The regularized free fall I -- Index computations
TThe regularized free fallI – Index computations
Urs Frauenfelder Joa Weber ∗ Universit¨at Augsburg UNICAMPFebruary 3, 2021
Abstract
Main results are, firstly, a generalization of the Conley-Zehnder indexfrom ODEs to the delay equation at hand and, secondly, the equality ofthe Morse index and the clockwise normalized Conley-Zehnder index µ CZ .We consider the non-local Lagrangian action functional B discoveredby Barutello, Ortega, and Verzini [BOV20] with which they obtained anew regularization of the Kepler problem. Critical points of this functionalare regularized periodic solutions q of the Kepler problem. In this articlewe look at period 1 only and at dimension one (gravitational free fall).Via a non-local Legendre transform regularized periodic Kepler orbits q can be interpreted as periodic solutions ( q, p ) of a Hamiltonian delayequation. In particular, regularized 1-periodic solutions of the free fall arerepresented variationally in two ways: As critical points q of a non-localLagrangian action functional and as critical points ( q, p ) of a non-localHamiltonian action functional.As critical points of the Lagrangian action the 1-periodic solutions havea finite Morse index which we compute first.As critical points of the Hamiltonian action A H one encounters the obsta-cle, due to non-locality, that the 1-periodic solutions are not generated anymore by a flow on the phase space manifold. Hence the usual definitionof the Conley-Zehnder index as intersection number with a Maslov cycleis not available. In the local case Hofer, Wysocki, and Zehnder [HWZ95]gave an alternative definition of the Conley-Zehnder index by assigning awinding number to each eigenvalue of the Hessian of A H at critical points.In this article we show how to generalize the Conley-Zehnder index tothe non-local case at hand. On one side we discover how properties fromthe local case generalize to this delay equation, and on the other side wesee a new phenomenon arising. In contrast to the local case the windingnumber is not any more monotone as a function of the eigenvalues. ∗ Email: [email protected] [email protected] a r X i v : . [ m a t h . S G ] F e b ontents B and L q inner product . . . . . . . 113.2 Critical points and Hessian operator . . . . . . . . . . . . . . . . 123.3 Eigenvalue problem and Morse index . . . . . . . . . . . . . . . . 15 A H . . . . . . . . . . . . . . . . . . 205.4 Lagrangian action B dominates Hamiltonian one A H . . . . . . . 205.5 Critical points and Hessian . . . . . . . . . . . . . . . . . . . . . 225.6 Eigenvalue problem and Conley-Zehnder index . . . . . . . . . . 245.7 Disjoint families and winding numbers . . . . . . . . . . . . . . . 29 References 33 Introduction
In celestial mechanics, as well as in the theory of atoms, collisions play anintriguing role. There are many geometric ways how to regularize collisions, seefor instance [LC20,Mos70]. In both regularizations, Levi-Civita and Moser, onehas to change time. A quite new approach to the regularization of collisions wasdiscovered in the recent paper [BOV20] by Barutello, Ortega, and Verzini wherethe change of time leads to a delayed functional (meaning that the critical pointequation is a delay equation). In Section 2 we explain starting from the physics1-dimensional Kepler problem how one arrives at this functional B : W , × := W , ( S , R ) \ { } → R q (cid:55)→ (cid:107) q (cid:107) (cid:107) ˙ q (cid:107) + 1 (cid:107) q (cid:107) where (cid:107)·(cid:107) is the L norm associated to the L inner product (cid:104)· , ·(cid:105) . One mightinterpret the functional B as a non-local Lagrangian action functional , anon-local mechanical system, consisting of kinetic minus potential energy B ( q ) = (cid:104) ˙ q, ˙ q (cid:105) q + 1 (cid:107) q (cid:107) Here we use the following metric on the tangent bundle of the loop space (cid:104)· , ·(cid:105) q := 4 (cid:107) q (cid:107) (cid:104)· , ·(cid:105) which from the perspective of the manifold is non-local as it depends on thewhole loop. Also the potential q (cid:55)→ − (cid:107) q (cid:107) is only defined on the loop space.The set Crit B of critical points of the non-local action consists of the smoothsolutions q ∈ C ∞ ( S , R ) \ { } of the second order delay, or non-local, equation¨ q = αq, α = α q := (cid:18) (cid:107) ˙ q (cid:107) (cid:107) q (cid:107) − (cid:107) q (cid:107) (cid:19) < L extensionof) the cotangent bundle of the loop space, namely H : W , × × L → R ( q, p ) (cid:55)→ (cid:104) p, p (cid:105) q − (cid:107) q (cid:107) where we use the dual metric on the cotangent bundle of the loop space (cid:104)· , ·(cid:105) q := (cid:107) q (cid:107) (cid:104)· , ·(cid:105) Associated to the non-local Hamiltonian H there is the non-local Hamilto-nian action functional , namely A H := A − H : W , × × L → R ( q, p ) (cid:55)→ (cid:90) p ( t ) ˙ q ( t ) dt − H ( q, p )3he natural base point projection and the following injection π : W , × × L → W , × ι : W , × → W , × × L ( q, p ) (cid:55)→ q q (cid:55)→ (cid:0) q, (cid:107) q (cid:107) ˙ q (cid:1) (1.2)are along the sets of critical points inverses of one another – which of courseexplains the choice of the factor 4 (cid:107) q (cid:107) . Therefore the setsCrit A H Crit B π ι are in one-to-one correspondence. Whereas B at any critical point has a finitedimensional Morse index, in sharp contrast both the Morse index and coindexare infinite at the critical points of A H .Since the action functional A H is non-local its critical points cannot beinterpreted as fixed points of a flow. Therefore the usual definition of the Conley-Zehnder index as a Maslov index does not make sense. However, in [HWZ95]Hofer, Wysocki, and Zehnder gave a different characterization of the Conley-Zehnder index in terms of winding numbers of the eigenvalues of the Hessian.We show in this paper that this definition of the Conley-Zehnder index makessense, too, for the critical points of the non-local functional A H . Our main resultis equality of Morse and Conley-Zehnder index of corresponding critical points. Theorem A.
For each critical point ( q, p ) of A H the canonical (clockwise nor-malized) Conley-Zehnder index µ CZ ( q, p ) = Ind( q ) is equal to the Morse index of B at q .Proof. Proposition 3.7 and Proposition 5.10.The theorem generalizes the local result, see [Web02], to this non-local case.Note that [Web02, Thm. 1.2] uses the counter-clockwise normalized Conley-Zehnder index µ CZ = − µ CZ . Sign conventions are discussed at large in theintroduction to [Web17]. The local result was a crucial ingredient in the proofthat Floer homology of the cotangent bundle is the homology of the loopspace [Vit98, SW06, AS06]. Notation.
We define S := R / Z and consider functions q : S → R as functionsdefined on R that satisfy q ( t + 1) = q ( t ) for every t ∈ R . Throughout (cid:104)· , ·(cid:105) isthe standard L inner product on L ( S , R ) and (cid:107)·(cid:107) is the induced L norm. Outlook.
Since 2018 first steps have been taken to study Floer homology fordelay equations, see [AFS20, AFS19a, AFS19b]. The present article is the firstin a series of four dealing with the free fall – the simplest instance which mightalready exhibit all novelties that occur in comparison to ode Floer homologyof the cotangent bundle (which still represents the homology of a space, loopspace). The other parts will deal with II “Homology computation via heat flow”,III “Floer homology”, and IV “Floer homology and heat flow homology”.
Acknowledgements.
UF acknowledges support by DFG grant FR 2637/2-2.4
Free gravitational fall
Section 2 is to motivate and explain regularization. Readers familiar with reg-ularization could go directly to subsequent sections.
For r > v ∈ R let L ( r, v ) := | v | − V ( r ) where V ( r ) := − /r . Then d v L ( r, v ) = v , d r L ( r, v ) = − /r Due to the potential one cannot allow r = 0. Thus the classical action functional S L ( x ) := (cid:90) L ( x ( τ ) , x (cid:48) ( τ )) dτ = (cid:90) (cid:18) | x (cid:48) ( τ ) | + 1 x ( τ ) (cid:19) dτ (2.3)is defined on the space W , := W , ( S , (0 , ∞ )) (2.4)that consists of absolutely continuous positive functions x : [0 , → (0 , ∞ ) whichare periodic, that is x (1) = x (0), and whose derivative is L integrable, that is (cid:107) x (cid:48) (cid:107) := (cid:82) x (cid:48) ( τ ) dτ < ∞ .The Euler-Lagrange (or critical point) equation is given by the second order ode ddτ d L ( x, x (cid:48) ) = d L ( x, x (cid:48) ) ⇔ x (cid:48)(cid:48) = − x (2.5)pointwise at τ . Unfortunately, there are no periodic solutions of this equation.In other words, there are no critical points of S L on the space W , , in symbolsCrit S L = ∅ (2.6)All solutions x with zero initial velocity v end up in collision with the origin.Set x := x (0) and v := x (cid:48) (0). Multiply (2.5) by 2 x (cid:48) and integrate to get x (cid:48) ( τ ) − v = (cid:90) τ x (cid:48)(cid:48) x (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) dds x (cid:48) ds (2 . = (cid:90) τ − x (cid:48) x (cid:124) (cid:123)(cid:122) (cid:125) dds x (cid:48)− ds = 2 x ( τ ) − x hence x (cid:48) ( τ ) = ± (cid:115) x ( τ ) + 2 E, E := v − x ≡ x (cid:48) ( τ ) − x ( τ )This shows that collision and bouncing off happen at infinite velocity. If theparticle starts with zero initial velocity, collision happens in finite time since theacceleration x (cid:48)(cid:48) is bounded away from zero.5 .2 Non-local regularization In this section we explain why Barutello, Ortega, and Verzini [BOV20] discov-ered their functional B defined on the larger (than W , ) space W , × := W , × ( S , R ) \ { } It consists of absolutely continuous maps q : [0 , → R which are periodic q (1) = q (0), not identically zero q (cid:54)≡
0, and whose derivative ˙ q is L integrable.The key observation is that if on the subset W , – the domain (2.4) ofthe classical action S L – one defines an operation R that takes the squareand appropriately rescales time, then the composition ( S L ◦ R )( q ) is given by aformula, let’s call it B ( q ), see Figure 1, that makes perfectly sense on the ambientspace W , × whose elements are real-valued and so allow for origin traversing. R R R W , × ( S , R ) W , ( S , (0 , ∞ )) W , ( S , (0 , ∞ )) q q ◦ t q B ⊃ R B := R ∗ S L S L Figure 1: Pull-back R ∗ S L produces formula B that makes sense on a larger spaceon which critical points exist and represent time rescaled physical trajectoriesThe key fact is that B admits critical points q on the ambient space W , × ,see (3.26), in which case x q := R ( q ) solves by Proposition 2.4 the classical freefall equation (2.5) away from a finite set T q ⊂ S of collision times. Time rescaling on the small space W , Pick a loop q ∈ W , , see (2.4). We call the variable of the map q : [0 , → (0 , ∞ )the regularized time , usually denoted by t . Classical time we call the valuesof the map τ q : [0 , → [0 ,
1] defined by τ q ( t ) := (cid:82) t q ( s ) ds (cid:107) q (cid:107) (2.7)Classical time has the following properties˙ τ q ( t ) = q ( t ) (cid:107) q (cid:107) > , τ q ∈ C , τ rq = τ q , τ q (0) = 0 , τ q (1) = 1 (2.8)for every real r >
0. Since, moreover, the map τ q : [0 , → [0 ,
1] is strictlyincreasing, it is a bijection and we denote its inverse by t q : [0 , → [0 , t q := τ q − of classical time inherits the property t rq = t q ∀ r > t (cid:48) q ( τ ) = 1˙ τ q ( t q ( τ )) = (cid:107) q (cid:107) q ( t q ( τ )) , t q ∈ C , t q (0) = 0 , t q (1) = 1 (2.9) Definition 2.1.
The rescale-square operation is defined by R : W , → W , , q (cid:55)→ q ◦ t q We abbreviate x q := R ( q ). Note that R ( rq ) = r R ( q ) for r > Lemma 2.2 (Well defined bijection) . For q ∈ W , the image R ( q ) lies in W , , too, and the map R : W , → W , is bijective with inverse (2.11).Proof. We must show that both are in L , namely a) x q ( τ ) = q ( t q ( τ )) and b) x (cid:48) q ( τ ) = 2 q ( t q ( τ )) ˙ q ( t q ( τ )) t (cid:48) q ( τ ) (2 . = 2 (cid:107) q (cid:107) ˙ q ( t q ( τ )) q ( t q ( τ ))Use that q is continuous, so (cid:107) q (cid:107) L ∞ < ∞ , to get a) (cid:107) x q (cid:107) = (cid:90) q ( t q ( τ )) dτ = (cid:107) q (cid:107) L ∞ < ∞ In what follows change the variable to σ := t q ( τ ) using (2.9) to get b) (cid:107) x (cid:48) q (cid:107) = (cid:90) (cid:18) (cid:107) q (cid:107) ˙ q ( t q ( τ )) q ( t q ( τ )) (cid:19) q ( σ ) dσ (cid:107) q (cid:107) = 4 (cid:107) q (cid:107) (cid:104) ˙ q, ˙ q (cid:105) < ∞ (2.10)Here the value is finite since q is of class W , . This proves that R ( q ) ∈ W , . Surjective.
Given x ∈ W , , set S ( x ) := q x := x ◦ t / √ x (2.11)Then for t ∈ [0 ,
1] we obtain τ q x ( t ) def. = (cid:82) t q x ( s ) ds (cid:82) q x ( s ) ds = (cid:82) t x ◦ =: σ (cid:122) (cid:125)(cid:124) (cid:123) t / √ x ( s ) ds (cid:82) x ◦ τ / √ x ( s ) ds = (cid:82) t / √ x ( t )0 dσ (cid:107) / √ x (cid:107) (cid:82) dσ (cid:107) / √ x (cid:107) = t / √ x ( t )by change of variables. With this result we get that (cid:0) R ◦ S ( x ) (cid:124) (cid:123)(cid:122) (cid:125) q x (cid:1) ( τ ) def. R = q x ◦ t q x ( τ ) def. q x = x ◦ t / √ x (cid:124) (cid:123)(cid:122) (cid:125) τ qx ◦ t q x (cid:124)(cid:123)(cid:122)(cid:125) τ qx − ( τ ) = x ( τ ) Injective.
For q ∈ W , set x q := R ( q ) := q ◦ t q . Then for τ ∈ [0 ,
1] we get (cid:90) τ x q ( σ ) dσ = (cid:90) τ q ◦ t q ( σ ) dσ = (cid:90) t q ( τ )0 q ( s ) q ( s ) ds (cid:107) q (cid:107) = t q ( τ ) (cid:107) q (cid:107)
7y change of variables s = t q ( σ ) using (2.9). Pick τ = 1 to obtain (cid:90) x q ( τ ) dτ = 1 (cid:107) q (cid:107) (2.12)Therefore we get for τ ∈ [0 ,
1] the formula t q ( τ ) = (cid:107) q (cid:107) (cid:90) τ x q ( σ ) dσ (2 . = (cid:82) τ x q ( σ ) dσ (cid:82)
10 1 x q ( σ ) dσ (2 . =: τ / √ x q ( σ )With this result we obtain (cid:0) S ◦ R ( q ) (cid:124) (cid:123)(cid:122) (cid:125) x q (cid:1) ( t ) def. S = √ x q ◦ t / √ x q ( t ) def. x q = q ◦ t q (cid:124)(cid:123)(cid:122)(cid:125) τ / √ xq ◦ t / √ x q (cid:124) (cid:123)(cid:122) (cid:125) τ / √ xq − ( t ) = q ( t ) Lemma 2.3 (Pull-back yields an extendable formula) . Given q ∈ W , , then S L ( R ( q )) = (cid:104) ˙ q, ˙ q (cid:105) q + 1 (cid:107) q (cid:107) as illustrated by Figure 1. Note that in the previous formula q ( t ) may take on the value zero at will,even along intervals, as long as (cid:107) q (cid:107) (cid:54) = 0, i.e. as long as q is not constantly zero. Proof.
Set x q := R ( q ). Changing again the variable to σ := t q ( τ ) we obtain S L ( x q ) (2 . = (cid:107) x (cid:48) q (cid:107) + (cid:90) x q ( τ ) dτ = (cid:107) q (cid:107) (cid:107) ˙ q (cid:107) + 1 (cid:107) q (cid:107) (2.13)where we used (2.10) and (2.12). Whereas the classical action S L does not admit, see (2.6), on the small space W , any critical points, that is no 1-periodic free fall solutions, the for-mula (2.13) for S L evaluated on the regularization x q := R ( q ) of a loop q ∈ W , makes perfectly sense on the larger Sobolev Hilbert space W , × := W , ( S , R ) \ { } with the origin removed. This way one arrives, in case of thefree fall, at the Barutello-Ortega-Verzini [BOV20] non-local functional B : W , × := W , ( S , R ) \ { } → R , q (cid:55)→ (cid:104) ˙ q, ˙ q (cid:105) q + 1 (cid:107) q (cid:107) (2.14)For this functional B eventual zeroes of q cause no problem at all. It has evenlots of critical points - one circle worth of critical points for each natural number k ∈ N , see Lemma 3.3.But are the critical points q : S → R of B related to free fall solutions x : dom x → (0 , ∞ )? If so what is the domain of x ? Next we prepare to give theanswers in Proposition 2.4 below. From now on8 we only consider critical points q ∈ W , × of B and • we identify S with [0 , / { , } .A priori such q : [0 , → R might have zeroes and these might obstruct bijec-tivity of the classical time τ q defined as before in (2.7). Indeed the derivative˙ τ q = q ( t ) / (cid:107) q (cid:107) ≥
0, more properties in (2.8), might now be zero at some times– precisely the times of collision of the solution q with the origin. By continuityof the map q : [0 , → R the zero set T q := q − (0) is closed, thus compact. Onthe other hand, the set is discrete because being a critical point q (cid:54)≡ T q := { t ∈ [0 , | q ( t ) = 0 } = { t , . . . , t N } Thus τ q : [0 , → [0 ,
1] is still strictly increasing, hence a bijection. We denotethe inverse again by t q := τ q − and the set of classical collision times , thisterminology will become clear in a moment, by T q := { τ , . . . , τ N } , τ i := τ q ( t i )The derivative of t q : [0 , → [0 ,
1] is still given by t (cid:48) ( τ ) = (cid:107) q (cid:107) q ( t ( τ )) (2.15)but now only at non-collision times τ , that is τ ∈ S \ T q .The following proposition is a special case of a theorem due to Barutello,Ortega, and Verzini [BOV20]. Proposition 2.4.
Given a critical point q ∈ W , × of B , namely a solutionof (1.1), then the rescale-squared map x = x q := R ( q ) is a physical solution in the sense that x solves the free fall equation at all times x (cid:48)(cid:48) ( τ ) = − x ( τ ) , ∀ τ ∈ S \ T q except at the finitely many collision times which form the set T q = { τ , . . . , τ N } . We think of the critical points q of B as the regularized versions, “the reg-ularizations”, of the physical solutions x q . Note that when the solution q runsthrough the big mass sitting at the origin the physical solution x bounces back. Proof. At τ ∈ S \ T , set t ( τ ) := t q ( τ ), the derivative of x = x q is given by x (cid:48) ( τ ) = 2 q ( t ( τ )) ˙ q ( t ( τ )) t (cid:48) ( τ ) (cid:124)(cid:123)(cid:122)(cid:125) (2 . = 2 (cid:107) q (cid:107) ˙ q ( t ( τ )) q ( t ( τ )) (2.16)and via a change of variables from τ to σ := t q ( τ ) we get using (2.15) that (cid:107) x (cid:48) (cid:107) = 4 (cid:107) q (cid:107) (cid:107) ˙ q (cid:107) equivalently (cid:10) x (cid:48) q , x (cid:48) q (cid:11) = (cid:104) ˙ q, ˙ q (cid:105) q (2.17)9 • (0 , ∞ ) x ( τ ) R finite time collision with origin at ∞ speed q ( t ) physical solutionregularized solutionpasses through origin x ( τ ) = q ( t q ( τ )) Figure 2: Physical solution x and regularized solution q To calculate the second derivative x (cid:48)(cid:48) use the critical point equation (1.1) to get x (cid:48)(cid:48) ( τ ) = 2 (cid:107) q (cid:107) q ( t ( τ )) (cid:18) ¨ q ( t ( τ )) − ˙ q ( t ( τ )) q ( t ( τ )) (cid:19) = 1 x ( τ ) (cid:18) (cid:107) q (cid:107) (cid:107) ˙ q (cid:107) − (cid:107) q (cid:107) (cid:19) − x (cid:48) ( τ ) x ( τ )= 1 x ( τ ) (cid:18) (cid:107) x (cid:48) (cid:107) − (cid:90) dτx ( τ ) − x (cid:48) ( τ ) (cid:19) (2.18)To obtain equation three we used the identities (2.17) and (2.12). Let τ − , τ + ∈ T be neighboring collision times in the sense that the interval ( τ − , τ + ) lies in thecomplement of the collision time set T .Similarly to Barutello, Ortega, and Verzini [BOV20, Eq. (3.12)] we consider β = β q := x (cid:48)(cid:48) x as a function on the open non-collision interval ( τ − , τ + ). Differentiate the iden-tity x β = (cid:16) (cid:107) x (cid:48) (cid:107) − (cid:82) dτx ( τ ) − x (cid:48) ( τ ) (cid:17) to obtain that 2 xx (cid:48) β + x β (cid:48) = − x (cid:48) x (cid:48)(cid:48) = − xx (cid:48) β or equivalently x β (cid:48) = − βxx (cid:48) . Hence the logarithmic derivatives satisfy β (cid:48) β = − x (cid:48) x and therefore β = µx for some constant µ ∈ R that a priori might depend on the interval ( τ − , τ + ).By definition of β we conclude that x (cid:48)(cid:48) = µx
10n the interval ( τ − , τ + ). By (2.18) we get for the constant µ the expression µ = x ( τ ) (cid:18) (cid:107) x (cid:48) (cid:107) − (cid:90) dτx ( τ ) (cid:19) − x ( τ ) x (cid:48) ( τ ) = x ( τ ) (cid:18) (cid:107) x (cid:48) (cid:107) − (cid:90) dτx ( τ ) (cid:19) − (cid:107) q (cid:107) ˙ q ( t ( τ )) where in the second equation we used (2.16). Taking the limits τ → τ ± we get µ = − (cid:107) q (cid:107) ˙ q ( t ( τ ± )) ≤ µ takes the same value on the boundary of adjacent intervals.Hence the constant µ is independent of the interval.To see that µ = − x and use x (cid:48)(cid:48) = µ/x to obtain µx = 12 (cid:107) x (cid:48) (cid:107) − (cid:90) dτx ( τ ) − x (cid:48) ( τ ) We take the mean value of this equation to get that µ (cid:90) dτx ( τ ) = 12 (cid:107) x (cid:48) (cid:107) − (cid:90) dτx ( τ ) − (cid:107) x (cid:48) (cid:107) = − (cid:90) dτx ( τ )This shows that µ = − B and L q inner product In the novel approach [BOV20] to the regularization of collisions discoveredrecently by Barutello, Ortega, and Verzini the change of time leads to a delayed action functional B . Section 2 above explains this for the free fall. The delay,that is the non-local term, is best incorporated into the L inner product onthe loop space. More precisely, we introduce a metric on the following Hilbertspace take away the origin, namely W , × := W , \ { } , W , := W , ( S , R )Given a point q ∈ W , × and two tangent vectors ξ , ξ ∈ T q W , × = W , we define the L q inner product by (cid:104) ξ , ξ (cid:105) q := 4 (cid:107) q (cid:107) (cid:104) ξ , ξ (cid:105) , where (cid:104) ξ , ξ (cid:105) := (cid:90) ξ ( t ) ξ ( t ) dt (3.19)11here (cid:107) q (cid:107) := (cid:112) (cid:104) q, q (cid:105) is the L norm associated to the L inner product. In ourcase of the 1-dimensional Kepler problem the functional then attains the form B : W , × := W , ( S , R ) \ { } → R , q (cid:55)→ (cid:104) ˙ q, ˙ q (cid:105) q + 1 (cid:107) q (cid:107) One might interpret this functional as a non-local mechanical system consistingof kinetic minus potential energy.
Straightforward calculation provides
Lemma 3.1 (Differential of B ) . The differential d B : W , × × W , → R is d B ( q, ξ ) = 4 (cid:104) q, ξ (cid:105)(cid:107) ˙ q (cid:107) + 4 (cid:107) q (cid:107) (cid:104) ˙ q, ˙ ξ (cid:105) − (cid:104) q, ξ (cid:105)(cid:107) q (cid:107) = −(cid:104) ¨ q, ξ (cid:105) q + (cid:18) (cid:107) ˙ q (cid:107) (cid:107) q (cid:107) − (cid:107) q (cid:107) (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) =: α (cid:104) q, ξ (cid:105) q where identity two is valid whenever q is of better regularity W , ( S , R ) \ { } . Corollary 3.2 (Crit B ) . The set of critical points consists of the smooth solu-tions q ∈ C ∞ ( S , R ) \ { } of the second order delay equation ¨ q = αq , see (1.1). Note that α > α = 0 is impossible since otherwise q would vanish identically which isexcluded by assumption.By Lemma 3.1 the L q gradient of B at a loop q ∈ W , × is given bygrad q B ( q ) = − ¨ q + αq, q ∈ W , × := W , ( S , R ) \ { } (3.20) Solutions of the critical point equation
To find the solutions q : S → R of equation (1.1) we set β := − α > q = − βq, q ( t ) = C cos (cid:112) βt + D sin (cid:112) βt where C and D are constants. Thus˙ q = − C (cid:112) β sin (cid:112) βt + D (cid:112) β cos (cid:112) βt Since our solutions q is periodic with period 1 we must have (cid:112) β = 2 πk, k ∈ N (3.21)Thus q ( t ) = C cos 2 πkt + D sin 2 πkt q ( t ) = − πkC sin 2 πkt + 2 πkD cos 2 πkt Note that integrating the identity 1 = cos + sin we get1 = (cid:90) dt = (cid:90) cos dt + (cid:90) sin dt = 2 (cid:90) cos (2 πkt ) dt (3.22)as is well known from the theory of Fourier series. Moreover, from the theoryof Fourier series it is known that cosine is orthogonal to sine, that is0 = (cid:90) cos(2 πkt ) sin(2 πkt ) dt Therefore (cid:107) q (cid:107) = C + D and (cid:107) ˙ q (cid:107) = C + D (2 πk ) and so we get that(2 πk ) . = β . = 12 (cid:107) q (cid:107) − (cid:107) ˙ q (cid:107) (cid:107) q (cid:107) = 4( C + D ) − (2 πk ) , k ∈ N (3.23)We fix the parametrization of our solution by requiring that at time zero thesolution is maximal. Therefore D = 0 and we abbreviate c k := C ( k ) >
0. Then c k is uniquely determined by k via the above equation which becomes4 c k = 8 π k , c k = 2( πk ) (3.24)Hence c k = 12 ( πk ) ∈ (0 , , c k = 12 ( πk ) (3.25)Note that c k <
1. This proves
Lemma 3.3 (Critical points of B ) . The maps q k ( t ) = c k cos 2 πkt, k ∈ N (3.26) are solutions of (1.1). Each map q k is worth a circle of solutions via time shift σ ∗ q k := q k ( · + σ ) where σ ∈ S . All solutions of (1.1) are given by Crit B = (cid:91) k ∈ N { σ ∗ q k | σ ∈ S } The Hessian operator with respect to the L q inner product The
Hessian operator A q of the Lagrange functional B is the derivative of the L q gradient equation 0 = − ¨ q + αq at a critical point q . Varying this equationwith respect to q in direction ξ we obtain that A q ξ = − ¨ ξ + αξ + (cid:18) − (cid:104) ¨ q, ξ (cid:105)(cid:107) q (cid:107) − (cid:107) ˙ q (cid:107) (cid:104) q, ξ (cid:105)(cid:107) q (cid:107) + 3 (cid:104) q, ξ (cid:105)(cid:107) q (cid:107) (cid:19) q = − ¨ ξ + αξ − (cid:18) α (cid:107) q (cid:107) + 2 (cid:107) ˙ q (cid:107) (cid:107) q (cid:107) − (cid:107) q (cid:107) (cid:19) (cid:104) q, ξ (cid:105) q = − ¨ ξ + αξ − (cid:107) q (cid:107) (cid:18) α − (cid:107) q (cid:107) (cid:19) (cid:104) q, ξ (cid:105) q q = αq and in step 3 wereplaced 2 (cid:107) ˙ q (cid:107) by 2 α (cid:107) q (cid:107) + (cid:107) q (cid:107) according to the definition of α . This proves Lemma 3.4.
The Hessian operator of B at a critical point q is given by A q : W , ( S , R ) → L ( S , R ) ξ (cid:55)→ − ¨ ξ + αξ − (cid:107) q (cid:107) (cid:18) α − (cid:107) q (cid:107) (cid:19) (cid:104) q, ξ (cid:105) q Recall that by (3.26) the critical points of B are of the form q k ( t ) = c k cos 2 πkt, k ∈ N (3.27)with c k given by (3.25). Taking two t derivatives we conclude that¨ q k = − (2 πk ) q k Since ¨ q k = αq k we obtain α = α ( q k ) = − (2 πk ) = − c k where the last equality is (3.24). The formula of the Hessian operator A q k in-volves the L norm of q k and, in addition, the formula of the non-local Lagrangefunctional B involves (cid:107) ˙ q k (cid:107) . By (3.26) and (3.22) we obtain that (cid:107) q k (cid:107) = c k ( πk ) , (cid:107) ˙ q k (cid:107) = (2 πk ) c k ( πk ) Thus B ( q k ) = 2 (cid:107) q k (cid:107) (cid:107) ˙ q k (cid:107) + 1 (cid:107) q k (cid:107) = 2 πk ) To calculate the formula of A q k we write ξ as a Fourier series ξ = ξ + ∞ (cid:88) n =1 ( ξ n cos 2 πnt + ξ n sin 2 πnt )and we use the orthogonality relation (cid:104) cos 2 πn · , ξ (cid:105) = 12 ξ n to calculate the product (cid:104) q k , ξ (cid:105) = c k ξ k Putting everything together we obtain
Lemma 3.5 (Critical values and Hessian) . The critical points of B are of theform q k ( t ) = c k cos 2 πkt for k ∈ N , see (3.27). At any such q k the value of B is B ( q k ) = 2 πk ) and the Hessian operator of B is A q k ξ = − ¨ ξ − (2 πk ) ξ + 12(2 πk ) ξ k cos 2 πkt for every ξ ∈ W , ( S , R ) . .3 Eigenvalue problem and Morse index Recall that k ∈ N is fixed since we consider the critical point q k . We are lookingfor solutions of the eigenvalue problem A q k ξ = µξ for µ = µ ( ξ ; k ) ∈ R and ξ ∈ W , ( S , R ) \ { } . Observe that − ¨ ξ = ∞ (cid:88) n =1 (2 πn ) ( ξ n cos 2 πnt + ξ n sin 2 πnt ) − (2 πk ) ξ = − (2 πk ) ξ − (2 πk ) ∞ (cid:88) n =1 ( ξ n cos 2 πnt + ξ n sin 2 πnt )Comparing coefficients in the eigenvalue equation A q k ξ = µξ we obtaincos 2 πnt (cid:40) µξ n = 4 π (cid:0) n − k (cid:1) ξ n , ∀ n ∈ N \ { k } µξ k = 12(2 πk ) ξ k , n = k sin 2 πnt (cid:110) µξ n = 4 π (cid:0) n − k (cid:1) ξ n , ∀ n ∈ N This proves
Lemma 3.6 (Eigenvalues) . The eigenvalues of the Hessian A q k are given by µ n := 4 π ( n − k ) , n ∈ N \ { k } and by µ := − π k , µ k := 0 , (cid:98) µ k := 12(2 πk ) Moreover, their multiplicity (the dimension of the eigenspace) is given by m ( µ n ) = 2 , n ∈ N \ { k } , m ( µ ) = m ( µ k ) = m ( (cid:98) µ k ) = 1Observe that the eigenvalue (cid:98) µ k (cid:54) = µ n is different from µ n for every n ∈ N .Indeed, suppose by contradiction that (cid:98) µ k = µ n for some n ∈ N , that is48 π k = 4 π ( n − k ) ⇔ k = n which contradicts that n is an integer. Proposition 3.7 (Morse index of q k ) . The number of negative eigenvalues ofthe Hessian A q k , called the Morse index of q k ∈ Crit B , is odd and given by Ind( q k ) = 2 k − , k ∈ N Note that by the lemma the bounded below functional B > B > Conley Zehnder index via spectral flow
In the local case the Conley-Zehnder index can be defined as a version of thecelebrated Maslov index [Mas65,Arn67]. In the non-local case we do not have aflow and therefore the interpretation as an intersection number of the linearizedflow trajectory with the Maslov cycle is not available.In Section 4 we recall the approach in the local case of Hofer, Wysocki, andZehnder [HWZ95] to the Conley-Zehnder index via winding numbers of eigen-values of the Hessian. In Section 5 we will see that this theory can be generalizedto the non-local case, namely for the Hessian of the non-local functional A H .Throughout we identify the unit circle S ⊂ R with R / Z and R (cid:39) C via( x, y ) (cid:55)→ x + iy . Multiplication by i on C is expressed by J on R , that is J = (cid:20) −
11 0 (cid:21)
Moreover, we think of maps f with domain S as 1-periodic maps defined on R ,that is f ( t + 1) = f ( t ), ∀ t ∈ R .To a continuous path of symmetric 2 × S : [0 , → R × , t (cid:55)→ S ( t ) = S ( t ) T , we associate the operator L S : L ( S , R ) ⊃ W , → L ( S , R ) , ζ (cid:55)→ − J ˙ ζ − Sζ (4.28)where ˙ ζ := ddt ζ . This operator is an unbounded self-adjoint operator whose re-solvent is compact (due to compactness of the embedding W , (cid:44) → L ). There-fore the spectrum is discrete and consists of real eigenvalues of finite multiplicity.Given an eigenvalue λ ∈ spec L S , then an eigenvector ζ corresponding to λ is anon-constantly vanishing solution ζ (cid:54)≡ ζ = J ( S + λ ζ for absolutely continuous 1-periodic maps ζ : R → R . In particular, since bydefinition an eigenvector is not vanishing identically, it does not vanish any-where, in symbols ζ ( t ) (cid:54) = 0 for every t . Therefore we can associate to eigenvec-tors ζ a winding number w ( ζ ) ∈ Z by looking at the degree of the map S → S , t (cid:55)→ ζ ( t ) | ζ ( t ) | This winding number only depends on the eigenvalue λ and not on the particulareigenvector for λ . Indeed if the geometric multiplicity of λ is 1, then a differenteigenvector for λ is of the form rζ for some nonzero real r (cid:54) = 0. If the geometricmultiplicity of λ is bigger than 1, then the space E λ of eigenvectors to λ equalsa vector space of dimension at least 2 minus the origin, in particular, it is path Because the elements of the target L of L S are not necessarily continuous, a requirementto close up after time 1 would be meaningless, hence paths S : [0 , → R × are fine. E λ , because it is discreteand depends continuously on the eigenvector. In view of these findings we write w ( λ ) = w ( λ ; S )for the winding number associated to an eigenvalue λ of the operator L S associated to the family S of symmetric matrices. Remark 4.1 (Case S = 0) . Each integer (cid:96) ∈ Z is realized as the windingnumber of the eigenvalue λ = 2 π(cid:96) of L of geometric multiplicity 2.To see this note that the spectrum of L = − J ddt is 2 π Z . Indeed pick aninteger (cid:96) ∈ Z . Then the eigenvectors of L corresponding to 2 π(cid:96) are of the form t (cid:55)→ ze πi(cid:96)t where z ∈ C \ { } . Therefore the winding number associated to (cid:96) is w ( (cid:96) ; 0) = (cid:96) The geometric multiplicity of each eigenvalue (cid:96) is 2 (pick z = 1 or z = i ). Remark 4.2 (General S ) . Each integer (cid:96) ∈ Z is realized either as the windingnumber of two different eigenvalues of L S of geometric multiplicity 1 each (cid:96) = w ( λ ; S ) = w ( λ ; S ) or as the winding number of a single eigenvalue of geometric multiplicity 2.To see this pick (cid:96) ∈ Z . Consider the family { rS } r ∈ [0 , . By Kato’s perturbationtheory we can choose continuous functions λ ( r ) and λ ( r ), r ∈ [0 , λ (0) = 2 π(cid:96) and λ (0) = 2 π(cid:96) (ii) λ ( r ), λ ( r ) ∈ spec L rS and the total geometric multiplicity remains 2, i.e.dim (cid:0) E λ ( r ) ⊕ E λ ( r ) (cid:1) = 2where E λ i ( r ) is the eigenspace of the eigenvalue λ i ( r )(iii) w ( λ ( r ); rS ) = (cid:96) and w ( λ ( r ); rS ) = (cid:96) The third assertion follows from the fact that the function r (cid:55)→ w ( λ ( r ); rS ) iscontinuous and takes value in the discrete set Z and, furthermore, since w ( (cid:96) ; 0) = (cid:96) by Remark 4.1.Moreover, the winding number continues to be monotone in the eigenvalueas the next lemma shows. Lemma 4.3 (Monotonicity of w ) . λ < λ ∈ spec L S ⇒ w ( λ ) ≤ w ( λ ) .Proof. By contradiction suppose that there are λ < λ ∈ spec L S such thattheir winding numbers satisfy (cid:96) := w ( λ ; S ) > w ( λ ; S ) =: (cid:96) Because eigenvalues depend continuously on the operator L S by Kato’s per-turbation theory, looking at the family { rS } r ∈ [0 , , we can choose continuousfunctions λ ( r ) and λ ( r ) with r ∈ [0 ,
1] having for every r ∈ [0 ,
1] the followingproperties 17i) λ (1) = λ and λ (1) = λ (ii) λ ( r ), λ ( r ) ∈ spec L rS (iii) w ( λ ( r ); rS ) = (cid:96) and w ( λ ( r ); rS ) = (cid:96) From the case S = 0 it follows that λ (0) = w ( λ (0); 0), which is (cid:96) by (iii), andthat λ (0) = w ( λ (0); 0), which is (cid:96) . Thus λ (0) = (cid:96) > (cid:96) = λ (0). Because (cid:96) is different from (cid:96) , it follows from (iii) as well that λ ( r ) is different from λ ( r )for every r ∈ [0 , λ ( r ) > λ ( r )for every r ∈ [0 , λ > λ . Contradiction. Definition 4.4.
Denote the largest winding number among negative eigenvalues of L S by α ( S ) := max { w ( λ ) | λ ∈ ( −∞ , ∩ spec L S } ∈ Z If both eigenvalues of L S (counted with multiplicity) whose winding number is α ( S ) are <
0, then S has parity p ( S ) := 1. Otherwise, define p ( S ) := 0. Definition 4.5 (Conley-Zehnder index of path S ) . Following Hofer, Wysocki,and Zehnder [HWZ95] the counter-clockwise normalized Conley-Zehnder index µ CZ of a continuous family of symmetric 2 × S : [0 , → R × is µ CZ ( S ) := 2 α ( S ) + p ( S ) Recall from (2.14) that the functional B : W , × → R , q (cid:55)→ (cid:104) ˙ q, ˙ q (cid:105) q + (cid:107) q (cid:107) , extendsthe classical action S L : W , → R . Then the functional defined by L : W , × × L → R , ( q, v ) (cid:55)→ (cid:104) v, v (cid:105) q + 1 (cid:107) q (cid:107) (5.29)naturally extends B ( q ) = L ( q, ˙ q ) in the same way as in the classical case S L ( q ) = L L ( q, ˙ q ) is extended by a corresponding functional L L ( q, v ).In the classical case a fiberwise strictly convex Lagrange function L on thetangent bundle determines a function H on the cotangent bundle: resolve p := d v L ( r, v ) for v and substitute the obtained v = v ( p ) in the Legendre identity (cid:104) p , v (cid:105) = L ( r, v ) + H ( r, p )Returning to the non-local situation where the manifold is loop space and q, v, p are loops, an analogous approach yields p := d v L ( q, v ) = 4 (cid:107) q (cid:107) v, v = (cid:107) q (cid:107) p, d vv L ( q, v ) = 4 (cid:107) q (cid:107) > non-local Hamiltonian function is then defined by H ( q, p ) := (cid:104) p, v (cid:105) − L ( q, v )and given by the formula H : W , × × L → R ( q, p ) (cid:55)→ (cid:104) p, p (cid:105) q − (cid:107) q (cid:107) = (cid:107) q (cid:107) (cid:0) (cid:107) p (cid:107) − (cid:1) The Hamiltonian equations of the non-local Hamiltonian H are the following ˙ q = ∂ p H = 14 (cid:107) q (cid:107) p ˙ p = − ∂ q H = q (cid:107) q (cid:107) (cid:18) (cid:107) p (cid:107) − (cid:19) (5.30) Lemma 5.1 (Correspondence of Hamiltonian and Lagrangian solutions) . a) If ( q, p ) solves the Hamiltonian equations (5.30), then q solves the La-grangian equations (1.1). b) Vice versa, if q solves the Lagrangian equations (1.1), then ( q, p q ) where p q := 4 (cid:107) q (cid:107) ˙ q solves the Hamiltonian equations (5.30).Proof. a) The first equation in (5.30) leads to (cid:107) ˙ q (cid:107) = (cid:107) p (cid:107) / (cid:107) q (cid:107) which weresolve for (cid:107) p (cid:107) and then plug it into the second equation to obtain˙ p = q (cid:107) q (cid:107) (cid:0) (cid:107) q (cid:107) (cid:107) ˙ q (cid:107) − (cid:1) Now take the time t derivative of the first equation to get indeed¨ q = 14 (cid:107) q (cid:107) ˙ p = 14 (cid:107) q (cid:107) q (cid:107) q (cid:107) (cid:0) (cid:107) q (cid:107) (cid:107) ˙ q (cid:107) − (cid:1) = (cid:107) ˙ q (cid:107) (cid:107) q (cid:107) q − (cid:107) q (cid:107) q = αq b) Suppose q solves (1.1) and define p = p q := 4 (cid:107) q (cid:107) ˙ q , hence (cid:107) p (cid:107) = 16 (cid:107) q (cid:107) (cid:107) ˙ q (cid:107) .Resolve for ˙ q to obtain the first of the Hamilton equations (5.30). To obtain thesecond equation take the time t derivative of p and use the Lagrange equationfor ¨ q to get˙ p = 4 (cid:107) q (cid:107) ¨ q = 4 (cid:107) q (cid:107) (cid:18) (cid:107) ˙ q (cid:107) (cid:107) q (cid:107) − (cid:107) q (cid:107) (cid:19) q = (cid:18) (cid:107) p (cid:107) (cid:107) q (cid:107) − (cid:107) q (cid:107) (cid:19) q = q (cid:107) q (cid:107) (cid:18) (cid:107) p (cid:107) − (cid:19) where in step three we substituted (cid:107) ˙ q (cid:107) according to (cid:107) p (cid:107) = 16 (cid:107) q (cid:107) (cid:107) ˙ q (cid:107) .19 .3 Non-local Hamiltonian action A H The symplectic area functional is defined by A : W , × L → R , ( q, p ) (cid:55)→ (cid:90) p ( t ) ˙ q ( t ) dt and the non-local Hamiltonian action functional by A H := A − H : W , × × L → R ( q, p ) (cid:55)→ (cid:90) p ( t ) ˙ q ( t ) dt − H ( q, p )The derivative d A H ( q, p ) : W , × L → R is given by d A H ( q, p )( ξ, η ) = (cid:90) (cid:16) p ˙ ξ + η ˙ q (cid:17) dt − (cid:104) p, η (cid:105) (cid:107) q (cid:107) + (cid:0) (cid:107) p (cid:107) − (cid:1) (cid:104) q, ξ (cid:105)(cid:107) q (cid:107) = −(cid:104) ˙ p, ξ (cid:105) + (cid:104) ˙ q, η (cid:105) − (cid:104) p, η (cid:105) (cid:107) q (cid:107) + (cid:0) (cid:107) p (cid:107) − (cid:1) (cid:104) q, ξ (cid:105)(cid:107) q (cid:107) = (cid:28) ˙ q − p (cid:107) q (cid:107) , η (cid:29) + (cid:28) − ˙ p + (cid:107) p (cid:107) − (cid:107) q (cid:107) q, ξ (cid:29) (5.31)and this proves Lemma 5.2.
The critical points of A H are precisely the -periodic solutions ofthe Hamiltonian equations (5.30) of H . In view of Lemma 5.1 and 5.2 we have a one-to-one correspondence betweencritical points of B and A H , namely q (cid:55)→ ( q, p q ). Under this correspondence thevalues of both functionals coincide along critical points, see Lemma 5.3. B dominates Hamiltonian one A H Lemma 5.3 (Lagrangian domination) . There are the identities B ( q ) = A H ( q, p ) + 12 (cid:13)(cid:13)(cid:13)(cid:13) (cid:107) q (cid:107) ˙ q − p (cid:107) q (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) = A H ( q, p ) + 12 (cid:13)(cid:13) (cid:107) q (cid:107) ˙ q − p (cid:13)(cid:13) (cid:107) q (cid:107) for every pair of loops ( q, p ) ∈ W , × × W , Proof.
Just by definition of A H and multiplying out the inner product we obtain A H ( q, p ) + 12 (cid:13)(cid:13)(cid:13)(cid:13) (cid:107) q (cid:107) ˙ q − p (cid:107) q (cid:107) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:104) p, ˙ q (cid:105) − (cid:107) p (cid:107) (cid:107) q (cid:107) + 1 (cid:107) q (cid:107) + 12 (cid:28) (cid:107) q (cid:107) ˙ q − p (cid:107) q (cid:107) , (cid:107) q (cid:107) ˙ q − p (cid:107) q (cid:107) (cid:29) = 2 (cid:107) q (cid:107)(cid:107) ˙ q (cid:107) + 1 (cid:107) q (cid:107) = B ( q ) 20 orollary 5.4 (Equal values on critical points) . Both functionals coincide B ( q ) = A H ( q, p q ) , p q := 4 (cid:107) q (cid:107) ˙ q on critical points, i.e. the solutions q of (1.1), equivalently ( q, p q ) of (5.30). In terms of the projection π and injection ι in (1.2) the corollary tells that B = A H ◦ ι, B ◦ π = A H along critical points of B , respectively of A H . The diffeomorphism L Given the functional L in (5.29), define the non-local analogue of the diffeomor-phism introduced in [AS15, p. 1891], in the local context, by the formula L : ( T L × ) = W , × × L → W , × × L = ( T ∗ L × ) ( q, v ) (cid:55)→ ( q, d v L ( q, ˙ q + v )) =: ( q, p )where ( T L × ) = ( L × ) × ( L ) is scale calculus notation, cf. [FW18, § p := d v L ( q, ˙ q + v ) = 4 (cid:107) q (cid:107) ( ˙ q + v )Note that since the inverse is given by L − ( q, p ) = (cid:18) q, p (cid:107) q (cid:107) − ˙ q (cid:19) =: ( q, v )the solutions ( q, p ) of the Hamiltonian equations (5.30) are zeroes of L − .As in the ode case [AS15] also in the present delay equation situation bothfunctionals are related through the maps ι and π (Lemma 5.3) in the form B ◦ π ( q, p ) = A H ( q, p ) + U ∗ ( q, p ) , U ∗ ( q, p ) := (cid:107) ι ( q ) − p (cid:107) (cid:107) q (cid:107) for every ( q, p ) ∈ W , × × W , . Observe that the map U ∗ ≥ U defined and given by U ( q, v ) := U ∗ ◦ L ( q, v ) = (cid:104) v, v (cid:105) q ≥ A H and B are related by the formula A H ◦ L ( q, v ) = B ( q ) − U ( q, v )= (cid:104) ˙ q, ˙ q (cid:105) q + 1 (cid:107) q (cid:107) − (cid:104) v, v (cid:105) q .5 Critical points and Hessian Defining a := 14 (cid:107) q (cid:107) > , b := (cid:18) − (cid:107) p (cid:107) (cid:19) (cid:107) q (cid:107) > (cid:40) ˙ q = ap ˙ p = − bq (5.32)Since our solution has to be periodic we conclude that b has to be non-negative.We claim that b has to be actually positive. Otherwise, we have the ode ˙ q = ap and ˙ p = 0. Therefore ¨ q = 0, thus q is linear. But since the solution has to beperiodic q has to be constant. This implies that ˙ q and p are zero, in particular (cid:107) p (cid:107) = 0. Hence b (cid:54) = 0. Contradiction. This shows that b > q = − abq, q ( t ) = c cos √ abt + d sin √ abt Thus ˙ q = − c √ ab sin √ abt + c √ ab cos √ abt and p ( t ) = ˙ q ( t ) a = − c (cid:112) b/a sin √ abt + c (cid:112) b/a cos √ abt Since our solutions q, p are periodic with period 1 we must have √ ab = 2 πk, k ∈ N (5.33)Thus q ( t ) = C cos 2 πkt + D sin 2 πkt (5.34)where C and D are constants and p ( t ) = − C πka sin 2 πkt + D πka cos 2 πkt (5.35)Note that integrating the identity 1 = cos + sin we get1 = (cid:90) dt = (cid:90) cos dt + (cid:90) sin dt = 2 (cid:90) cos (2 πkt ) dt (5.36)as is well known from the theory of Fourier series. Moreover, from the theoryof Fourier series it is known that cosine is orthogonal to sine, that is0 = (cid:90) cos(2 πkt ) sin(2 πkt ) dt Therefore (cid:107) q (cid:107) = ( C + D ) / a the value a = 12 ( C + D )22imilarly we get (cid:107) p (cid:107) = C + D (cid:0) πka (cid:1) = 2( C + D ) (2 πk ) and so for b we get b = 8( C + D ) − C + D )(2 πk ) Using (5.33) we get that(2 πk ) = ab = 4( C + D ) − (2 πk ) , k ∈ N (5.37)We fix the parametrization of our solution by requiring that at time zero thesolution is maximal. Therefore D = 0 and we abbreviate c k := C ( k ) >
0. Then c k is uniquely determined by k via the above equation which becomes4 c k = 8 π k , c k = 2( πk ) (5.38)Hence c k = 12 ( πk ) ∈ (0 , , c k = 12 ( πk ) (5.39)Note that c k <
1. Thus for each k ∈ N there is a solution of (5.32), namely (cid:40) q k ( t ) = c k cos 2 πktp k ( t ) = − c k (2 πk ) sin 2 πkt k ∈ N (5.40) Linearization
Linearizing the Hamilton equations (5.30) at a solution ( q, p ) ∈ D we get ˙ ξ = − p (cid:107) q (cid:107) (cid:104) q, ξ (cid:105) + η (cid:107) q (cid:107) ˙ η = (cid:18) ξ (cid:107) q (cid:107) − q (cid:107) q (cid:107) (cid:104) q, ξ (cid:105) (cid:19) (cid:18) (cid:107) p (cid:107) − (cid:19) + 12 q (cid:107) q (cid:107) (cid:104) p, η (cid:105) (5.41)for function pairs ζ = ( ξ, η ) ∈ W , ( S , R ) = T ( q,p ) D Consider the linear operator defined for ( q, p ) ∈ D by S = S ( q,p ) : W , ( S , R ) → W , ( S , R ) (cid:18) ξη (cid:19) (cid:55)→ (cid:32)(cid:16) ξ (cid:107) q (cid:107) − q (cid:107) q (cid:107) (cid:104) q, ξ (cid:105) (cid:17) (cid:16) (cid:107) p (cid:107) − (cid:17) + q (cid:107) q (cid:107) (cid:104) p, η (cid:105) p (cid:107) q (cid:107) (cid:104) q, ξ (cid:105) − η (cid:107) q (cid:107) (cid:33) and the linear operator defined by A ( q,p ) = L S : L ( S , R ) ⊃ W , → L ( S , R ) ζ := (cid:18) ξη (cid:19) (cid:55)→ − J ˙ ζ − I S ζ I : W , ( S , R ) (cid:44) → L ( S , R ) is the compact operator given by inclusion.The kernel of the operator L S is composed of the solutions to the linearizedequations (5.41). If ( q, p ) ∈ Crit A H is a critical point, then L S is equal to theHessian operator A ( q,p ) of A H at ( q, p ). Fix k ∈ N and let ( q k , p k ) be the solution (5.40) of the Hamiltonian equa-tion (5.30). The square of the L norm of the solution is given by (cid:107) q k (cid:107) = c k πk ) , (cid:107) p k (cid:107) = 2(2 πk ) c k = 4 (5.42)Abbreviating ( q, p ) := ( q k , p k ), we look for reals λ and functions ζ = ( ξ, η ) with L S ζ := − J ˙ ζ − I S ζ = λζ, S = S ( q k ,p k ) Apply J to both sides of the eigenvalue problem to obtain equivalently (cid:18) − ληλξ (cid:19) = J λζ = J L S ζ = (cid:0) ∂ t − J I S ( q,p ) (cid:1) ζ = ˙ ξ + p (cid:107) q (cid:107) (cid:104) q, ξ (cid:105) − η (cid:107) q (cid:107) ˙ η − (cid:16) ξ (cid:107) q (cid:107) − q (cid:107) q (cid:107) (cid:104) q, ξ (cid:105) (cid:17) (cid:16) (cid:107) p (cid:107) − (cid:124) (cid:123)(cid:122) (cid:125) = − (cid:17) − q (cid:107) q (cid:107) (cid:104) p, η (cid:105) Resolving for the first order terms and substituting (cid:107) p (cid:107) = 4 the ode becomes˙ ζ = (cid:18) ˙ ξ ˙ η (cid:19) = (cid:32) − λη − p (cid:107) q (cid:107) (cid:104) q, ξ (cid:105) + η (cid:107) q (cid:107) λξ − (cid:16) ξ (cid:107) q (cid:107) − q (cid:107) q (cid:107) (cid:104) q, ξ (cid:105) (cid:17) + q (cid:107) q (cid:107) (cid:104) p, η (cid:105) (cid:33) = J λζ + J I S ζ Substitute first (cid:107) q (cid:107) = c k and then ( q, p ) := ( q k , p k ) according to (5.40) to get (cid:18) ˙ ξ ˙ η (cid:19) = (cid:32) − λη − pc k (cid:104) q, ξ (cid:105) + η c k λξ − (cid:16) ξc k − qc k (cid:104) q, ξ (cid:105) (cid:17) + qc k (cid:104) p, η (cid:105) (cid:33) = − (cid:16) λ − c k (cid:17) η + 8 πk sin 2 πkt (cid:104) cos 2 πk · , ξ (cid:105) (cid:16) λ − c k (cid:17) ξ + cos 2 πktc k (cid:104) cos 2 πk · , ξ (cid:105) − πk cos 2 πkt (cid:104) sin 2 πk · , η (cid:105) We write the periodic absolutely continuous maps ξ, η : S → R as Fourier series (cid:40) ξ = ξ + (cid:80) ∞ n =1 ( ξ n cos 2 πnt + ξ n sin 2 πnt ) η = η + (cid:80) ∞ n =1 ( η n cos 2 πnt + η n sin 2 πnt )We set ξ = η = 0. Take the derivative to get that (cid:40) ˙ ξ = (cid:80) ∞ n =1 ( − πn · ξ n sin 2 πnt + 2 πn · ξ n cos 2 πnt )˙ η = (cid:80) ∞ n =1 ( − πn · η n sin 2 πnt + 2 πn · η n cos 2 πnt )24y the orthogonality relation and (5.36) we obtain (cid:104) cos 2 πn · , ξ (cid:105) = 12 ξ n , (cid:104) sin 2 πn · , η (cid:105) = 12 η n , n ∈ N Let n ∈ N . Comparing coefficients we obtain from the first equations abovesin 2 πnt − πn · ξ n = − (cid:16) λ − c k (cid:17) η n , n (cid:54) = k − πk · ξ k = − (cid:16) λ − c k (cid:17) η k + 4 πk · ξ k , n = k cos 2 πnt (cid:110) πn · ξ n = − (cid:16) λ − c k (cid:17) η n , ∀ n and from the second equations cos 2 πnt πn · η n = (cid:16) λ − c k (cid:17) ξ n , n (cid:54) = k πk · η k = (cid:16) λ − c k (cid:17) ξ k + c k ξ k − πkη k , n = k sin 2 πnt (cid:110) − πn · η n = (cid:16) λ − c k (cid:17) ξ n , ∀ n Simplifying we get from the first equations sin 2 πnt a) 2 πn · ξ n = (cid:16) λ − c k (cid:17) η n , n (cid:54) = k b) 6 πk · ξ k = (cid:16) λ − c k (cid:17) η k , n = k cos 2 πnt (cid:110) c) 2 πn · ξ n = − (cid:16) λ − c k (cid:17) η n , ∀ n and from the second equations sin 2 πnt (cid:110) d) − πn · η n = (cid:16) λ − c k (cid:17) ξ n , ∀ n cos 2 πnt e) 2 πn · η n = (cid:16) λ − c k (cid:17) ξ n , n (cid:54) = k f) 6 πk · η k = (cid:16) λ + c k (cid:17) ξ k , n = k Eigenvalues.
We obtain from equations c) and d) that (cid:18) λ n − c k (cid:19) (cid:18) λ n − c k (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) polynomial p k ( x ) in variable x = λ n = 4 π n , n ∈ N (5.43)The polynomial p k ( x ) is illustrated by Figure 3. Equivalently to (5.43) we obtainthe quadratic equation for λ n given by λ n − β k λ n + γ k,n = 025 ∈ R π n c k R p k c k λ + n λ − n eigenvalues ( n ∈ N ) λ − k = 04 π k λ + k eigenvalues ( n = k ) Figure 3: The parabola p k and the eigenvalues λ ∓ n for each n ∈ N where β k := 4 c k + 12 c k = 8 + c k c k , γ k,n := 2 c k − π n = 4 π (cid:0) k − n (cid:1) The solutions are λ − n = β k − (cid:113) β k − γ k,n , λ + n = β k (cid:113) β k − γ k,n Note that γ k,k = 0 and therefore λ − k = 0 (5.44)is zero as well and λ + k = β k . In case n = 0 the quadratic equation (5.43) isalready factorized, so we read off λ − = 12 c k , λ +0 = 4 c k , λ − < λ +0 both of which are real numbers. So the argument of the square root is positivefor n = 0 and therefore for all n (since − γ k,n is monotone increasing in n ). Thus Lemma 5.5 (Monotonicity) . The sequence ( λ − n ) n ∈ N is strictly monotone de-creasing and ( λ + n ) n ∈ N is strictly monotone increasing. Eigenvectors.
Recall that we had fixed k ∈ N , in other words the solution( q k , p k ) given by (5.40) of the Hamiltonian equation (5.30). We assume inaddition n (cid:54) = 0, that is n ∈ N . Eigenvectors to the eigenvalues λ ± n , notation u ± n , can be found by setting η n := 1, then according to equation c) we define ξ n ± := − πn (cid:0) λ ± n − / c k (cid:1) . The other Fourier coefficients we define to be equal0. With these choices an eigenvector for λ ± n is given by the function u ± n : S → R , t (cid:55)→ (cid:18) − πn (cid:18) λ ± n − c k (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ξ n ± sin 2 πnt, cos 2 πnt (cid:19) There appear new phenomena in case n = 0. For instance, the geometric multiplicities of λ ∓ are 1, as opposed to 2 in case n > ξ n + is strictly negative and ξ n − is strictly positive since λ + n − c k > λ +0 − c k > λ − − c k = 0and similarly λ − n − c k < λ − − c k = 0Since ξ n + > u + n winds n times counter-clockwisearound the origin, while u − n winds n times clockwise around the origin since ξ n − <
0. Therefore the winding numbers equal ± n , in symbols w ( u ± n ) = ± n Note that in the ode (local) case Lemma 5.5 would tell us that w ( λ + n ) = n , butin the non-local case we cannot yet conclude independence of the choice of aneigenvector. Remark 5.6 (Case n = 0) . Eigenvalue λ − = 1 / c k . In c) we choose η := 1 and set all other Fouriercoefficients zero. Then u − = (0 ,
1) is an eigenvector to the eigenvalue λ − . Sincethe function u − is constant, its winding number vanishes, in symbols w ( u − ) = 0. Eigenvalue λ +0 = 4 /c k . By e) we can choose ξ := 1 and all other Fouriercoefficients equal zero. For these choices u +0 = (1 ,
0) is an eigenvector to theeigenvalue λ +0 . By constancy the winding number is 0, in symbols w ( u +0 ) = 0. Remark 5.7 (Geometric multiplicity of eigenvalues λ ± n is ≥ n (cid:54) = 0 , k ) . Instead of using c) and d) one can use a) and e). Setting η n := 1 equation a)motivates to define ξ ± n := − πn (cid:0) λ + n − / c k (cid:1) . With these choices a furthereigenvector for λ ± n is given by the function v ± n : S → R , t (cid:55)→ (cid:18) πn (cid:18) λ ± n − c k (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ξ ± n cos 2 πnt, sin 2 πnt (cid:19) We observe that, just as above, the winding number of the eigenvector v + n is n ,and of v − n it is − n , in symbols w ( v ± n ) = ± n Case n = k and the eigenvalues (cid:98) λ ± k In the case n = k we obtain from equations b) and f) the quadratic equation (cid:18)(cid:98) λ k − c k (cid:19) (cid:18)(cid:98) λ k + 12 c k (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) polynomial (cid:98) p k ( x ) in variable x = (cid:98) λ k = 36 π k (5.45)27 (cid:98) p k π k (cid:98) λ + k (cid:98) λ − k − c k c k R Figure 4: Parabola (cid:98) p k in the variable x = (cid:98) λ k given by (5.45)The polynomial (cid:98) p k ( x ) is illustrated by Figure 4. Eigenvalues.
Equivalentlywe obtain the quadratic equation for (cid:98) λ k given by (cid:98) λ k − B k (cid:98) λ k + C k = 0where B k := 12 c k − c k = β k − c k , β k := 4 c k + 12 c k = 8 + c k c k C k := − c k − π k = − π k with c k < (cid:98) λ − k = B k − (cid:113) B k − C k , (cid:98) λ + k = B k (cid:113) B k − C k Eigenvectors.
Eigenvectors to the eigenvalues (cid:98) λ ± k , notation u ± k , can be foundby setting η k := 1, then equation b) motivates to define ξ k := 16 πk (cid:16)(cid:98) λ ± k − / c k (cid:17) The other Fourier coefficients we define to be equal 0. With these choices aneigenvector for (cid:98) λ ± k is given by the function v ± k : S → R , t (cid:55)→ (cid:18) πk (cid:18)(cid:98) λ ± k − c k (cid:19) cos 2 πkt, sin 2 πkt (cid:19) As one sees from Figure 4 the following inequalities hold (cid:98) λ − k < − c k < < c k < (cid:98) λ + k Therefore (cid:98) λ + k − c k > (cid:98) λ − k − c k < w ( v ± k ) = ± k (cid:98) p k π k (cid:98) λ + k (cid:98) λ − k − c k c k R p k c k λ +3 k λ − k − k − k k k ≤ > < < winding numberseigenvalues Figure 5: The parabolas (cid:98) p k and p k , eigenvalues and winding numbers Consider the two quadratic polynomials p k and (cid:98) p k in the variable x = λ n givenby the left hand sides of (5.43) and (5.45), namely p k ( x ) := (cid:18) x − c k (cid:19) (cid:18) x − c k (cid:19) and (cid:98) p k ( x ) := (cid:18) x − c k (cid:19) (cid:18) x + 12 c k (cid:19) These two polynomials have a common zero at x = 1 / c k , they are sketchedin Figure 5. For n = 3 k there is equality 4 π n = 36 π k and the intersectionof (cid:98) p k and p k with the horizontal line { π k } consists of 4 points whose x -coordinates are the following eigenvalues in the following order (cid:98) λ − k < λ − k < (cid:98) λ + k < λ +3 k Proposition 5.8.
For any n ∈ N the λ ∓ n are different from (cid:98) λ − k and from (cid:98) λ + k ,in symbols λ ∓ n (cid:54) = (cid:98) λ − k , λ ∓ n (cid:54) = (cid:98) λ + k , n ∈ N Proof.
Note that (cid:98) λ − k < λ + n >
0, for n ∈ N , aswell as λ − > (cid:98) λ − k (cid:54) = λ − n forevery n ∈ N .Suppose by contradiction that there are i, j ∈ { + , −} such that (cid:98) λ ik = λ jn =: λ for some n ∈ N . The idea is to construct two polynomials P ( z ) and Q ( z ) whichhave a common zero at z = c k and then use algebra to show that there cannotbe two such polynomials. Step 0.
It is useful to consider the field extension Q ( π ) of Q which is a subfieldof R , that is Q ⊂ Q ( π ) ⊂ R . Elements of Q ( π ) have the following form. Givenrational polynomials p, q ∈ Q [ x ] with q (cid:54)≡ Q ( π ) are given by p ( π ) /q ( π ). Note that by the theorem of Lindemann [Lin82],29ee [Hil93] for an elegant proof by Hilbert, the number π is transcendental andtherefore q ( π ) (cid:54) = 0 is non-zero. Step 1.
The definition of the polynomial Q ( z ) := z + a , a := − ( c k ) . = − π k ∈ Q ( π ) (5.46)is motivated by the goal that it has a zero at the point z = c k given by (5.39).Here Q ( π ) is the field extension of Q by adjoining π from Step 0. The fieldextension Q ( π ) is isomorphic to the field Q ( x ) of rational functions. Step 2.
Divide the polynomial identity (cid:98) p k ( λ ) = 36 π k by p k ( λ ) = 4 π n toget that λ + c k λ − c k = 36 π k π n = 9 k n Now multiply by the denominator to get that λ + 12 c k = 9 k n (cid:18) λ − c k (cid:19) Resolving for λ yields λ = 4 c k · k n + 39 k n − c k · k + 3 n k − n Consequently λ ∈ c k Q (5.47)Now evaluate (cid:98) p k at λ to get (since λ := (cid:98) λ ik ) that0 = (cid:98) p k ( λ ) − π k = (cid:18) λ − c k (cid:19) (cid:18) λ + 12 c k (cid:19) − π k (5.48)Multiplication by c k and division by − π k leads to0 = λ c k + λc k (cid:0) − c k / (cid:1) − c k − π k + c k = − π k (cid:18) k + 3 n k − n (cid:19) + 118 π k (cid:18) k + 3 n k − n (cid:19) c k − π k (cid:18) k + 3 n k − n (cid:19) + 118 π k (cid:18) k + 3 n k − n (cid:19) c k + c k = P ( c k ) (5.49) A real number is called transcendental if it is not a zero of a polynomial with rationalcoefficients. Transcendental implies irrational. Note that √ x − P is given by P ( z ) = z + b z + b , b , b ∈ Q ( π )and the coefficients b and b – according to (5.49) – by b = − π k k + 3 n k − n (cid:18) k + n k − n + 1 (cid:19) = − π k + n (9 k − n ) < b = 19 π k (cid:18) k + 3 n k − n (cid:19) We define a linear polynomial in z with coefficients in the field Q ( π ) by theformula R ( z ) := P ( z ) − zQ ( z ) = ( b − a ) z + b Since z = c k is a zero of P by (5.49) and of Q by (5.46), it is a zero of the linearpolynomial R .Since b (cid:54) = 0 it follows that b − a (cid:54) = 0: otherwise R ≡ b (cid:54) = 0 would not have azero at all. Since 0 = R ( c k ) = ( b − a ) c k + b we get that c k = − b / ( b − a ) ∈ Q ( π ), that is c k is of the form p ( π ) /q ( π ) where p, q ∈ Q [ z ].We derive a contradiction: Evaluate (5.46) at z = p ( π ) q ( π ) = c k ∈ (0 ,
1) to get0 = Q ( c k ) = p ( π ) q ( π ) − π k Multiply the identity by q ( π ) (cid:0) π k (cid:1) to get that0 = p ( π ) (cid:0) π k (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) deg=2 mod 3 − q ( π ) (cid:124) (cid:123)(cid:122) (cid:125) deg=0 mod 3 Consider the polynomial s := p r − q ∈ Q [ z ] , where r ( z ) := 2 k z . Claim. s (cid:54)≡ Proof of claim.
This follows from considering the degrees. Note thatdeg r = 2 mod 3 , deg p = 0 mod 3 , deg q = 0 mod 3Thereforedeg p r = deg p + deg r = 2 mod 3 , deg q = 0 mod 3Hence p r (cid:54) = q and consequently s (cid:54)≡
0. This proves the claim.In view of the claim we found a non-zero polynomial s with rational coef-ficients and the property that s ( π ) = 0. But this contradicts the theorem ofLindemann as explained earlier. Lindemann showed that π is transcendental: There is no non-zero polynomial with rationalcoefficients having π as a zero. orollary 5.9 (Well-defined winding number) . For every n ∈ N we define thewinding numbers w ( λ − n ) := − n, w ( λ + n ) := n, w ( (cid:98) λ − k ) := − k, w ( (cid:98) λ + k ) := k In view of Proposition 5.8 these winding numbers are well-defined and, in viewof the discussion before, correspond to the winding number of an arbitrary eigen-vector of the eigenvalue. spec L S λ − n λ − k = 0 λ − λ +0 λ + k λ + n (cid:98) λ − k (cid:98) λ + k m w − n − k k n Figure 6: Multiplicities m and winding numbers w of eigenvalues, n ∈ N \ { k } Proposition 5.10.
At a critical point ( q k , p k ) of A H , see (5.40), it holds that α ( S ( q k ,p k ) ) = w ( (cid:98) λ − k ) = − k, p ( S ( q k ,p k ) ) = 1 and with the definitions µ CZ := 2 α + p and µ CZ := − µ CZ we obtain µ CZ ( q k , p k ) = − k + 1 , µ CZ ( q k , p k ) = 2 k − Proof.
With S := S ( q k ,p k ) we recall the definition of α , namely α ( S ) := max { w ( λ ) | λ ∈ ( −∞ , ∩ spec L S } ∈ Z Observe that (cid:98) λ − k <
0, see Figure 4, and w ( (cid:98) λ − k ) = − k , see Figure 6. Therefore α ( S ) ≥ − k . To show the reverse inequality α ( S ) ≤ − k we need to checknon-negativity of all eigenvalues with winding number > − k . By Figure 6 theeigenvalues of winding number > − k are of three types:(i) λ − n for n < k : In this case λ − n > λ − k = 0 for every n < k by monotonicity,see Lemma 5.5, and (5.44).(ii) λ + n for all n ∈ N : In this case λ + n > n ∈ N by Figure 3(iii) (cid:98) λ + k : In this case (cid:98) λ + k > α ( S ) = − k .Because there exists a non-negative eigenvalue, namely λ − k = 0, with thesame winding number − k as the negative eigenvalue (cid:98) λ − k < α ( S ( q k ,p k ) ) among negative eigenvalues, we get that p ( S ( q k ,p k ) ) = 132 eferences [AFS19a] Peter Albers, Urs Frauenfelder, and Felix Schlenk. A compactnessresult for non-local unregularized gradient flow lines. J. Fixed PointTheory Appl. , 21(1):34–61, 2019. arXiv:1802.07445.[AFS19b] Peter Albers, Urs Frauenfelder, and Felix Schlenk. An iterated graphconstruction and periodic orbits of Hamiltonian delay equations.
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