian casesurely justifies considering this special case. Moreoverthis will help keeping the discussion and notation simpler,focusing on the key points.[32]We can then consider the Taylor expansion of V around q = 0; we thus have, in view of the non-degeneracy as-sumption, V ( q ) = V + 12 ( q , A q ) + h . o . t . , where V = V ( ) is a constant – which can be set to zerowith no loss of generality – and A is a symmetric tensorof order two, i.e. a symmetric matrix, A ij = (cid:18) ∂ V∂q i ∂q j (cid:19) . (2)We will now truncate the potential at order two, i.e.omit the higher order terms and deal just with V =1 / q , A q ); the relevant case for normal modes is thatwhere the origin is a stable fixed point, hence A is a pos-itive definite matrix.This matrix A will have eigenvalues λ i and correspond-ing (say normalized) eigenvectors φ ( i ) ; these in turn cor-respond to normal modes in a very well known way, whichwe briefly recall to fix notation [1–4].1. The eigenvectors φ ( i ) identify invariant lines : ifa motion has initial conditions q (0) = α (0) φ ( i ) ,˙ q (0) = β (0) φ ( i ) , then q ( t ) = α ( t ) φ ( i ) (and hence˙ q ( t ) = β ( t ) φ ( i ) ) for all t .2. The eigenvectors φ ( i ) ( i = 1 , ..., n ) span the linearspace R n ;3. A general motion, i.e. one with generic initial con-ditions q (0), ˙ q (0) can then be decomposed in termsof normal modes. In fact, by the property just men-tioned, there will exist constants α i (0), β i (0) suchthat q (0) = n X i =1 α i (0) φ ( i ) ; ˙ q (0) = n X i =1 β i (0) φ ( i ) . The evolution of the system will then just be de-scribed by q ( t ) = n X i =1 α i ( t ) φ ( i ) ; ˙ q ( t ) = n X i =1 β i ( t ) φ ( i ) , with α i ( t ) and β i ( t ) being exactly as in the nor-mal mode solution with initial data { α i (0) , β i (0) } .In other words, any solution for this quadraticHamiltonian will be a linear superposition of nor-mal modes solutions .4. As a consequence of the previous item, in the casethere exist several normal modes associated withthe same frequency, the whole linear space spannedby the associated eigenvectors is made of eigenvec-tors and hence of normal modes with the given fre-quencies.As anticipated, we will find that properties (1) and (2)extend to the higher order case, while properties (3) and(4), related to linearity, do not. III. EIGENVECTORS OF TENSORS
We now consider the case where q = 0 is an isolatedcritical point, but we will drop the hypothesis it is hy-perbolic. Actually, we want to assume that the matrix A defined in (2) is identically zero , i.e. that the potential V is fully nonlinear at the considered critical point.Thus we have to go further in the Taylor expansion –even if we want to stop at the first significant term – and actually if we want the critical point to be a stable oneit is needed to consider a fourth order[33] tensor T , T ijkℓ := 124 (cid:18) ∂ H∂q i ∂q j ∂q k ∂q ℓ (cid:19) . (3)Thus, disregarding higher order terms, we want to con-sider a potential expressed in local coordinates which ishomogeneous of order four and which has an isolatedequilibrium at the origin. After getting rid of inessen-tial (additive and multiplicative) constants, this can bewritten as V = T ijkℓ q i q j q k q ℓ . (4)The “direct extension” of properties (1) and (2) re-called above for usual normal modes would requireto consider eigenvectors for the fourth order tensor T (rather than the matrix A ) and see if these span thewhole space.The notion of eigenvector of tensors is not so wellknown in general, but it can be defined and it has beenstudied (with a revival of interest in recent times) bothfrom the algebraic point of view [13–15] and in connec-tion with dynamics [16–22] (see also [23, 24]); as alreadymentioned, it has also been recently considered in connec-tion with critical points of a constrained potential withapplications in the Physics of Liquid Crystals [9–12].We will not introduce and discuss the notion of eigen-vectors of tensors right away, but we will start with ageneral discussion of one dimensional eigenspaces of ho-mogeneous polynomial maps. A. Eigenspaces of homogeneous polynomial vectorfields
We will simultaneously discuss the real and the com-plex case in this section. We first recall a classical resultfrom Algebraic Geometry; see [25] for more about this.
Theorem 1 (Bezout).
Let { f , ...f m } be homogeneouspolynomials of degree n , f i : C m → C . Then the numberof common zeros of the f i in projective space P m is eitherinfinite or equal to n m , counting multiplicities. For real polynomials this theorem provides only limitedinformation about the nature of these critical points; inparticular, we cannot infer how many of these are real .[34]But, since the complex conjugate of every solution is alsoa solution (with the same multiplicity, as can be shown),we conclude that a real solution exists whenever n is odd(and there are only finitely many solutions).We now denote by K the real or complex numbers andconsider a homogeneous polynomial map B : K q → K q , x B ( x )... B q ( x ) (5)with each B i homogeneous of degree p ≥
2. In coordi-nates we have B i = B ij ...j q x j ...x j q . (6)We stipulate that the coefficients B ij ...j q are symmet-ric with respect to permutations of j , . . . , j q ; this choicemakes the coefficients unique. It is possible to identify B with its coefficients, and consider it as an element ofthe coefficient space (which is just some K N .) A formaldefinition follows next. Definition.
Let B be given as in (5). Then a nonzero v ∈ K q is called an eigenvector of B if there exists an α ∈ K such that B ( v ) = αv . Some remarks are in order here:1. Every nonzero scalar multiple of an eigenvector v is also an eigenvector. Therefore it makes sense tocall K v an eigenspace of B .2. On the other hand, the notion of eigenvalue is prob-lematic for homogeneous maps of degree >
1, since B ( βv ) = β p αv = ( β p − α ) · ( βv )for any β , and thus one may replace the “eigen-value” α by β p − α .3. One may use this property to scale “eigenvalues”to be either 0 or 1 in the complex case, and also inthe real case when the degree p is even; for the realcase with odd p one may achieve “eigenvalue” 0, 1or − Theorem 2.
Let B be as in (5), and K = C . Then thefollowing hold.(i) The number of one-dimensional eigenspaces of B iseither infinite or equal to N R = p q − p − , counting multiplicities.(ii) Test for multiplicity one: Let v ∈ C q be nonzeroand B ( v ) = αv with some α = 0 . Then C v corre-sponds to a solution of multiplicity one if and onlyif the Jacobian DB ( v ) does not admit the eigen-value α . (iii) If the equation B ( x ) = 0 has only the trivial so-lution x = 0 then the number of one-dimensionaleigenspaces of B is finite.(iv) There is an open and dense subset of coefficientspace such that every B with coefficients in this sub-set admits a basis for C q of eigenvectors.(v) There is an open and dense subset of coefficientspace such that every B with coefficients in thissubset admits exactly N R different one-dimensionaleigenspaces.(vi) If the coefficients B ij ...j q are algebraically inde-pendent over the rational number field then theequation B ( x ) = 0 has only the trivial solution,and B admits exactly N R different one-dimensionaleigenspaces. Proof.
We just sketch some arguments for the proofs,and give references. (See also the review [24].) The firstassertion is due to Rohrl [16, 18], the second is derivedfrom the familiar criterion for multiplicity one (i.e. in-vertibility of the linearization). The third assertion isshown e.g. in [21], based on the fact that any projectivevariety of positive dimension intersects every hyperplane.The fourth and fifth assertion go essentially back to Rohrl[19], although the full statement given in this paper is notcorrect, and the proof has to be modified. See the Ap-pendix of [22] for a full discussion. The last statementis again due to Rohrl [16]; the algebraic independencecondition guarantees that the multiplicity one criterionis always satisfied. ⊙ We turn to the real setting. For proofs and referencesconcerning the following statements we refer to [24]. (Insome of the proofs analytic techniques enter the picture.)
Theorem 3.
Let B be as in (5), and K = R . Then thefollowing hold.(i) If the dimension q is odd then there exists a one-dimensional real eigenspace of B .(ii) If the dimension q is even and the degree p of B is even then there exists a one-dimensional realeigenspace of B .(iii) If the complexification admits finitely many one-dimensional eigenspaces, then the number of realeigenspaces is congruent to N R modulo . B. Radial solutions of fully nonlinear dynamicalsystems
Rohrl was interested in (one-dimensional) eigenspacesof homogeneous polynomial maps because they give riseto special solutions of an associated differential equation,similar to the linear case. Rohrl considered first orderdifferential equations, and we paraphrase his result here(see the original work in [16] and [17]).
Theorem 4.
Let B be as in (5), and consider the ordi-nary differential equation ˙ x = B ( x ) in K n . Then every one-dimensional eigenspace of B isan invariant set for this differential equation.For nonzero v with B ( v ) = αv , some α ∈ K , oneobtains solutions with the ansatz x ( t ) = ξ ( t ) · v , whichleads to the one-dimensional equation ˙ ξ = αξ p . There is a straightforward extension of this approachto second order equations.
Theorem 5.
Let B be as in (5), and consider the secondorder ordinary differential equation in K n ¨ x = B ( x ) . (i) Then every nonzero v with B ( v ) = αv , some α ∈ K , gives rise to special solutions of the differentialequation, via the ansatz x ( t ) = γ ( t ) · v , which leadsto the one-dimensional second order equation ¨ γ = αγ p .(ii) In case K = R a phase plane analysis of the asso-ciated system ˙ y = y , ˙ y = α (cid:0) y (cid:1) p yields the first integral ψ = 2 α (cid:0) y (cid:1) p +1 − ( p +1) (cid:0) y (cid:1) . When α = 0 then the level sets of ψ arebounded if and only if α < and p is odd. In thiscase, the level sets are orbits of periodic solutionsof the second order system; in all other cases ev-ery nonconstant solution obtained by the ansatz isunbounded. Proof (Sketch). Note that 0 is the only stationary pointof the system. It is elementary to see that the level setsare bounded, hence compact, if and only if α < p is odd. By standard Poincar´e-Bendixson theory forplanar systems, the only possible limit sets of points ona compact level set containing more than one point areclosed orbits, thus they must coincide with the level setsby connectedness. In every other case, each level set thatcontains more than one point is unbounded, and any α or ω limit point of a solution starting on such a level setconsists of a stationary point, by Poincar´e-Bendixson.Unboundedness of the solution follows. ⊙ Remark 1.
It is worth noting that in the case p = 2the special solutions from the theorem correspond to awell-known class of special functions. Indeed, the secondorder equation ¨ z = α z becomes, upon employing the first integral,˙ z = 23 α z + c with some constant c , and this is the differential equationfor a Weierstrass ℘ -function. So, elliptic functions appearin a natural manner. ⊙ C. Critical points on the unit sphere
In this section we are only interested in the real case K = R . We consider a symmetric tensor T i ...i n of order n on R m , and we associate with this a polynomial P n ( x ) := T i ...i n x i ...x i n ;note that here the dimension m of the ambient space andthe degree n of the polynomial are not related. In the fol-lowing, P n will also be called the potential ; it will also bejust denoted as P , when we do not need to emphasize itsdegree. Conversely, as is well known, the algebra of ho-mogeneous polynomials of degree n in R m is isomorphicto the algebra of symmetric tensors of the same order n over R m .Consider now the gradient of P n , i.e. the m -dimensional vector ∇ P n = (cid:18) ∂P n ∂x , ... , ∂P n ∂x m (cid:19) ;here of course each component is a homogeneous functionof degree ( n −
1) in the x i .We define an eigenvector of the tensor T to be an eigen-vector of ( ∇ P n ).For eigenvectors of tensors we obtain an improvementof earlier results concerning the real case. We collectthem in the following. Theorem 6.
Let v ∈ R m be of Euclidean norm one.Then(i) v is an eigenvector of ∇ P n if and only if v is acritical point of P n on the unit sphere S m − ⊂ R m .(ii) The real homogeneous gradient map ∇ P n admits areal eigenvector.(iii) If we have finitely many critical points x k , then thesum of the indices of all critical points is equal tothe (Euler-Poincar´e) characteristic χ ( S m − ) of theambient sphere, thus: X k ι ( x k ) = χ ( S m − ) = (cid:26) if m is odd if m is even (7) Proof.
To prove the first assertion ( i ), introduce a La-grange multiplier λ and consider the modified potential b P ( x ) := P ( x ) − λ | x | . (8)The gradient of b P is given by ∇ b P = ∇ P − λ x ;hence the solutions to ∇ b P = 0 are exactly the points onthe unit sphere such that ( ∇ P )( x ) is collinear to x ;these identify (unit length) eigenvectors of T and henceeigenspaces. The second assertion ( ii ) is then clear since P n attains maximum and minimum on the compact unitsphere. The notion of index of a critical point a is definedin [26] via the Brouwer degree. When a is nondegenerate(i.e., the derivative at a is invertible) it is equal to thesign of its Jacobian determinant. For the proof of thelast assertion ( iii ) see [26, 27]. ⊙ In view of the results above we will adopt the con-vention that whenever reference is made to eigenvaluesassociated with eigenvectors, it is understood that theseare associated with eigenvectors of unit length.[35]We also note that (like for matrices) if T depends onparameters then the eigenvectors and eigenvalues willin general depend on these parameters. We anticipatethat also the number of independent eigenvectors (thatis, eigenspaces) can vary depending on such parameters.This situation is met already in the simplest nontriv-ial case, i.e. for completely symmetric cubic tensors inthree-dimensional space[36]. For a full discussion of thiscase we refer to [10] (with a more physical approach) andespecially to [12].The situation is specially simple when the ambientspace is R (a case of clear physical interest!) and hencewe work in S ⊂ R and all critical points are non-degenerate. In this case maxima and minima have index+1 while saddle points have index -1.In any case, it is elementary to classify all possible com-binations of non-degenerate critical points compatiblewith the formula (7) and with Bezout’s theorem, whichprovides the maximal number of real critical points[37].In view of the special nature of the potential consideredhere (homogeneous of degree n ), it is either even or odd,depending on the parity of n , hence all critical points aredual to each other under reflection. Thus in the odd casethere will be as many maxima as minima while in theeven one these numbers will necessarily be even, as wellas (in all cases) the number of saddles of any given index.For example, in [12] it is argued that in the case of a cu-bic potential (and hence a quadratic gradient mapping)in three-dimensional space (and hence a two-dimensionalambient sphere) with non-degenerate extremals (but pos-sibly degenerate saddles) only the possibilities listed inTable I arise.[38]Here we are more interested in even degree, and es-pecially in quartic potentials, and hence cubic gradientmappings. In this case, already for ambient space R we get up to (3 −
1) = 26 critical points and a com-plete classification would make little sense. We remark,however, that Table I is still valid in that it concernstopological features; on the other hand, in this context
Max Min S S S NCP1 1 0 0 0 22 2 2 0 0 63 3 4 0 0 103 3 0 2 0 84 4 6 0 0 144 4 2 2 0 124 4 0 0 2 10TABLE I: Different possibilities for the number and type ofcritical points in the case of a cubic potential in three dimen-sions; here “Max”and “Min” represent the number of max-ima and minima, while “ S k ” represents the number of saddlepoints of index − k . Finally, “NCP” is the total number ofcritical points. it does not provide a complete classification of the possi-ble situations, but only of those with no more than fourmaxima or minima.We also note that when the ambient space is R , andhence the relevant sphere is just a circle S , then wealways have as many maxima as minima, whose numberis of course limited by the degree of the potential; and ofcourse no saddles. E.g. for p = 3 and q = 2 we have atmost four maxima and four minima.Some simple Examples will be considered in detail inSection IV. IV. EXAMPLE. INVARIANT LINES;EXISTENCE AND NUMBER OF HIGHERORDER NORMAL MODES
As the simplest possible example of the situation wehave been studying, we consider a point particle of mass m = 1 in R (with cartesian coordinates x, y ) evolv-ing under the action of a quartic potential (a cubic onewould not satisfy the requirement that the origin is astable equilibrium); in order to reduce the complexity ofthe potential (and hence of the analysis) we assume itdepends on x and y only through their squares. That is, V ( x, y ) = a x + b y + 2 c x y . (9)Note that this is symmetric under Z × Z (these actingas reflections in x and in y ); if we wish to require this tobe also invariant under the Z involution exchanging x and y then we should require b = a . In this case it wouldbe convenient, with a suitable redefinition of constants,to rewrite this as V ( x, y ) = α ( x + y ) + β x y . (10)We will refer to these cases as the lower symmetry andthe higher symmetry cases respectively. A. The higher symmetry case
We will first consider the case where the system is Z × Z × Z symmetric, i.e. the potential is in the form (10).Note that in order to have a stable point at the origin,the coefficients α should be positive; we will assume thisto be the case from now on.Moreover, we can always rescale V (which amounts toa rescaling of time) and choose α = 1. Then in order tohave a minimum at the origin we can ask β > −
4; thisis obtained by looking at the behavior of the potential V m ( x ) := V ( x, mx ) along all lines y = mx (including the m = ∞ case, i.e. the y axis). We will moreover assume β = 0 to avoid the fully degenerate case with rotationalsymmetry.When we pass to polar coordinates (we consider θ ∈ ( − π, π ], and of course ρ ∈ [0 , ∞ )) x = ρ cos( θ ) , y = ρ sin( θ )and constrain the potential on the unit circle ρ = x + y = 1, call it W ( θ ), we get W = 18 [8 − β cos(4 θ ) + β ] . (11)Thus we have dWdθ = 12 β sin(4 θ ) ;critical points are obtained for θ = θ k := k π , | k | ≤ . That is, we get eight critical points on the unit circle,corresponding to four invariant lines; this applies for any nonzero value of β (as already remarked β = 0 is thefully rotationally invariant and hence infinitely degener-ate case; we excluded this from our considerations).The stability of the critical points is controlled by (cid:20) d Wdθ (cid:21) θ k = 2 β cos(4 θ k ) . Thus we have a bifurcation at β = 0; for this value of β all the stabilities are exchanged. In particular, for β < θ = ± π/ θ = 0 , π/
2) are unstable, while for β > θ = ± π/ V θ where θ is the (invariant) angular coordinatein the ( x, y ) plane, is always of the form V θ ( r ) = c θ r with c θ a constant. We actually get (recall we assumed β > − c = c π/ = 1 ; c π/ = c − π/ = 1 + β/ . This is coherent, of course, with the stability of the equi-librium point at the origin. -3 -2 -1 1 2 30.80.91.11.2
FIG. 1: The potential W ( θ ) as in eq.(11) for different valuesof β ; here β = − , − . , . ,
1. The exchange of stability takesplace at β = 0. -1 -0.5 0.5 1-1-0.50.51 -1 -0.5 0.5 1-1-0.50.51 ( a ) ( b ) -1 -0.5 0.5 1-1-0.50.51 -1 -0.5 0.5 1-1-0.50.51 ( c ) ( d )FIG. 2: Numerical integration of the motion generated by thepotential (10) with the choice β = 1 for initial conditions nearto normal modes. In all cases, initial data correspond to zerospeed and position at r = 1 along eigenvectors, with an offsetof 0.001 from the latter. The simulation show the outcome,for t ∈ (0 , θ =0, (b) near the eigenvector θ = π , (c) near the eigenvector θ = π/
4, (d) near the eigenvector θ = − π/ We stress that, apart from the stability exchange for W ( θ ), in this case there are no qualitative changes as theparameters (which in this case means just the parameter β ) are varied: we always have four critical lines and hencefour normal modes; two of them are stable and two ofthem unstable.Some numerical simulations of this dynamics for initialdata near to the (higher order) normal modes are shownin Figure 2; they confirm stability as discussed above.[40] B. The lower symmetry case
We will now consider the general form (which has only Z × Z as symmetry), i.e. the potential (9). With noloss of generality, we can assume a > b (if not, just switch x and y ). Here again stability requires that both a and b are positive; by a rescaling we can set a = 1, and hence0 < b <
1, and deal with the potential V ( x, y ) = x + α y + 2 β x y , (12)where 0 < α < y = mx we get V m ( x ) := V ( x, mx ) = (1 + 2 βm + αm ) x ;stability of the origin requires therefore that1 + 2 β m + α m > m , and this implies β > − √ α , which we assume from now on.After passing to polar coordinates, the restriction of V given by (12) to the unit sphere reads W ( θ ) = cos ( θ ) + α sin ( θ ) + 2 β sin ( θ ) cos ( θ ) , (13)and from this we get at once dWdθ = − [1 − α + (1 + α − β ) cos(2 θ )] sin(2 θ ) . Thus critical points of W are identified either bysin(2 θ ) = 0, i.e. by θ = 0 , ± π/ , ± π ; or bycos(2 θ ) = α − α + 1 − β . Solutions to this equation exist only in the regions β < α and β > < α < α < β < β lies in the range β > −√ α ,and that for β taking the values β = α and β = 1 thereare bifurcations changing the number of critical pointsfor W , i.e. of invariant lines for our potential V ( x, y ).More precisely, we always have four critical points at θ = 0 , ± π/ , π ; moreover for suitable β we also have fourmore critical points at θ = ±
12 arccos (cid:20) − α α − β (cid:21) := ± θ ∗ . That is, we have either eight or four critical points forthe potential W restricted on the sphere, correspondingto four or two critical lines and hence normal modes,depending on the value of β . See Figure 3.The stability of critical points is controlled by d Wdθ = − − α ) cos(2 θ ) + (1 + α − β ) cos(4 θ )] ;in particular at the various critical points identified abovewe have( d /dθ ) θ =0 = ( d /dθ ) θ = π = − − β ) ;( d /dθ ) θ = π/ = ( d /dθ ) θ = − π/ = − α − β ) ;( d /dθ ) θ = θ ∗ = ( d /dθ ) θ = − θ ∗ = 8 (cid:20) ( α − β ) (1 − β )1 + α − β (cid:21) . -1 -0.5 0.5 10.20.40.60.811.21.4 -1 -0.5 0.5 10.20.40.60.811.21.4 -1 -0.5 0.5 10.20.40.60.811.21.4 β = − / β = − / β = 0 -1 -0.5 0.5 10.20.40.60.811.21.4 -1 -0.5 0.5 10.20.40.60.811.21.4 -1 -0.5 0.5 10.20.40.60.811.21.4 β = 1 / β = 1 / β = 3 / -1 -0.5 0.5 10.20.40.60.811.21.4 -1 -0.5 0.5 10.20.40.60.811.21.4 -1 -0.5 0.5 10.20.40.60.811.21.4 β = 1 β = 5 / β = 3 / W ( θ ), see (13), for α = 1 / β . Here θ is measured in units of π . This immediately shows that the critical line corre-sponding to θ = 0 (equivalently, to θ = π ) undergoes achange of stability at β = 1; and the critical line corre-sponding to θ = π/ θ = 3 π/
4) under-goes a change of stability at β = α . As for the criticallines corresponding to θ = ± θ ∗ , existing in the regions β < α and β >
1, noting that α < (1 + α ) / <
1, wehave that these are stable for β < α and unstable for β >
1. (Numerical simulations, not shown for the sakeof brevity, confirm again our analysis.)
V. DISCUSSION AND CONCLUSIONS
In this note we have considered natural Hamiltoniansfor point particle, H = K + V , for which the dynamicsin local coordinates is simply ¨ x = −∇ V . We have con-sidered the neighborhood of a stable equilibrium x , andstudied the case where the Taylor expansion of V startswith terms of order k higher than two (we have of coursegiven special attention to the case k = 4).We have discussed how the notion of normal modes ismodified in this case; this is based essentially on knownresults concerning eigenvalues of (homogeneous) tensors .It results that higher order normal modes do exist, butwhile some of their properties extend from the standard(i.e. k = 2) case to the present one, other do not. Inparticular, their number is not fixed and can exceed thedimensionality of the ambient space; moreover the mostgeneral dynamics near the equilibrium is in general[41] not a superposition of normal modes, at difference withthe standard case.It should be mentioned that a question which arisenaturally has not been studied here, and should be con-sidered in the future. This is of course the persistence of these higher order normal modes under perturbations,i.e. when one considers also higher order terms in theseries expansion for the potential around the equilibriumpoint. We recall that for standard normal modes a the-ory of persistence exists [5, 6]; this is based on variationalanalysis and guarantees persistence of some of the normalnodes under certain conditions. Apart from an extensionto the new higher order normal modes along this lineof attack, one could also consider an approach based onthe theory of Poincar´e-Birkhoff normal forms [28, 29] (orsome generalization thereof).It is worth noting that for the examples discussed inSection IV, persistence of some modes can be guaranteedby symmetry arguments if the symmetry is preserved by higher order perturbations: In both cases, the potentialthen admits symmetries sending x x, y
7→ − y , resp. x
7→ − x, y y (analogously for the time derivatives),hence the fixed point spaces of these symmetries are nec-essarily invariant. The fixed point spaces correspond tothe cases θ = 0 , ± π above. In the higher symmetry caseone also has symmetries exchanging x and y , resp. x and − y , with invariant fixed point spaces corresponding to θ = π/ θ = 3 π/ [1] L.D. Landau & I.M. Lifhsitz, Mechanics , Pergamon 1960[2] F. Gantmacher,
Lectures in Analytical Mechanics , MIR1970[3] V.I. Arnold,
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Physica D (1987),95-127; addendum, Physica D (1988), 488[30] J.F. Cari˜nena & A. Ramos, “A new geometric ap-proach to Lie systems and physical applications”, ActaAppl.Math. (2002), 43-69[31] J. F. Cari˜nena, J. Grabowski & G. Marmo, Lie-ScheffersSystems. A Geometric Approach , Bibliopolis 2000[32] In particular, all the discussion in this section, i.e. refer-ring to the quadratic case, can easily be set in terms ofa general Hamiltonian H ( p , q ).[33] This is quite different from the case of liquid crystalsmentioned above: there the relevant tensor is of orderthree [9–12].[34] This is of course the same situation met in the Funda-mental Theorem of Algebra: we know that a polynomialof degree n in one variable always has n roots (count-ing multiplicities) but, with no further study, we do notknow how many of these are real.[35] Note that for n odd this still leaves an ambiguity, as theeigenvalues for v and − v differ by sign; it would actuallybe convenient to consider these as two distinct eigenval-ues, corresponding to distinct eigenvectors, in view of thediscussion in Sect. III C, see [10, 12]. [36] For completely symmetric cubic tensors in two-dimensional space we get a degenerate situation; see thediscussion in [9]. We will see later on, in Section IV, thata similar degeneration is met for quartic tensors in two-dimensional space.[37] Note that while the characteristic is an intrinsic propertyof the sphere we work on, the bound provided by Bezout’sand by Rohrl’s theorems depends on the degree of themapping.[38] It should be noted that not all of these are realized whenwe consider, as the Physics of the problem studied in[10, 12] requires, completely traceless tensors. See also the discussion in [24] in this respect.[39] We stress that here the stability means stability for thepotential restricted to the unit circle, while the originis always stable for the full two-dimensional potential V ( x, y ).[40] Here we show the situation for β >
0; simulations for β < nonlinear superposition principlenonlinear superposition principle