Higher Elastica: Geodesics in the Jet Space

Carnot groups are subRiemannian manifolds. As such they admit geodesic flows, which are left-invariant Hamiltonian flows on their cotangent bundles. Some of these flows are integrable. Some are not. The space of k-jets for real-valued functions on the real line forms a Carnot group of dimension $k+2$. We show that its geodesic flow is integrable and that its geodesics generalize Euler's elastica, with the case $k=2$ corresponding to the elastica, as shown by Sachkov and Ardentov.


Introduction
The space of k-jets of real functions of a single real variable, denoted here by J k , is a k + 2-dimensional manifold endowed with a canonical rank 2 distribution, by which we mean a linear sub-bundle of its tangent bundle. This distribution is framed by two vector fields, denoted X 1 , X 2 below, whose iterated Lie brackets give J k the structure of a Carnot group. Declaring X 1 and X 2 to be orthonormal endows J k with the structure of a subRiemannian manifold, one which is (left-) invariant under the Carnot group multiplication. Like any subRiemannian structure, the cotangent bundle T * J k is endowed with a Hamiltonian system whose underlying Hamiltionian H is that whose solution curves project to the subRiemannian geodesics on J k . We call this Hamiltonian system the geodesic flow on J k . Theorem 1.1. The geodesic flow for the subRiemannian structure on J k is integrable. J 1 is isometric to the Heisenberg group where this theorem is well-known see [2] and [3]. J 2 is isometric to the Engel group and Ardentov and Sachkov showed that its subRiemannian geodesics correspond to Euler elastica. Their result inspired our next theorem.
J k comes with a projection Π : J k → R 2 = R 2 x,u k onto the Euclidean plane which projects the frame X 1 , X 2 projects onto the standard coordinate frame ∂ ∂x , ∂ ∂u k of R 2 . (See SETUP below for the meaning of the coordinates.) As a consequence, a horizontal curve γ in J k is parameteried by (subRiemannian ) arclength if and only if its planar projection Π • γ to R 2 is parameterized by arclength. We will characterize geodesics on J k in terms of their planar projections. As alluded to already, Ardentov and Sachkov, [1], proved that when k = 2 the planar projections of geodesics are Euler elastica. These elastica have "'directrix" the u 2 -line, the line orthogonal to the x-axis. There are many ways to characterize Euler's elastica, see [4], [5] and [6], [7]. The one we will use is as follows. Take a planar curve c(s) = (x(s), y(s)) and consider its curvature κ = κ(s), where s is arclength. Then the curve c is an Euler elastica with directrix a line parallel to the y-axis if and only if κ(s)) = P (x(s)) for P (x) some linear polynomial in x -that is P (x) = ax + b for some constants a and b. See   Let γ : I → J k be a subRiemannian geodesic parameterized by arclength s and π • γ = c(s) = (x(s), u k (s)) its planar projection. Let κ be the curvature of c. Then κ(s) = p(x(s)) for some degree k − 1-polynomial p(x) in the coordinate x. Conversely, any plane curve c(s) in the (x, u k ) plane which is parameterized by arclength s and whose curvature κ(s) equals p(x(s)) for some polynomial p(x) of degree at most k−1 in x is the projection of such a subRiemannian geodesic.
Example For the case k = 1 of the Heisenberg group the theorem asserts that κ = P (x) where P a degree 0 polynomial -i.e. a constant function. The only curves having constant curvature are lines and circles, and these are well-known to be the projections of the Heisenberg geodesics.

set-up
The k-jet of a smooth function f : R → R at a point x 0 ∈ R is its kth order Taylor expansion at x 0 . We will encode this k-jet as a k + 2-tuple of real numbers as follows: As f varies over smooth functions and x 0 varies over R, these k-jets sweep out the k-jet space, denoted by J k . J k is diffeomorphic to R k+2 and its points are coordinatized according to (x, u k , u k−1 , . . . , u 1 , y) ∈ R k+2 := J k .
Rearranging these equations into dy = u 1 dx, du j = u j+1 dx we see that J k is endowed with a natural rank 2 distribution D ⊂ T J k characterized by the k Pfaffian equations The typical integral curves of D are the k-jet curves x → (j k f )(x). In addition to these integral curves we have a distinguished family of curves which arise by varying only the highest derivative u k , and which are the integral curves of the vector field X 2 below (eq (2.2). These latter curves are C 1 -rigid in the sense of Bryant-Hsu, [8], and they exhaust the supply of C 1 -rigid curves.
A subRiemannian structure on a manifold consists of a non-integrable distribution together with a smoothly varying family of inner products on the distribution. We have our distribution D on J k . We arrive at our subRiemannian structure by observing that D is globally framed by the two vector fields and then declaring these two vector fields to be orthonormal . Now the restrictions of the one-forms dx, du k to D form a global co-frame for D * which is dual to our frame (eq (2.2). It follows that an equivalent way to describe our subRiemannian structure is to say that its metric is dx 2 + du 2 k restricted to D.
For the purposes of theorem 1.2 the following alternative characterization of the subRiemannian metric is crucial. Consider the projection Its fibers are transverse to D and we have that Π * X 1 = ∂ ∂x , Π * X 2 = ∂ ∂u k , so our frame pushes down to the standard frame for R 2 The metric on each twoplane D p , p ∈ J k is characterized by the condition that dΠ p (which is just Π since Π is linear), restricted to D p is a linear isometry onto R 2 , where R 2 is endowed with the standard metric dx 2 + du 2 k . It follows immediately that the length of any horizontal path equals the length of its planar projection, that Π is a "submetry": Π(B r (p)) = B r (Π(p)), where B r (p) denotes the metric ball of radius r about q, and that the horizontal lift a Euclidean line in R 2 is a geodesic in J k .

2.1.
Hamiltonian. Let P 1 , P 2 : T * J k → R be the 'power functions' of the vector fields X 1 , X 2 above. (REF: [3], 8 pg.) . In terms of traditional cotangent coordinates (x, u k , u k−1 , . . . , u 1 , y, p x , p k , p k−1 , . . . , p 1 , p y ) for T * J k , with p i short for p u i we have Then the Hamiltonian governing the subRiemannian geodesic flow on J k is See [3]; 8 pg. If we want our geodesics to be parameterized by arclength then we set H = 1/2, and this we will do in what follows. Remark.
[C 1 -rigidity.] The u k curves are C 1 -rigid for D, and form what Liu-Sussmann christened as the "regular-singular" curves for D. As such, they are geodesics for any subRiemannian metric Edx 2 + 2F dxdu k + Gdu 2 k , restricted to D. Such that ds 2 is positive definite, for E, F, G any functions of the jet coordinates (x, u k , u k−1 , . . . , y), regardless of whether or not they satsify the corresponding (normal) geodesic equations. For our metric each u k -curve is indeed the projection to J k of a solution to our H, so we do not go to extra effort to account for these abnormal geodesics. (REF [3], chapter 3).

Carnot Group structure
Under iterated bracket our frame {X 1 , X 2 } generates a k + 2-dimensional nilpotent Lie algebra which can be identified pointwise with the tangent space to J k . Specifically, if we write then we compute that and that all other Lie brackets [X i , X j ] are zero. The span of the X i thus form a k + 2-dimensional graded nilpotent Lie algebra Like any graded nilpotent Lie algebra, this algebra has an associated Lie group which is a Carnot group G, and by using the flows of the X i we can identify G with J k , and the X i with left-invariant vector fields on G ∼ = J k .
The arrows J R , J L are the momentum maps for the actions of G on itself by right and left translation, lifted to T * G. The subscripts ± are for a plus or minus sign in front of the Lie-Poisson (=Kostant-Kirrilov-Souriau) bracket on g * . J R corresponds to left translation back to the identity and realizes the quotient of T * G by the left action. J L corresponds to right translation of a covector back to the identity and forms the components of the momentum map for left translation, lifted to the cotangent bundle. In our case, g * = R k+2 and J R = (P 1 , P 2 , P 3 , . . . , P k+2 ) with P i the power function associated to X i , so that P 3 = p k−1 , P 4 = p k−2 , . . . , P k+2 = p y in terms of standard canonical coordinates as above.
When the Hamiltonian H : T * G → R is left-invariant it can be expressed as a function of the components of J R , that is H = h•J R for some h : g * → R, and H Poisson commutes with every component of the left momentum map J L , so that these left-components are invariants. J L and J R are related by J L (g, p) = (Ad g ) * J R (g, p) where we have written p ∈ T * g G and where Ad g is the adjoint action of g.
The reason underlying the integrability of our system is a simple dimension count. Recall that the symplectic reduced spaces for left translation action are the co-adjoint orbits, for g * + , and that J R realizes this symplectic reduction procedure, mapping each J −1 L (µ) onto the co-adjoint orbit through µ. The hypothesis of the Proposition asserts that the symplectic reduced spaces associated to the G-action are zero or two dimensional, so, morally speaking, the system is automatically integrable by reasons of dimension count.
Proof of Proposition. We must produce n commuting integrals in involution, where n = dim(G). The hypothesis asserts that there are n − 2 Casimirs C 1 , . . . , C n−2 for g * , these being the functions whose common level sets at a generic value define a generic co-adjoint orbit. These Casimirs are a functional basis for the Ad * G invariant polynomails on g * . When viewed as functions on T * G via C i • J R the Casimirs Poisson commute with any leftinvariant function on T * G, and in particular with H and with each other. Thus, H, C 1 , C 2 , . . . , C n−2 yield n − 1 integrals. To get the last commuting integral take any component of J L . QED.
Proof of theorem 1.1. In order to use the proposition, we need to verify that the co-adjoint orbits are generically 2-dimensional. We have the Poisson brackets with all other Poisson brackets {P i , P j }, 1 < i < j ≤ k + 2 being zero. Thus the Poisson tensor B at a point Z = (P 1 , P 2 , P 3 , . . . , P k+2 ) ∈ g * + is : which has rank 2 generically and rank 0 if and only if Z k = 0, i.e. if and only if P i = 0 for 2 < i. QED Thanks to this information we now that the system has k Casimir functions.
Now for k + 2 > i > 2 we have that {P i , P 1 } = −P i+1 , {P i , P 2 } = 0, and {P j , P k+2 } = 0 for all j so thatṖ Proof of theorem 1.2. Consider a geodesic γ and an arc of the geodeisc for whichẋ = 0. Instead of arclength t = s use x to parameterize this arc. From eq (5.1) we have, along this arc, that d ds so that the equations for the evolution of P 3 , P 4 , . . . , P k+2 along the curve become These equations can be summarized by d k P 3 dx k = 0 which asserts that the curvature P 3 of the projected curve c = π • γ, is a polynomial p(x) of degree k − 1 in x, at least along our arc. Finally, since γ is an analytic function of s, so are c(s) and κ(s), so that if κ(s) enjoys a relation κ(s) = p(x(s)) along some subarc of c(s), it enjoys this same relation everywhere along c.
To prove the converse, first consider a general smooth curve in the x − u plane along which dx > 0. We can parameterize the curve either by arclength s → (x(s), u(s)) or as graph, u = u(x). Define the function F (x), with −1 ≤ F (x) ≤ 1 by way of relating the two parameterizations: It follows that Now the curvature of our curve, when viewed as a graph, is well known to be To finish the proof, suppose that we are given a curve c in the x − u plane, with u = u k , whose curvature κ is a degree k − 1 polynomial in x. Define F (x) by eq (5.4) along an arc of c for which dx > 0 From eq (5.5) we know that F is an anti-derivative of κ and so a polynomial of degree k in x. (The constant term in the integration F (x) = x κdx is fixed by choosing any point (x * , u * ) = (x(s * ), u(s * )) along c for which dx/ds > 0 so that −1 < du/ds| s=s * < 1 and setting F (x * ) = du/ds| s=s * .) By the preceding analysis, c has curvature κ(x(s)) along the entire arc dx > 0 of our curve which contains (x * , u * ). Moreover u (x) = F (x)/ 1 − F (x) 2 . Set (5.6) (P 1 (x), P 2 (x)) := ( 1 − (F (x)) 2 , F (x)), , View the P i as momentum functions. Reparameterize the momentum functions by s using dx/ds = P 1 (x). Then we verify that the P i satisfy 5.2 and 5.3, so that the horizontal curve γ(x(s)) in J k with these momenta satisfies the geodesic equations and projects on our given curve c. QED Proof. The equilibria of equations (5.2) and (5.3) are the points with P 1 = 0 and P 3 = 0, as long as H = 0. If H = 1 2 , the condition P 1 = 0 forces P 2 = ±1 but P 2 = F (x). Finally P 3 = F (x).

Structure of higher Elastica
As we see in the last prove we have an option to select a primitive of p(x), then given F k (x) = p(x)dx the dynamics is trivial when F −1 k ([−1, 1]) is empty or isolated points, i.e. F ( x(s)) is constant for all s. Then we can take , by the last corollary the points x 0 and x 1 are equilibrium points if and only if they are critical points of the function F k (x), then it takes infinite time to arrive to them.
Theorem 6.1. The curve (x, u) with curvature k((x(s)) = p(x) is bounded in the x-direction, and generically the curve is periodic in x and the period L given by we also define τ = .
Let x 0 be a regular point, we will answer the question how to extend the curve c(s) as a function of x such that its lift is a smooth solution of geodesics equation, set (P 1 , P 2 ) = (cos θ, sin θ) andθ = p(x) since F k (x 0 ) = ±1 define θ(x 0 ) = ± π 2 andθ(x 0 ) = 0, then P 1 has a change of sign, while, P 2 does not change. Therefore if x(s 0 ) = x 0 we define Therefore, the curve stays in the interval [x 0 , x 1 ], same with x 1 . If both are regular point, we can read the equation ..,C k } and consider action function I given by the area under the graph 1 − F k (x) going from x 0 to x 1 and the area of − 1 − F k (x) going from x 1 to x 0 , i.e.
Finally, the period is given by ∂I ∂H | {H= 1 2 ,C 1 ,...,C k } = L, (see [9] chapter 10). The period goes to infinite when x 0 or x 1 are critical points, much like the very well known homolinic connection of a pendulum. Let us consider (x 0 , u * ) the initial point of the curve and x(s) ∈ [x 0 , x 1 ] and 2s ≤ L, then Where again we use the fact that . QED Here, we have three cases; • Periodic case -p(x 0 ) = 0 and p(x 1 ) = 0 • Asymptotic behavior to one line -p(x 0 ) = 0 and p(x 1 ) = 0 or p(x 0 ) = 0 and p(x 1 ) = 0 . • Asymptotic behavior to two line -p(x 0 ) = 0 and p(x 1 ) = 0 . 6.1. General Convict curve. The elastica equation has a distinguished solution which we call the Euler Kink. Other names for it are the Euler soliton or Convict's curve. See figure and see 1.1, see [1], [6], [7] and [4]. We define the Convict's curve at the level k in the sense that the curvature of the curve (x, u k (x)) it is always proportional to x k−1 . See figure 6.1 Theorem 6.2. If 1 < k then the level k has a convict curve.
Consider the polynomial F k (x) = x k a k − α. Set x = a k α + cos(t) , we can find the next expressions u(t) =  The case k = 2 is the classic solution for Elastica equation, see [7] page 436. If α = 1, then we have the explicit expression ).
We can find a explicit second order ODE forθ, p u k−1 = ∂P ∂x (x) = kx k−1 andθ 2 k 2 a 2k = (cos θ + α) In the case k = 2 is the pendulum equation define in [1], the ODE equation can be extend to k = 1. Also, α = 1 degenerate a system with a degenerated equilibrium point with a homoclinic connection.
6.2. Infinite geodesic graph. Here, we will define a infinite geodesic graph like the curve whose projection to the plane (x, u) is always a graph of x. See figure 6.1 and 6.2.
Theorem 6.3. If k > 2 then J k has a geodesic graph.