ON THE SUBRIEMANNIAN GEOMETRY OF CONTACT ANOSOV FLOWS SLOBODANN.SIMIC´ Abstract. We investigate certain natural connections between subriemannian geometry and hy- 5 perbolic dynamical systems. In particular, we study dynamically defined horizontal distributions 1 which split intotwointegrable onesandask: howis theenergy ofasubriemannian geodesic shared 0 between itsprojections ontotheintegrable summands? Weshowthat ifthehorizontaldistribution 2 is the sum of the strong stable and strong unstable distributions of a special type of a contact n Anosov flow in three dimensions, then for any short enough subriemannian geodesic connecting a points on the same orbit of the Anosov flow, the energy of the geodesic is shared equally between J its projections onto the stable and unstable bundles. The proof relies on a connection between 4 thegeodesic equations andtheharmonicoscillator equation,andits explicit solution bytheJacobi 1 elliptic functions. Using a different idea, we prove an analogous result in higher dimensions for the ] geodesic flow of a closed Riemannian manifold of constant negative curvature. G D . h t 1. Introduction a m The goal of this paper is to investigate certain natural but insufficiently explored connections [ between hyperbolic dynamical systems and subriemannian (or Carnot-Carath´eodory) geometry. A 1 subriemannian geometry on a smooth connected manifold M is a geometry defined by a nowhere v 2 integrable distribution E, called a horizontal distribution, equipped with a Riemannian metric g. 7 Both E and g are required to be at least continuous but in most scenarios they are usually C . ∞ 4 3 Since we can extend any partially defined Riemannian metric to the entire tangent bundle and the 0 extension does not affect the properties of the subriemannian geometry, we will always assume that . 1 g is defined on the entire tangent bundle. 0 If γ : [a,b] M is a horizontal (i.e., tangent to E) path, its length is defined in the usual way by 5 → 1 b : v γ = γ˙(t) dt, | | Z k k i a X where v = g(v,v), for any vector v E. r a k k ∈ A horizontapl distribution E on M is called nowhere integrable if for every p M and every ε > 0 ∈ there exists a neighborhood U of p in M such that every point in U can be connected to p by a horizontal pathof length < ε. Inparticular, every two pointsof M can beconnected by ahorizontal path. This definition avoids certain undesirable pathological behavior which can arise if E is not smooth; see [Sim10]. The subriemannian distance between x,y M is given by ∈ d (x,y) = inf γ : γ is a horizontal path from x to y . H {| | } A subriemannian geodesic from x to y is any horizontal path γ which minimizes length among all horizontal paths connecting x and y. Thus γ = d (x,y). H | | Date: January 15, 2015. Keywordsandphrases. Subriemanniangeodesic;contactAnosovflow;harmonicoscillator;Jacobiellipticfunction. 1 2 S.N.SIMIC´ Recall that a C horizontal distribution E is called bracket generating if any local smooth frame ∞ X ,...,X for E together with all its iterated Lie brackets span the entire tangent bundle of M. 1 k { } (In the PDE literature, the bracket generating condition is called the H¨ormander condition.) By the Chow-Rashevskii theorem [Mon02] any bracket generating distribution is nowhere integrable. It is sometimes the case that a horizontal distribution E splits into two integrable orthogonal distributions, E = E E , and E is in turn orthogonal to a globally defined vertical distribution 1 2 ⊕ V, with TM = E V. If E is bracket-generating, then any motion in the vertical direction ⊕ is due to the fact that iterated Lie brackets of vector fields in E and those in E generate the 1 2 entire tangent bundle. Given a “vertical” curve c tangent to V with endpoints x and y and a unit speed subriemannian geodesic γ connecting x and y, it is natural to ask the following question (see Figure 1). V y c γ E x 1 E E 2 Figure 1. A subriemannian geodesic γ connecting x = c(0) and y = c(1), where c is a path tangent to the vertical bundle V. Question. If TM = E V and E = E E , how is the energy of a subriemannian geodesic γ 1 2 ⊕ ⊕ connecting endpoints of a curve tangent to V shared between its projections onto E and E ? 1 2 Stated more precisely, if γ : [0,ℓ] M is a unit speed horizontal path and E = E E , then 1 2 → ⊕ γ˙(t)= w (t)+w (t), with w (t) E . We define 1 2 i i ∈ 1 ℓ (γ) = w (t) 2 dt, i i E ℓ Z k k 0 for i = 1,2, and think of (γ) as the energy of the projection of γ to E . Clearly, 0 (γ) 1 i i i E ≤ E ≤ and (γ)+ (γ) = 1. If (γ) = (γ) we call γ a (E ,E )-balanced horizontal path. The above 1 2 1 2 1 2 E E E E question therefore asks if every subriemannian geodesic connecting endpoints of a vertical path is (E ,E )-balanced. 1 2 Example 1 (The Heisenberg group). The Heisenberg group is a subriemannian geometry on M = R3 definedby thehorizontal distributionE whichis thekernel ofthe1-form α = dz 1(xdy ydx). −2 − The Riemannian metric on E is defined by ds2 = dx2+dy2. The vector fields ∂ y ∂ ∂ x ∂ X = and X = 1 2 ∂x − 2∂z ∂y − 2∂z form a global orthonormal frame for E. It is not hard to check that [X ,X ] = ∂ =: X , so 1 2 ∂z 0 E is bracket-generating. Since [X ,X ] = [X ,X ] = 0, the Heisenberg group is nilpotent. We 0 1 0 2 SUBRIEMANNIAN GEOMETRY OF CONTACT ANOSOV FLOWS 3 will show in 2.2 that every Heisenberg subriemannian geodesic whose endpoints differ only in the § z-coordinate is (E ,E )-balanced. This follows easily from the fact that Heisenberg geodesics are 1 2 lifts of circles in the xy-plane. Subriemannian geometries whose horizontal distributions have a natural splitting into two inte- grable distributions occur frequently in hyperbolic and partially hyperbolicdynamical systems. For instance, if f is a partially hyperbolic diffeomorphism of a compact manifold M, then f preserves two invariant bundles called the stable Es and the unstable Eu bundles, both uniquely integrable. Transversetothemisthecenter bundleEc, whichisnotalways integrable. AlthoughEs andEu are usuallynotsmooth,their sumEsu = Es Eu frequently hasthesocalled accessibilityproperty. This ⊕ meansthatanytwopointsinM canbejoinedbyacontinuous piecewisesmoothpathwhosesmooth legs are alternately tangent to Es and Eu. Thus Esu naturally defines a subriemannian geometry on M and we can take H = Esu and V = Ec. Accessibility plays an important role in partially hyperbolic dynamics where it is an essential ingredient in the study of stably ergodic systems and the Pugh-Shub conjecture [PS04]. The main difficulty with the subriemannian geometry defined by Esu is that it lacks smoothness, so it is not amenable to analysis using standard techniques. In this paper we consider the case of contact Anosov flows, where the natural horizontal distri- bution is always at least C1. This is a scenario which is in a sense diametrically opposite to that of the Heisenberg group. Recall that a non-singular smooth flow Φ = f on a closed (compact and without boundary) t { } Riemannian manifold M is called an Anosov flow if there exists an invariant splitting TM = Ess ⊕ Ec Euu such that Ec is spanned by the infinitesimal generator X of the flow, Ess is uniformly ⊕ exponentially contracted and Euu is uniformly exponentially expanded by the flow in positive time. We call Ess and Euu the strong stable and strong unstable bundles; Ec is the center bundle. A contact structure on a manifold M of dimension 2n+1 is a C1 hyperplane field E which is as far from being integrable as possible [MS99]. This means that there exists a C1 1-form α such that Ker(α) = E and α (dα)n is a volume form for M; α is called a contact form for E. Contact ∧ structures are always bracket-generating. A vector field X is called the Reeb vector field of α if α(X) = 1 and X is in the kernel of dα, i.e., i dα = 0. An Anosov flow is called contact if E = Esu is a contact structure (and in particular X C1) and the infinitesimal generator X of the flow is the Reeb vector field for the contact form α for Esu with α(X) = 1. Our goal is to understand the subriemannian geometry defined by the distribution Esu associated with a contact Anosov flow. We will call subriemannian geodesics of this geometry su-subriemannian geodesics. An su-subriemannian geodesic will be called su-balanced if it is balanced with respect to the splitting Ess Euu. ⊕ Contact Anosov flows have good dynamical properties; in particular, they exhibit exponential decay of correlations (cf., [Liv04]). Until recently however, the only known contact Anosov flows were the geodesic flows of Riemannian or Finsler manifolds; in [FH13] Foulon and Hasselblatt used surgery near a transverse Legendrian knot to construct many new contact Anosov flows on 3-manifolds which are not topologically orbit equivalent to any algebraic flow. AssumenowthatΦisacontactAnosovflowona3-manifoldM. Denoteitsinfinitesimalgenerator byX andletY andZ beunit(withrespecttosomeRiemannianmetricg whosevolumeformequals the contact volume form) vector fields in Ess and Euu respectively. Then T f (Y) = µ(x,t)Y and x t T f (Z) = λ(x,t)Z, for some 1-cocycles µ,λ : M R R, with λµ = 1, where Tf denotes the x t t × → 4 S.N.SIMIC´ tangent map (i.e., derivative) of the time-t map f of the flow. Thus t [X,Y] = aY and [X,Z] = aZ, (1.1) − where a(x) = µ˙(x,0) = λ˙(x,0), for all x M. Since Esu is contact, it follows that [Y,Z] is − ∈ transverse to Esu. Definition. An Anosov flow Φ on a 3-dimensional closed manifold will be called a special contact Anosov flow if there exists a C1 Riemannian metric g and a C1 global orthonormal frame (X,Y,Z) relative to g such that: (a) Ec = RX, Ess = RY and Euu = RZ. (b) [X,Y] = Y, [Y,Z] = X and [Z,X] = Z. Note that X,Y,Z, and g are required to be only C1. The following lemma shows that must in fact be C . ∞ Lemma 1.1. If Φ is a special contact Anosov flow, then (with the notation as above), X,Y and Z are all C . ∞ Proof. Let (α,β,γ) be the coframe dual to (X,Y,Z). Since X,Y and Z are C1, so are α,β and γ. We have: 1 = α(X) = α([Y,Z]) = Yα(Z) Zα(Y) dα(Y,Z) − − = dα(Y,Z). − We can show in a similar way that dα(X,Y)= dα(X,Z) = 0. Therefore, dα is C1 relative to a C1 frame, hence α is, in fact, C2. It follows analogously that β and γ are also C2. Hence X,Y and Z are all C2 as well. By bootstrap, it follows that X,Y and Z are in fact C . (cid:3) ∞ Remark. Observe that if Φ is a special contact Anosov flow, then X,Y and Z span a copy of the Lie algebra sl(2,R), so the universal cover of M is the Lie group SL(2,R) and the lift of the Anosov flow to the universal cover is an algebraic one. In particular, M is a quotient of SL(2,R) by a discrete cocompact subgroup. In other words, special contact Anosov flows are precisely algebraic Anosov flows. Ghys [Ghy87] showed that in three dimensions every contact Anosov flow Φ with C strong ∞ bundles is C equivalent to an algebraic flow on a quotient N = Γ SL(2,R), in the sense that ∞ \ there exists a C diffeomorphism h : N M that sends the orbits of the “diagonal” flow on ∞ → f N to the orbits of Φ. Therefore, every contact Anosov flow with C strong bundles is C orbit ∞ ∞ equivalent to a special Anosov flow. Our main results are the following. Theorem A. Let Φ = f be a special contact Anosov flow on a closed Riemannian 3-manifold t { } M. Then there exists a δ > 0 such that for all x M and t < δ, every su-subriemannian geodesic ∈ | | connecting x and f (x) is su-balanced. t In higher dimensions we prove a result analogous to Theorem A if the contact Anosov flow is the geodesic flow on the unit tangent bundle of a manifold with constant negative sectional curvature. SUBRIEMANNIAN GEOMETRY OF CONTACT ANOSOV FLOWS 5 Theorem B. Let Φ = f be the geodesic flow of a closed Riemannian manifold N with constant t { } negative sectional curvature on its unit tangent bundle M. Then there exists a δ > 0 such that for all x M and t < δ, every su-subriemannian geodesic connecting x and f (x) is su-balanced. t ∈ | | Outline of the paper. In Section 2 we review some basic results on Anosov and geodesic flows, § subriemannian geodesics, and the solution of the harmonic oscillator equation via Jacobi elliptic functions. Theorem A is proved in Section 3 and Theorem B in Section 4. We conclude the paper § § with a list of open questions in Section 5. § Acknowledgments. We are grateful to Alan Weinstein, who generously offered the main idea of proof of Theorem B. We would also like to thank him for many inspiring conversations and moral support over the years. 2. Preliminaries 2.1. Anosov flows. A non-singular smooth flow Φ = f on a closed Riemannian manifold M t { } is called an Anosov flow if there exists an invariant splitting TM = Ess Ec Euu such that ⊕ ⊕ Ec is spanned by the infinitesimal generator of the flow and there exist uniform constants c > 0, 0< µ µ < 1 and λ λ > 1 such that for all v Ess, w Euu, and t 0, we have + + − ≤ ≥ − ∈ ∈ ≥ 1 µt v Tf (v) cµt v , (2.1) c −k k ≤ k t k ≤ +k k and 1 λt w Tf (w) cλt w . (2.2) c −k k ≤ k t k ≤ +k k The strong stable Ess and strong unstable bundles Euu are in general only H¨older continuous [HPS77], but they are nevertheless always uniquely integrable giving rise to the strong stable and strongunstablefoliations denoted by Wss and Wuu, respectively. Thecodimension onedistribution Esu = Ess Euu is generally not integrable; if it is, then by Plante [Pla72], the flow admits a global ⊕ cross section and is therefore topologically conjugate to a suspension of an Anosov diffeomorphism. The bundles Ecs = Ec Ess and Ecu = Ec Euu are called the center stable and center unstable ⊕ ⊕ bundles. They are generically only H¨older continuous [HPS77], but are always uniquely integrable [Ano67]. However, if dimM = 3 and the flow is C3 and preserves the Riemannian volume, then it follows from the work of Hurder and Katok [HK90] that Ecs and Ecu are both of class C1 and the transverse derivatives of both bundles are Cθ-H¨older, for all 0 < θ < 1. WithoutlosswewillalwaysassumethatalltheinvariantbundlesofanAnosovflowareorientable. (If not, we can pass to a double cover.) We will need the following easy lemma. Lemma 2.1. Let Φ be a contact Anosov flow on a 3-manifold M. Then Euu and Ess are both C1. Proof. Since Esu is C1 by assumption and Ecs and Ecu are C1 by [HK90], it follows that Ess = Ecs Esu and Euu = Ecu Esu are also C1. (cid:3) ∩ ∩ Geodesic flows. In this section we briefly review some basic facts about geodesic flows. If N is a Riemannian manifold, then its geodesic flow Φ = f restricted to the unit tangent bundle t { } M = T1N of N admits a canonical contact form (cf., [Pat99]). If the sectional curvature K of N is negative, then Φ is known to be of Anosov type [Ano67, Ebe73], in which case Esu is a contact structure and TM = Ec Esu is an orthogonal splitting with respect to the Sasaki metric [Pat99]. ⊕ 6 S.N.SIMIC´ If the sectional curvature K is constant (and negative), then Ess and Euu are C , but if K is ∞ variable, then Ess and Euu are only of class C1+θ, for some 0 < θ < 1 [HP75]. Assume now that K is constant and negative. Without loss we can assume that K = 1. Then − (cf., [Ano67]) there exists a constant c > 0 such that Tf (v) = e ct v and Tf (w) = ect w , t − t k k k k k k k k for all t R, v Ess and w Euu. In other words, the flow contracts all stable directions and ∈ ∈ ∈ expands all unstable directions at the same rates at all points of M. For simplicity, we will assume that c= 1; this can always be achieved by a constant time change. Let F be an isometry of N and denote by F the restriction of TF to the unit tangent bundle ∗ M = T1N of N. Then for any unit-speed geodesic t c(t) in N, we have F (c˙(t)) = F (f c˙(0)) = t 7→ ∗ ∗ f (F (c˙(0)), which implies that F preserves the geodesic vector field X. We claim that F also t ∗ ∗ ∗ preserves the strong stable Wss and strong unstable Wuu foliations of the geodesic flow. Indeed, since F f = f F , for any v ,v in the same Wss-leaf, we have: t t 1 2 ∗◦ ◦ ∗ d(f (F (v )),f (F (v ))) = d(F (f (v )),F (f (v ))) t 1 t 2 t 1 t 2 ∗ ∗ ∗ ∗ Kd(f (v ),f (v )) t 1 t 2 ≤ 0, → as t , where K is the Lipschitz constant of F (which is finite, since M is compact and F is → ∞ ∗ ∗ smooth) and d denotes the distance function on M induced by the Sasaki metric. Therefore, F (v ) 1 ∗ andF (v ) lie in the same Wss-leaf. ThusF preserves Wss and TF preserves Ess. It can similarly 2 ∗ ∗ ∗ beshownthat Euu is also invariant with respectto TF . ThusTF preserves thesplitting Ec Esu. An analogous statement is true for any lift F˜ of F to∗the univers∗al Riemannian covering sp⊕ace M˜ ∗ ∗ of M. Recall that if K = 1, then N = Hn/Γ, where Γ is a group of isometries of Hn acting freely and − properlydiscontinuouslyonit[Boo03]. ItisclearthatΓalsoactsfreelyandproperlydiscontinuously on the unit tangent bundle T1Hn of Hn and that (T1Hn)/Γ is isometric to T1(Hn/Γ) = T1N = M. ThusM˜ is isometric to theuniversal Riemannian covering space of (T1Hn)/Γ. Since n 3, T1Hn is ≥ simply connected, so it is the universal covering space of (T1Hn)/Γ. Thus M˜ is isometric to T1Hn. Lemma 2.2. For all u˜,v˜ M˜ there exists an isometry F of M˜ such that F(u˜) = v˜ and F leaves ∈ the lift X˜ of the geodesic vector field X invariant. Proof. By the above observation, M˜ is isometric to T1Hn, so we can identify u˜,v˜ with unit tangent vectors to Hn at some points x,y Hn, respectively. There exists an isometry f of Hn such that ∈ f(x) = y and Tf(u˜) = v˜ (see [Boo03]). Since isometries map geodesics to geodesics, Tf leaves X˜ invariant. Thus F = Tf↾T1Hn has the desired properties. (cid:3) 2.2. Subriemannian geodesics. In this section we briefly review subriemannian geodesic equa- tions. We follow [Mon02]. Let E be a bracket-generating distribution on a smooth manifold M. For each smooth vector field X on M we define the momentum function P :T M R by X ∗ → P (p) = p(X), X for any p T M, where T M is the cotangent bundle of M. Thus the momentum function of X is ∗ ∗ ∈ just the evaluation of any covector on M at X. SUBRIEMANNIAN GEOMETRY OF CONTACT ANOSOV FLOWS 7 The subriemannian Hamiltonian H of E is the map H : T M R defined by ∗ → 1 H(p) = p,p , 2h i where , denotesthecometriconT M inducedbytheRiemannianmetricg onE (see[Mon02]). If ∗ h· ·i (X ,...,X )is a local horizontal frame andg = g(X ,X ), then theHamiltonian can beexpressed 1 k ij i j as 1 H = gijP P , 2 Xi Xj Xi,j where gij are the entries of the inverse of the matrix [g ]. In particular, if (X ,...,X ) is a local ij 1 k orthonormal frame for E, then k 1 H = P2 . 2 Xi Xi=1 The normal geodesic equation for E is the equation f˙= f,H , (2.3) { } where f : T M R is a smooth function and f,H denotes the Poisson bracket of f and H. ∗ → { } Projections of the solutions to (2.3) to M are called normal geodesics. Recall that f,H = ω(X ,X ), f H { } where ω is the (canonical) symplectic form on T M and X ,X are the Hamiltonian vector fields ∗ f H definedbyf,H,respectively. Itiswell-knownthatthePoissonbracketdefinesaLiealgebrastructure ontheringofsmoothfunctionsonT M andthatthemapf f,H satisfiestheLeibnizrulethus ∗ 7→ { } definingavectorfieldonT M (whichofcourseisexactlyX ). Recallalsothat P ,P = P , ∗ H X Y [X,Y] { } − for any smooth vector fields X,Y on M. The equation (2.3) is to be interpreted in the following way: if t p(t) is an integral curve of 7→ the Hamiltonian vector field X and if f : T M R is any smooth function, then H ∗ → d f(p(t))= f,H (p(t)). dt { } In canonical coordinates (x ,...,x ;p ,...,p ), where (x ,...,x ) are local coordinates on M and 1 n 1 n 1 n p = P , the geodesics equations assume the familiar form: i ∂/∂xi ∂H ∂H x˙ = , p˙ = . i i ∂p −∂x i i Theorem 2.3 ([Mon02]). Let t Γ(t) be a solution to the normal geodesic equation (2.3) and let 7→ γ be its projection to M. Then every sufficiently short arc of γ is a subriemannian geodesic. If E is a 2-step distribution, then every subriemannian geodesic is normal. Recall that E is a 2-step distribution if for any local frame X ,...,X for E, the vector fields 1 k X ,...,X together with their first-order Lie brackets [X ,X ] (1 i,j k) generate the entire 1 k i j ≤ ≤ tangent bundle. Assume now dimM = 3 and E is a contact structure. It is easy to see that E is a 2-step distribution. Let (X ,X ) be a local orthonormal frame for E and α a contact form for E. Denote 1 2 8 S.N.SIMIC´ the Reeb field of α by X . Clearly, (X ,X ,X ) is a local frame for TM. The structure contants 0 0 1 2 of the frame (X ,X ,X ) are smooth functions ck defined by 0 1 2 ij 2 [X ,X ] = ckX . i j ij k Xk=0 It follows that 2 P ,P = ckP . { Xi Xj} − ij Xk Xk=0 The subriemannian Hamiltonian corresponding to the frame (X ,X ) is 1 2 1 H = (P2 +P2 ). 2 X1 X2 Introduce fiberwise coordinates (P ,P ,P ) on T M. In these coordinates the normal geodesic X0 X1 X2 ∗ equations are x˙ = P X +P X X1 1 X2 2 P˙ = P ,H , Xi { Xi } for i= 0,1,2. Example 2 (TheHeisenberggroup,continued). WewillshowthateveryHeisenbergsubriemannian geodesic whose endpoints differ only in the z-component is balanced with respect to the splitting E = E E . See Example 1. Since (X ,X ) is an orthonormal frame for E, the subriemannian 1 2 1 2 ⊕ Hamiltonian is 1 H = (P2 +P2 ) 2 X1 X2 and the subriemannian geodesic equation is f˙= f,H . { } Using [X ,X ] = ∂/∂z =: X , [X ,X ] = [X ,X ] = 0, we obtain P ,P = P , 1 2 0 0 1 0 2 { X1 X2} − X0 P ,P = P ,P = 0. Therefore the subriemannian geodesic equations are { X0 X1} { X0 X2} p˙ = P X +P X X1 1 X2 2 P˙ = 0 X0 P˙ = P P X1 − X0 X2 P˙ = P P , X2 X0 X1 where p = (x,y,z). Since geodesics travel at constant speed, we can restrict the equations to the level set P2 +P2 = 1 of H and reparametrize P and P by X1 X2 X1 X2 P = cosθ, P = sinθ. X1 X2 It is not hard to check that the last three geodesic equations are equivalent to θ˙ = P , P˙ = 0. X0 X0 Thus θ¨ = 0, so θ(t) = v t +θ , where v = θ˙(0) = P (0) and θ = θ(0). It follows that every 0 0 0 X0 0 Heisenberg geodesic satisfies p˙ = cos(v t+θ )X +sin(v t+v )X , 0 0 1 0 0 2 with real parameters v and θ as above. Note that x˙ = cos(v t+θ ) and y˙ = sin(v t+θ ). 0 0 0 0 0 0 SUBRIEMANNIAN GEOMETRY OF CONTACT ANOSOV FLOWS 9 Now assume that a subriemannian geodesic γ :[0,ℓ] R3 connects two points which differ only → in the z-coordinate, i.e., they lie on an orbit of the flow of X . Projecting to the xy-plane we obtain 0 ℓ ℓ cos(v t+θ )dt = x(ℓ) x(0) = 0, sin(v t+θ )dt = y(ℓ) y(0) = 0. 0 0 0 0 Z − Z − 0 0 Thus v ℓ must be an integer multiple of 2π. Let us show that γ is balanced with respect to the 0 splitting E = RX RX . We have 1 2 ⊕ 1 ℓ 1 ℓ (γ) = cos2(v t+θ )dt and (γ) = sin2(v t+θ )dt. 1 0 0 2 0 0 E ℓ Z E ℓ Z 0 0 Hence 1 ℓ (γ) (γ) = cos2(v t+θ ) sin2(v t+θ ) dt 1 2 0 0 0 0 E −E ℓ Z { − } 0 1 ℓ = cos2(v t+θ )dt 0 0 ℓ Z 0 = 0, since v ℓ = 2πn, for some integer n. 0 2.3. Harmonic oscillator and Jacobi elliptic functions. To make the paper as self-contained as possible, we review in some detail the method of explicitly solving the harmonic oscillator (i.e., unforced undamped pendulum) equation θ¨+ω2sinθ = 0 (2.4) by the Jacobi elliptic functions sn and cn, defined below. We closely follow [Mey01], adding results we need along the way. Let 0 < k < 1. TheJacobi ellipticfunctions sn(t,k), cn(t,k) and dn(t,k) are definedas the unique solutions x(t),y(t) and z(t) of the system of differential equations x˙ = yz y˙ = zx − z˙ = k2xy, − satisfying the initial conditions x(0) = 0, y(0) = 1, z(0) = 1. The paramater k is called the modulus. Some basic properties of sn,cn and dn are listed in the following proposition whose proof can be found in [Mey01]. Proposition 1. (a) The Jacobi elliptic functions sn, cn and dn are analytic and defined for all real t. (b) sn2(t,k)+cn2(t,k) = 1 and k2sn2(t,k)+dn2(t,k) = 1, for all t R and 0< k < 1. ∈ (c) Let K = K(k) > 0 be the unique number such that cn(K,k) = 0 and cn(t,k) > 0, for all 0 < t < K. That is, K is the time it takes cn(t,k) to decrease to 0 from its initial value 1. Then sn(t,K) and dn(t,k) are even about K and cn(t,k) is odd about K. (d) sn(t,k) and cn(t,k) are 4K-periodic in t and dn(t,k) is 2K-periodic in t. (e) The function x(t) = sn(t,k) is the unique solution to the initial value problem (x˙)2 = (1 x2)(1 k2x2), x(0) = 0, x˙(0) = 1. − − 10 S.N.SIMIC´ Observe that sn(t,k) and cn(t,k) have the same symmetries with respect to K as sint and cost have with respect to π/2. Corollary 2.4. If b b sn(t;k)dt = cn(t;k)dt = 0, Z Z a a then b a is an integer multiple of 4K. Moreover, − b sn(t;k)cn(t;k)dt =0. Z a Proof. The proof follows from parts (c) and (d) of the previous Proposition. The calculations are analogous to those proving similar properties for the functions sin and cos. (cid:3) Now consider the pendulum equation θ¨+ω2sinθ = 0, with ω > 0. It is not hard to check that the “energy” of the oscillator given by 1 ω2 I = θ˙2+ (1 cosθ) 4 2 − is constant along solutions. We can rewrite I as 1 θ I = θ˙2+ω2sin2 . 4 2 θ(t) 1 θ Let y(t) = sin . Then y˙ = θ˙cos = θ˙ 1 y2. Squaring both sides and solving for θ˙2 from 2 2 2 ± − the equation for I, we obtain p (y˙)2 = (1 y2)(I ω2y2). (2.5) − − There are four possibilities. Case 1: I = 0. Then θ(t) 0 (mod 2π) and the pendulum is in the stable downward ≡ equilibrium. Case 2: 0 < I < ω2. We look for a solution in the form y(t) = Asn(B(t t );k), for some 0 − constants A,B,t and 0< k < 1, and obtain 0 √I y(t)= ksn(ω(t t );k), with k = . 0 − ω Therefore, for any t R, 0 ∈ θ(t)=2arcsin ksn(ω(t t );k) , 0 { − } is a solution to (2.4). This case corresponds to the pendulum swinging back and forth. Given a particular solution θ(t), we can compute t using θ(t )= 0. 0 0 Case 3: I = ω2. If θ˙(0) = 0, then θ(t) π (mod 2π) and the pendulum is in the unstable ≡ upward equilibrium. If θ˙(0) = 0, then 6 y˙ = ω(1 y2), ± − and it is not hard to check that y(t) = tanh(ω(t t )) 0 ± − satisfies (2.5) for any t , so the solution to the pendulum equation in this case is 0 θ(t)= 2arcsintanh(ω(t t )). 0 ± −