An Intrinsic Version of the k-Harmonic Equation

Abstract

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^{k}M$ to the cotangent bundle $T^{*}{T}^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves.

1. Introduction

Polyharmonic curves of order k in Riemannian manifolds are the critical points of the k-energy functional

$${\displaystyle J_k\left(\gamma \right)=\frac{1}{2}{\int }_0^T〈\frac{D^k\gamma }{d{t}^k},\frac{D^k\gamma }{d{t}^k}〉dt}$$

(1)

and are described by the Euler–Lagrange equation

$$\frac{D^{2k}\gamma }{d{t}^{2k}}+\sum _{j=2}^k{(-1)}^{j}R\left(\frac{D^{2k-j}\gamma }{d{t}^{2k-j}},\frac{D^{j-1}\gamma }{d{t}^{j-1}}\right)\frac{d\gamma }{dt}=0.$$

(2)

Notice that the functional (1) is considered a higher-order version of the energy functional

$${\displaystyle J_1\left(\gamma \right)=\frac{1}{2}{\int }_0^T〈\frac{d\gamma }{dt},\frac{d\gamma }{dt}〉dt}$$

(3)

and, in this sense, k-harmonic curves, also referred to as polyharmonic curves, higher-order geodesics, or Riemannian polynomials, are seen as a natural generalization of geodesic curves, the extremal curves of the functional (3).

The study of polyharmonic curves fits into the more general theory of polyharmonic maps between Riemannian manifolds, just as the theory of geodesics falls under that of harmonic applications. Polyharmonic maps have only recently became a subject of interest (see [1] and references therein), but biharmonic maps and, in particular, biharmonic submanifolds and curves have been extensively studied in the last decades (see, for instance, [2,3,4,5,6]).

There is a strong relationship between optimal control problems and variational problems, particularly concerning the variational problem associated with the k-energy functional (1). The main topic of this subject is the study of the dynamic interpolation problem, where the goal is to find the curves that minimize the 2-energy functional and satisfy some interpolation conditions. Applications to motion planning and tracking problems for nonlinear systems were the special motivation for the analysis of this second-order problem. The first steps in this direction were given by L. Noakes, G. Heinzinger, and B. Paden in [7] and by P. Crouch and F. Silva Leite in [8], where the authors obtained the necessary optimality conditions for the problem and called Riemannian cubic splines to the curves under these conditions. In the context of robotic motion planning, a natural extension of the dynamic interpolation problem to higher orders has also been developed, giving rise to the notion of higher-order splines in Riemannian manifolds [9,10].

Applications of polyharmonic curves to trajectory planning problems in robotics and computational anatomy, especially when the configuration space is a Lie group, also brought the subject to the field of geometric mechanics. The Hamiltonian structure and symmetry reductions of the polyharmonic equation have deserved special attention and have also been extended to the study of optimal control problems for mechanical control systems [11,12,13,14,15,16,17].

Polyharmonic curves depend on the choice of the parametrization, as happens with geodesics. From the point of view of differential geometry, these curves are studied by considering arclength parametrization. When they are seen as motion trajectories, arclength parametrization is not always possible (when motion reaches zero velocity) and is thought of as a constraint.

In this work, we present an intrinsic version of the k-harmonic equation based on the symplectic formalism for higher-order regular Lagrangians given in [18]. More specifically, we consider a geometric formulation of the kth-order variational problem on a Riemannian manifold using the framework of symplectic geomety and define a generalized Legendre transformation involving higher-order tangent and cotangent bundles. The corresponding Hamiltonian equation obtained via this Legendre transformation is also explained. This study covers some research topics of interest, such as the interpolation theories involving geodesics and cubic splines. In fact, these cases are explored in the present work as being free harmonic and biharmonic curves (without any constraints on the parameter). The relationship of the variational problem with the optimal control problem is also an interesting field of research and is presented for the case of biharmonic curves, always with emphasis on the intrinsic approach.

The structure of the paper is as follows. In Section 2, we recall some important notions from the geometry of higher-order tangent bundles. The variational problem associated with the k-harmonic curves is studied in Section 3. We begin by showing that the kth-order Lagrangian is regular and then adapt the Lagrangian formalism of higher order to the problem being studied. A higher-order Legendre transformation that allows relating the Lagrangian and the Hamiltonian formalisms is described. Section 4 is devoted to the first-order case, which corresponds to the classical geodesic problem. In Section 5, the formalism for biharmonic curves is explored in more depth and, in this case, the associated optimal control problem is also exposed.

2. Higher-Order Tangent Bundles

Let M be a differentiable manifold of finite dimension n. Consider a local coordinate system $(U,x^1,\dots ,x^n)$ on M, simply denoted by $\left(x^i\right)$. Throughout this paper, we use similar abbreviations for the coordinate notations. Let k be an integer greater than or equal to 1.

In this work, we are interested in the formalism of higher-order tangent bundles. In order to introduce the geometry of those bundles (see [18] for further details), we consider the well-defined equivalence relationship on the set of smooth curves in M, as follows:

We say that two smooth curves in M, ${\gamma }_1$ and ${\gamma }_2$, defined on an interval $(-a,a)$ with $a\in \mathbb{R}$, have a contact of order k at 0 if ${\gamma }_1\left(0\right)={\gamma }_2\left(0\right)=x$, and for a local coordinate system $(U,\phi )$ on M around x, the derivatives of $\phi \circ {\gamma }_1$ and $\phi \circ {\gamma }_2$ up to order k, included, coincide at 0.

The equivalence class determined by a curve $\gamma$ is represented by ${\left[\gamma \right]}_0^k$ and is called k-jet or k-velocity.

Definition 1.

The tangent bundle of order k of M is the set of all equivalence classes of curves in M that have contact of order k and is denoted by $T^{k}M$.

The following characteristics of the tangent bundle $T^{k}M$ should be emphasized:

• $T^{k}M$ is a $(k+1)n$-dimensional manifold and a fibered manifold over M with projection ${\pi }_k:T^{k}M\to M,{\left[\gamma \right]}_0^k\mapsto \gamma \left(0\right)=x$.
• $T^{k}M$ has natural local coordinates $\left({\pi }_k^{-1}\left(U\right),x_0^i;x_1^i;x_2^i;\dots ;x_k^i\right)$ induced by $\left(x^i\right)$, where

  $$x_l^i:{\pi }_k^{-1}\left(U\right)\subset T^{k}M\to \mathbb{R},{\left[\gamma \right]}_0^k\mapsto {\left.\frac{d^l}{d{t}^l}\left(x^i\circ \gamma \right)\left(t\right)\right|}_{t=0},$$

  for $l=0,\dots ,k$ and $i=1,\dots ,n.$
• If $k=0$, $T^{0}M$ is identified with the manifold M and for $k=1$, $T^{1}M$ is just the tangent bundle of M, $TM$.
• There are canonical projections ${\tau }_k^l:T^{k}M\to T^{l}M,{\left[\gamma \right]}_0^k\mapsto {\left[\gamma \right]}_0^l,l=0,\dots ,k$, which define several different fibered structures on $T^{k}M$. Locally,

  $${\tau }_k^l(x_0^i;x_1^i;x_2^i;\dots ;x_k^i)=(x_0^i;x_1^i;x_2^i;\dots ;x_l^i).$$

  (4)
  Note that ${\tau }_k^0={\pi }_k$. The tangent applications $T{\tau }_k^l:T\left(T^{k}M\right)\to T\left(T^{l}M\right)$ are defined by

  $$\left(T{\tau }_k^l\right)\left(X\right)=\sum _{i=1}^n\sum _{j=0}^l{X}_i^j\frac{\partial }{\partial x_j^i},$$

  (5)

  for each $X={\sum }_{i=1}^n{\sum }_{j=0}^k{X}_i^j(\partial /\partial x_j^i)\in T_{{\left[\gamma \right]}_0^k}{T}^{k}M$, with ${\left[\gamma \right]}_0^k\in T^{k}M$.

Definition 2.

Let γ be a smooth curve in M. The lift to $T^{k}M$ of γ is a smooth curve in $T^{k}M$ denoted by ${\gamma }_k$ and defined by ${\gamma }_k\left(t\right)={\left[{\gamma }_t\right]}_0^k$, where ${\gamma }_t\left(s\right)=\gamma (t+s)$.

If $\gamma$ is locally given by $\left(x^i\right)$, then $(x^i;d{x}^i/dt;\dots ;d^k{x}^i/d{t}^k)$ locally represents ${\gamma }_k$.

2.1. The Liouville Vector Field of Higher Order

In order to introduce the notion of a Liouville vector field of higher order, we begin by defining k vertical bundles of $T^{k}M$ determined by foliations of type (4) of $T^{k}M$. Let $r=1,\dots ,k$.

Definition 3.

The vertical bundle of $T^{k}M$ over $T^{r-1}M$, denoted by $V^{{\tau }_k^{r-1}}\left(T^{k}M\right)$, is the set of all tangent vectors to $T^{k}M$ that are projected onto zero by $T{\tau }_k^{r-1}$.

According to (4) and (5), if ${\left[\gamma \right]}_0^k\in T^{k}M$ and X is an element of $V^{{\tau }_k^{r-1}}\left(T^{k}M\right)$ at ${\left[\gamma \right]}_0^k$, then X is locally written as

$$X=\sum _{i=1}^n\sum _{j=r}^k{X}_i^j(\partial /\partial x_j^i).$$

Remark 1.

In the particular case when $k=1$ and $r=1$, the projection ${\tau }_k^{r-1}$ is just the canonical projection of the tangent bundle $TM$, ${\tau }_1^0={\pi }_M:TM\to M$. The only vertical bundle is $TM$ over M, usually denoted by $VTM$, whose elements are tangent vectors of $TM$ and which are projected onto zero for $T{\pi }_M$.

Now consider:

• The canonical applications

  $$\begin{array}{cccc}{j}_r:& T^{k}M& ⟶& T\left(T^{r-1}M\right)\\ & {\left[\gamma \right]}_0^k& ⟼& {\left[{\gamma }_{r-1}\right]}_0^1\end{array}.$$

  (6)
• The vector bundle isomorphisms over $T^{k}M$

  $$h_r:T^{k}M{\times }_{T^{r-1}M}T\left(T^{r-1}M\right)⟶V^{{\tau }_k^{k-r}}\left(T^{k}M\right)$$

  locally defined by

  $$\begin{array}{cc}\hfill & h_r\left(x_0^i;\dots ;x_k^i,x_0^i;\dots ;x_{r-1}^i,X_i^0,\dots ,X_i^{r-1}\right)\hfill \\ \hfill =& \sum _{i=1}^n\sum _{j=1}^r\frac{(k-r+j)!}{(j-1)!}{X}_i^{j-1}\frac{\partial }{\partial x_{k-r+j}^i},\hfill \end{array}$$

  (7)

  where $T^{k}M{\times }_{T^{r-1}M}T\left(T^{r-1}M\right)$ is the induced bundle of $T\left(T^{r-1}M\right)$ via ${\tau }_k^{r-1}$.

Remark 2.

If $k=1$, we have just one vectorial bundle isomorphism over $TM$,

$$h_1:TM{\times }_{M}TM⟶V\left(TM\right),$$

which is locally given by $h_1\left(x_0,x_1,x_0,X^0\right)=\left(x_0,x_1,0,X^0\right)$.

If $k=2$, we can define two vectorial bundle isomorphisms over $T^{2}M$, 

$$\begin{array}{cc}\hfill h_1& :T^{2}M{\times }_{M}TM⟶V^{{\tau }_2^1}\left(T^{2}M\right)\hfill \\ \hfill h_2& :T^{2}M{\times }_{TM}T\left(TM\right)⟶V^{{\tau }_2^0}\left(T^{2}M\right),\hfill \end{array}$$

which is locally defined by

$$\begin{array}{cc}\hfill h_1\left(x_0,x_1,x_2,x_0,X^0\right)& =\left(x_0,x_1,x_2,0,0,2X^0\right)\hfill \\ \hfill h_2\left(x_0,x_1,x_2,x_0,x_1,X^0,X^1\right)& =\left(x_0,x_1,x_2,0,X^0,2X^1\right).\hfill \end{array}$$

Definition 4.

The canonical vector field of order r on $T^{k}M$ is the vector field 

$$C_r:T^{k}M⟶V^{{\tau }_k^{r-1}}\left(T^{k}M\right)\subset T\left(T^{k}M\right)$$

defined by the composition $C_r=h_{k-r+1}\circ (Id\times j_{k-r+1})$

$$T^{k}M\stackrel{Id\times j_{k-r+1}}{\to }{T}^{k}M{\times }_{T^{k-r}M}T\left(T^{k-r}M\right)\stackrel{h_{k-r+1}}{\to }{V}^{{\tau }_k^{r-1}}\left(T^{k}M\right),$$

where $Id$ is the identity map in $T^{k}M$. The Liouville vector field of order k is the canonical vector field of order 1 on $T^{k}M$, $C_1$.

Locally, we have

$$C_r=\sum _{i=1}^n\sum _{j=1}^{k-r+1}\frac{(r+j-1)!}{(j-1)!}{x}_j^i\frac{\partial }{\partial x_{r+j-1}^i}.$$

(8)

For $r=1$, $C_1={\sum }_{i=1}^n{\sum }_{j=1}^{k}j{x}_j^i(\partial /\partial x_j^i)$.

Remark 3.

If $k=1$, then $j_1:TM\to TM$ is the identity map, and we have just the Liouville vector field on $TM$, $C_1:TM\to V\left(TM\right)\subset T\left(TM\right)$:

$$C_1=\sum _{i=1}^n{x}_1^i\frac{\partial }{\partial x_1^i}.$$

If $k=2$, we have two canonical vector fields on $T^{2}M$, the Liouville vector fields $C_1:T^{2}M\to V^{{\tau }_2^0}\left(T^{2}M\right)\subset T\left(T^{2}M\right)$ and $C_2:T^{2}M\to V^{{\tau }_2^1}\left(T^{2}M\right)\subset T\left(T^{2}M\right)$, which are locally given by

$${\displaystyle C_1=\sum _{i=1}^n\left(x_1^i\frac{\partial }{\partial x_1^i}+2x_2^i\frac{\partial }{\partial x_2^i}\right)and{C}_2=\sum _{i=1}^{n}2x_1^i\frac{\partial }{\partial x_2^i}.}$$

2.2. The Canonical Almost-Tangent Structure of Higher Order

We now generalize to higher order the notion of canonical almost-tangent structures. For $r=1,\dots ,k$, consider the following:

• The vector bundle isomorphisms over $T^{k}M$ defined by (7), $h_{k-r+1}$.
• The canonical inclusions $i_{k-r+1}:V^{{\tau }_k^{r-1}}\left(T^{k}M\right)\to T\left(T^{k}M\right)$.
• The vectorial bundle homomorphisms over $T^{k}M$ given by

  $$\begin{array}{cccc}{s}_r:& T\left(T^{k}M\right)& ⟶& T^{k}M{\times }_{T^{k-r}M}T\left(T^{k-r}M\right)\\ & X& ⟼& \left({\pi }_{T^{k}M}\left(X\right),T{\tau }_k^{k-r}\left(X\right)\right),\end{array}$$

  where ${\pi }_{T^{k}M}:T\left(T^{k}M\right)\to T^{k}M$ is the canonical projection. Note that

  $$Ker\left(s_r\right)=V^{{\tau }_k^{k-r}}\left(T^{k}M\right).$$

Definition 5.

The endomorphism $J_r:T\left(T^{k}M\right)\to T\left(T^{k}M\right)$ defined by

$$J_r=i_{k-r+1}\circ h_{k-r+1}\circ s_r,$$

is called the vertical endomorphism of order r of $T\left(T^{k}M\right)$,

$$T\left(T^{k}M\right)\stackrel{s_r}{\to }{T}^{k}M{\times }_{T^{k-r}M}T\left(T^{k-r}M\right)\stackrel{h_{k-r+1}}{\to }{V}^{{\tau }_k^{r-1}}\left(T^{k}M\right)\stackrel{i_{k-r+1}}{\to }T\left(T^{k}M\right).$$

.

The vertical endomorphism $J_1$ is called a canonical almost-tangent structure of order k on $T^{k}M$.

Locally, we have

$$J_r=\sum _{i=1}^n\sum _{j=1}^{k-r+1}\frac{(r+j-1)!}{(j-1)!}\frac{\partial }{\partial x_{r+j-1}^i}\otimes d{x}_{j-1}^i.$$

Proposition 1.

The vertical endomorphism $J_r$ of order r of $T\left(T^{k}M\right)$ has a constant rank equal to $(k-r+1)n$ and satisfies

$${\left(J_r\right)}^s=\left\{\begin{array}{ccc}0\hfill & if\hfill & rs\ge k+1\hfill \\ J_{rs}\hfill & if\hfill & rs<k+1.\hfill \end{array}\right.$$

According to the above proposition, $J_1$ is an almost-tangent structure on $T^{k}M$ since ${\left(J_1\right)}^{k+1}=0$ and $rank{J}_1=kn$.

Remark 4.

If $k=2$, we have two vertical endomorphisms of $T\left(T^{2}M\right)$, $J_1$ and $J_2$, whose matrix representations are, respectively, given by

$$J_1=\left[\begin{array}{ccc}0& 0& 0\\ I_n& 0& 0\\ 0& 2I_n& 0\end{array}\right]and{J}_2=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 2I_n& 0& 0\end{array}\right],$$

where $I_n$ and 0 are the identity matrix and the null matrix of order n, respectively. In this case, ${\left(J_1\right)}^3=0$ and $rank{J}_1=2n$, so $J_1$ determines an almost-tangent structure of order 2 on $T^{2}M$, the so-called canonical almost-tangent structure of order 2 on $T^{2}M$. Notice that ${\left(J_1\right)}^2=J_2$.

Proposition 2.

Let $J_i$ be the vertical endomorphism of order i of $T\left(T^{k}M\right)$ and let $C_i$ be the canonical vector field of order i on $T^{k}M$ (i = r, s). The following relationships are satisfied:

$$\begin{array}{c}{J}_r{C}_s=\left\{\begin{array}{ccc}0\hfill & if\hfill & r+s\ge k+1\hfill \\ C_{r+s}\hfill & if\hfill & r+s<k+1\hfill \end{array}\right.\hfill \\ [C_r,J_s]=\left\{\begin{array}{ccc}0\hfill & if\hfill & r+s>k+1\hfill \\ -s{J}_{r+s-1}\hfill & if\hfill & r+s\le k+1\hfill \end{array}\right.\hfill \\ [J_r,J_s]=0,\hfill \end{array}$$

with $r,s=1,\dots ,k$.

Notice that, on $T^{2}M$, we have $C_2=J_1{C}_1$.

Definition 6.

The vertical differentiation of order r on the exterior algebra of $T^{k}M$, denoted by $d_{J_r}:{\bigwedge }^p\left(T^{k}M\right)\to {\bigwedge }^{p+1}\left(T^{k}M\right)$, is given by the commutator

$$d_{J_r}=\left[i_{J_r},d\right]=i_{J_r}d-d{i}_{J_r},$$

where d is the exterior differentiation and $i_{J_r}$ is the inner product of $J_r$.

Proposition 3.

The vertical differentiation $d_{J_r}$ of order r on the exterior algebra on  $T^{k}M$ satisfies, for each function f on $T^{k}M$, the following relationship:

$$d_{J_r}f=J_r^{*}\left(df\right)and{d}_{J_r}df=-d\left(J_r^{*}df\right).$$

Locally,

$$\begin{array}{cc}\hfill {\displaystyle d_{J_r}f}& =\sum _{i=1}^n\sum _{l=r}^k\frac{l!}{(l-r)!}\frac{\partial f}{\partial x_l^i}d{x}_{l-r}^i\hfill \\ \hfill d_{J_r}\left(d{x}_l^i\right)& =0,\mathrm{for}i=1,\dots ,n\mathrm{and}l=0,\dots ,k.\hfill \end{array}$$

(9)

Remark 5.

Notice that on $TM$, we have

$$d_{J_1}f=\sum _{i=1}^n\frac{\partial f}{\partial x_1^i}d{x}_0^i,wherefisafunctiononTM.$$

Moreover, on $T^{2}M$, we obtain

$${\displaystyle d_{J_1}f=\sum _{i=1}^n\left(\frac{\partial f}{\partial x_1^i}d{x}_0^i+2\frac{\partial f}{\partial x_2^i}d{x}_1^i\right)and{d}_{J_2}f=\sum _{i=1}^{n}2\frac{\partial f}{\partial x_2^i}d{x}_0^i,}$$

where f is a function on $T^{2}M$.

2.3. The Tulczyjew Differential Operator

Definition 7.

The Tulczyjew differential operator or total time derivative operator on $T^{r}M$ is the operator $d_T$ that maps each function f on $T^{r-1}M$ to a function $d_{T}f$ on $T^{r}M$ such that

$$d_{T}f\left({\left[\gamma \right]}_0^r\right)=j_r\left({\left[\gamma \right]}_0^r\right)f,$$

for each ${\left[\gamma \right]}_0^r\in T^{r}M$, with $j_r:T^{r}M\to T\left(T^{r-1}M\right)$ defined in (6).

In local coordinates, we obtain

$$d_T=\sum _{i=1}^n\sum _{j=0}^{r-1}{x}_{j+1}^i\frac{\partial }{\partial x_j^i}.$$

(10)

We shall mention that the total time derivative $d_T$ may be naturally extended to an operator that acts on differentiable forms. This operator maps p-forms on $T^{k}M$ into p-forms on $T^{k+1}M$. Moreover, we have $d_{T}d=d{d}_T$, where d is the exterior differentiation defined on the exterior algebra on $T^{k}M$.

Definition 8.

Let X be a vector field along a curve γ in M. The kth-order lift $X_k$ of X is a vector field along the lifted curve ${\gamma }_k$, $X_k:T^{k}M\to T\left(T^{k}M\right)$, satisfying $(d/dt)\circ X_{k-1}=X_k\circ d_T$.

Note that $X_k$ is obtained by applying repeated lifts to X. Its local coordinate expression is given by

$$X_k\left(t\right)=\sum _{i=1}^n{\left.\sum _{j=0}^k\frac{d^j{X}^i}{d{t}^j}\frac{\partial }{\partial x_j^i}\right|}_{{\gamma }_k\left(t\right)},$$

where $X\left(t\right)={\sum }_{i=1}^n{X}^i{\left.(\partial /\partial x^i)\right|}_{\gamma \left(t\right)}$.

3. Higher-Order Variational Problem

From now on, we take M to be a Riemannian manifold with the Riemannian metric $\langle ·,·\rangle$. The Levi–Civita connection on M is denoted by ∇. Let $DX/dt$ represent the covariant derivative ${\nabla }_{(d\gamma /dt)}X$ along the curve $\gamma$ in M, with X being a vector field along $\gamma$. Set $D^j\gamma /d{t}^j=D(D^{j-1}\gamma /d{t}^{j-1})/dt$ as the jth-order covariant derivative of $\gamma$, where $j\ge 2$ and $D\gamma /dt=d\gamma /dt$. Consider the following sign convention for the curvature tensor field R:

$$R(X,Y)=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]}.$$

See [19] for more details about Riemannian geometry.

Remember that if a curve $\gamma$ in M is locally represented by $\left(x^i\right)$, then ${\gamma }_k$ is locally represented by $(x^i;d{x}^i/dt;\dots ;d^k{x}^i/d{t}^k)$. Thus, the velocity vector field along the curve $\gamma$ is $d\gamma /dt={\sum }_{i=1}^n(d{x}^i/dt){\left.(\partial /\partial x^i)\right|}_{\gamma \left(t\right)}$. Moreover, given a vector field $X={\sum }_{i=1}^n{X}^i(\partial /\partial x^i)$, the covariant derivative of X along $\gamma$ is given by

$$\frac{DX}{dt}=\sum _{k=1}^n\left(\frac{d{X}^k}{dt}+\sum _{i,j=1}^n{\mathrm{\Gamma }}_{ij}^k\frac{d{x}^i}{dt}{X}^j\right){\left.\frac{\partial }{\partial x^k}\right|}_{\gamma \left(t\right)}.$$

In particular, the covariant acceleration of $\gamma$ can be written as

$$\frac{D^2\gamma }{d{t}^2}=\sum _{k=1}^n\left({\ddot{x}}^k+\sum _{i,j=1}^n{\mathrm{\Gamma }}_{ij}^k{\dot{x}}^i{\dot{x}}^j\right){\left.\frac{\partial }{\partial x^k}\right|}_{\gamma \left(t\right)},$$

where we simplify the notations of the derivatives, using ${\dot{x}}^i$ for the first derivative $d{x}^i/dt$ and similar notations for the higher-order derivatives. Here, ${\mathrm{\Gamma }}_{ij}^k$ are the Christoffel symbols defining the Riemannian connection, which can be obtained using the identity

$${\mathrm{\Gamma }}_{ij}^k=\frac{1}{2}\sum _{l=1}^n{g}^{kl}\left(\frac{\partial g_{jl}}{\partial x^i}+\frac{\partial g_{li}}{\partial x^j}-\frac{\partial g_{ij}}{\partial x^l}\right),$$

where $g_{ij}$ are the components of the Riemannian metric and ${\left[g^{ij}\right]}_{1\le i,j\le n}$ is the inverse matrix of the matrix ${\left[g_{ij}\right]}_{1\le i,j\le n}$.

Using the Riemannian structure of M, we can also define the bundle morphism $A_k$ from $T^{k}M$ to $TM$ given by $A_k\left({\left[\gamma \right]}_0^k\right)=\left(D^k\gamma /d{t}^k\right)\left(0\right)$. The morphism $A_k$ can be expressed as follows.

$$A_k\left({\left[\gamma \right]}_0^k\right)=\sum _{r=1}^n{A}_k^r{\left.\frac{\partial }{\partial x^r}\right|}_{\gamma \left(0\right)},$$

where

$$A_k^r(x_0,\dots ,x_k)=x_k^r+B_{k-1}^r(x_0,\dots ,x_{k-1})$$

(11)

and

$$B_k^r(x_0,\dots ,x_k)=\sum _{l=0}^{k-1}\sum _{i=1}^n\frac{\partial }{\partial x_l^i}{B}_{k-1}^r(x_0,\dots ,x_{k-1})x_{l+1}^i+\sum _{i,j=1}^n{\mathrm{\Gamma }}_{ij}^k{x}_1^i{A}_{k-1}^j(x_0,\dots ,x_{k-1}).$$

3.1. The k-Energy Functional

Let ${\mathcal{C}}_k$ be the class of smooth curves $\gamma :[0,T]\to M$ satisfying the boundary conditions

$$\gamma \left(0\right)=x_0,\gamma \left(T\right)=x_T,\frac{D^j\gamma }{d{t}^j}\left(0\right)=y_{0j},\frac{D^j\gamma }{d{t}^j}\left(T\right)=y_{Tj},j=1,\dots ,k-1,$$

where $x_0,x_T\in M$, $y_{ij}$ are fixed n-vectors ($i=0,T$; $j=1,\dots ,k-1$) and $T\in {\mathbb{R}}^{+}$. Consider the kth-order variational problem described by the action functional $J_k$ defined by (1). From the point of view of intrinsic variational calculus, $J_k$ can be written as

$$J_k\left(\gamma \right)={\int }_0^T{\gamma }_k^{*}\left(L\right)dt={\int }_0^{T}L(x^i;d{x}^i/dt;\dots ;d^k{x}^i/d{t}^k)dt,$$

where L is the Lagrangian of order k associated with the problem. Therefore, the Lagrangian of the problem, $L:T^{k}M\to \mathbb{R}$, is defined, for each ${\left[\gamma \right]}_0^k\in T^{k}M$, by

$$L\left({\left[\gamma \right]}_0^k\right)=\frac{1}{2}〈A_k\left({\left[\gamma \right]}_0^k\right),A_k\left({\left[\gamma \right]}_0^k\right)〉,$$

(12)

where $A_k$ is given by (11). We may remark that (12) may be locally expressed by

$$L(x_0^i;x_1^i;x_2^i;\dots ;x_k^i)=\frac{1}{2}\sum _{i,j=1}^n{g}_{ij}{A}_k^i{A}_k^j.$$

Differentiating L, we obtain

$$\frac{\partial L}{\partial x_k^i}=\sum _{j=1}^n{g}_{ij}{A}_k^j\left(x_0,\dots ,x_k\right)=〈A_k\left(x_0,\dots ,x_k\right),\frac{\partial }{\partial x^i}〉.$$

Furthermore, ${\left[\left({\partial }^{2}L/\partial x_k^i\partial x_k^j\right)\right]}_{1\le i,j\le n}={\left[g_{ij}\right]}_{1\le i,j\le n}$ and, since this is the matrix that represents the Riemannian metric, we have the guarantee that the Lagrangian L is regular.

3.2. Intrinsic Version of the Euler–Lagrange Equation

Given a curve $\gamma$ in ${\mathcal{C}}_k$, the tangent space to ${\mathcal{C}}_k$ at $\gamma$, $T_{\gamma }{\mathcal{C}}_k$, is constituted by smooth vector fields X along $\gamma$ such that $X_j\left(0\right)=X_j\left(T\right)=0$ for $j=1,\dots ,k-1$, where $X_j$ is the jth-order lift of X. The variation of the curve $\gamma$ is given by a smooth 1-parameter family of curves ${\gamma }_{ϵ}\in {\mathcal{C}}_k$ with ${\gamma }_0=\gamma$, and the corresponding variation vector field $X\in T_{\gamma }\mathcal{C}$ is defined by $X\left(\gamma \left(t\right)\right)={\left.\left(d{\gamma }_{ϵ}/dϵ\right)\left(t\right)\right|}_{t=0}$. The first-order variation of $J_k$ associated with X takes the form

$$d{J}_k\left(\gamma \right)\left(X\right)={\left.\frac{d}{dϵ}{J}_k\left({\gamma }_{ϵ}\right)\right|}_{ϵ=0}={\int }_0^T{X}_k\left(L\right)dt.$$

Hamilton’s variational principle establishes that a curve $\gamma \in {\mathcal{C}}_k$ is a critical curve of $J_k:{\mathcal{C}}_k\to \mathbb{R}$ if, for an arbitrary variation vector $X\in T_{\gamma }{\mathcal{C}}_k$, we have $d{J}_k\left(\gamma \right)\left(X\right)=0$. If the Lagrangian L is regular (which is the case that we are considering), the arbitrariness of the variation vector field X in the condition for the action integral to be stationary,

$$d{J}_k\left(\gamma \right)\left(X\right)={\int }_0^T{X}_k\left(L\right)dt=0,\forall X\in T_{\gamma }{\mathcal{C}}_k,$$

gives the geometric version of the Euler–Lagrange equation

$$i_{X_E}{\omega }_L=dE,$$

(13)

where ${\omega }_L$ is the Poincaré–Cartan 2-form on $T^{2k-1}M$ and $E:T^{2k-1}M\to \mathbb{R}$ is the energy function associated with $L:T^{k}M\to \mathbb{R}$, defined, respectively, by

$${\omega }_L=\sum _{r=1}^k{(-1)}^r\frac{1}{r!}{d}_T^{r-1}d{d}_{J_r}L\mathrm{and}E=\sum _{r=1}^k{(-1)}^{r-1}\frac{1}{r!}{d}_T^{r-1}\left(C_{r}L\right)-L.$$

Consider the one-form

$${\alpha }_L=\sum _{r=1}^k{(-1)}^{r-1}\frac{1}{r!}{d}_T^{r-1}{d}_{J_r}L.$$

(14)

Proposition 4.

The one-form ${\alpha }_L$ on $T^{2k-1}M$ is semibasic of type k; that is, ${\alpha }_L\in Im\left(J_k^{*}\right)$.

One calls ${\alpha }_L$ the Jacobi–Ostrogradsky form associated with the Lagrangian L. We have ${\omega }_L=-d{\alpha }_L$. Locally,

$${\alpha }_L=\sum _{i=1}^n\sum _{r=0}^{k-1}{p}_r^{i}d{x}_r^i,{\omega }_L=\sum _{i=1}^n\sum _{r=0}^{k-1}d{x}_r^i\wedge d{p}_r^i\mathrm{and}E=\sum _{i=1}^n\sum _{r=0}^{k-1}{p}_r^i{x}_{r+1}^i-L,$$

where $p_r^i={\sum }_{l=0}^{k-r-1}{(-1)}^l{d}_T^l\left(\partial L/\partial x_{r+l+1}^i\right)$, $i=1,\dots ,n$.

The Euler–Lagrange Equation (13) uniquely defines the vector field $X_E$ on $T^{2k-1}M$ since, due to the regularity of the Lagrangian L, ${\omega }_L$ is sympletic (see [18]). Moreover, since $J_1{X}_E=C_1$, $X_E$ is a semispray on $T^{2k-1}M$ of type 1, which represents the kth-order differential Equation (2). This means that the integral curves of $X_E$ are lifts to $T^{k}M$ of the curves in M satisfying the Euler–Lagrange Equation (2).

We also remark that Equation (2) can be rewritten in local coordinates as follows:

$$\sum _{j=0}^k{(-1)}^j\frac{d^j}{d{t}^j}\left(\frac{\partial L}{\partial x_j^i}\right)=0,i=1,\dots ,n.$$

3.3. Generalized Legendre Transformation and the Hamiltonian Approach

Proposition 4 allows us to conclude that the Jacobi–Ostrogradsky form (14), ${\alpha }_L$, is semibasic of type k and consequently determines, via the identity

$${\alpha }_L=\ll Leg\circ {\pi }_{T^{2k-1}M},T{\tau }_{2k-1}^1\gg ,$$

the Legendre transformation $Leg$ (see [18] for more details),

where $\ll .,.\gg$ represents the pairing duality of vectors and covectors on $T^{k-1}M$, and ${\pi }_{T^{2k-1}M}:T{T}^{2k-1}M\to T^{2k-1}M,{\tau }_{2k-1}^{k-1}:T^{2k-1}M\to T^{k-1}M\mathrm{and}{\tau }_{T^{k-1}M}:T^{*}{T}^{k-1}M\to T^{k-1}M$ are the natural projections. Locally, we have

$$Leg(x_0^i;x_1^i;\dots ;x_{2k-2}^i;x_{2k-1}^i)=(x_0^i;x_1^i;\dots ;x_{k-1}^i;{\alpha }_i^0;{\alpha }_i^1;\dots ;{\alpha }_i^{k-1}),$$

where ${\alpha }_i^r$, $r=0,1,\dots ,k-1$, are the real functions defined by (14).

When $Leg$ is a diffeomorphism, we say that the Lagrangian L is hyper-regular, and we have a symplectomorphism from $(T^{2k-1}M,{\omega }_L)$ to $(T^{*}{T}^{k-1}M,{\omega }_1)$, where ${\omega }_1$ is the symplectic canonical form on $T^{*}{T}^{k-1}M$. Under the hyper-regularity condition, we can consider the Hamiltonian energy function associated with L given by

$$H_L=E\circ Le{g}^{-1},$$

and the system $(T^{2k-1}M,{\omega }_L,E)$ is associated with the Hamiltonian system $(T^{*}{T}^{k-1}M,{\omega }_1,H_L)$. The dynamics of the Hamiltonian system is described by

$$i_{X_{H_L}}{\omega }_1=d{H}_L,$$

(15)

and the Hamiltonian vector field $X_{H_L}$ defined by (15) verifies

$$X_{H_L}={\left(Leg\right)}_{*}{X}_E=T\left(Leg\right)\circ X_E\circ Le{g}^{-1}.$$

The Hamiltonian function $H_L:T^{*}{T}^{k-1}M\to \mathbb{R}$ is locally given by

$$H_L(x_0^i;x_1^i;\dots ;x_{k-1}^i;p_i^0;p_i^1;\dots ;p_i^{k-1})=\sum _{i=1}^n\sum _{r=0}^{k-1}{p}_i^r{x}_{r+1}^i-L(x_0^i;x_1^i;\dots ;x_{k-1}^i).$$

4. First-Order Variational Problem

4.1. The Energy Functional

We consider the first-order variational problem defined by the energy functional (3). The Euler–Lagrange equation is the geodesic equation

$$\frac{D^2\gamma }{d{t}^2}=0.$$

If we represent the curve $\gamma$ locally by $\left(x^i\right)$, the geodesic equation can be rewritten in local coordinates as

$${\ddot{x}}^k+\sum _{i,j=1}^n{\mathrm{\Gamma }}_{ij}^k{\dot{x}}^i{\dot{x}}^j=0,k=1,\dots ,n.$$

The lift of a curve $\gamma$ to $TM$, ${\gamma }_1$, is locally represented by $(x^i,{\dot{x}}^i)$, and the value of the energy functional (3) can be written as

$$J_2\left(\gamma \right)={\int }_0^T{\gamma }_2^{*}\left(L\right)dt={\int }_0^{T}L(x^i;{\dot{x}}^i;{\ddot{x}}^i)dt,$$

where L is the Lagrangian of order 2 associated with the problem. Therefore, the Lagrangian $L:TM\to \mathbb{R}$ is defined, for each $y\in TM$, by

$$L\left(y\right)=\frac{1}{2}〈y,y〉.$$

(16)

Using the canonical local coordinates $(x^i;y^i)$ of $TM$, the Lagrangian L may be locally expressed by

$$L(x^i;y^i)=\frac{1}{2}\sum _{i,j=1}^n{g}_{ij}{y}^i{y}^j,$$

4.2. Intrinsic Version of the Geodesic Equation

We may remark that we can associate with a Lagrangian $L:TM\to \mathbb{R}$ the following structures:

• The Poincaré–Cartan 2-form on $TM$ given by

  $${\omega }_L=-d{d}_{J_1}L,$$

  where d and $d_{J_1}$ represent, on the exterior algebra of differentiable forms on $TM$, the usual exterior differentiation and the vertical differentiation of order 1 defined by (9), respectively. In other words, ${\omega }_L=-d{\alpha }_L$, where ${\alpha }_L$ is the Jacobi–Ostrogradsky form on $TM$ associated with L by

  $${\alpha }_L=d_{J_1}L,$$

  with $d_{J_1}L={\sum }_{i=1}^n(\partial L/\partial y^i)d{x}^i$.
• The energy function $E:TM\to \mathbb{R}$ is defined by

  $$E=C_{1}L-L,$$

  with ${\displaystyle C_{1}L=\sum _{i=1}^n(\partial L/\partial y^i)y^i}$, where we have used (8).

The following result allows us to obtain an expression for the Poincaré–Cartan 2-form ${\omega }_L$ associated with the Lagrangian (16).

Proposition 5.

The Jacobi–Ostrogradsky form ${\alpha }_L$ associated with the Lagrangian (16) is given by

$${\alpha }_L=\sum _{i=1}^n{\alpha }_i^{0}d{x}^i,$$

where ${\alpha }_i^0$ are the real functions defined by

$$\begin{array}{cc}\hfill {\displaystyle {\alpha }_i^0\left(y\right)}& =〈y,\frac{\partial }{\partial x^i}〉.\hfill \end{array}$$

Corollary 1.

The Poincaré–Cartan 2-form ${\omega }_L$ associated with the Lagrangian (16) is given by

$${\omega }_L=\sum _{i=1}^{n}d{x}^i\wedge d{\alpha }_i^0.$$

The expression for the energy function E associated with the Lagrangian (16) is given in the following result.

Proposition 6.

The energy function $E:TM\to \mathbb{R}$ associated with the Lagrangian (16) is expressed by

$$E\left(y\right)=\frac{1}{2}〈y,y〉,$$

and coincides with the Lagrangian (16).

Remark 6.

The energy function E associated with the Lagrangian (16) is written along the lift ${\gamma }_1$ to $TM$ of the curve γ as

$${\gamma }_1^{*}\left(E\right)=\frac{1}{2}〈\frac{d\gamma }{dt},\frac{d\gamma }{dt}〉.$$

The Equation (13) uniquely defines the vector field $X_E$ on $TM$. $X_E$ is called the geodesic vector field. In local coordinates, the geodesic vector field $X_E$ can be written as

$$X_E(x^i;y^i)=\sum _{k=1}^n\left(y^k\frac{\partial }{\partial x^k}-\sum _{i,j=1}^n{\mathrm{\Gamma }}_{ij}^k{y}^i{y}^j\frac{\partial }{\partial y^k}\right).$$

The integral curves of $X_E$ are the lifts of the geodesics in M to $TM$. The corresponding flow is called the geodesic flow.

The classical geodesic equation can be deduced directly from Equation (13) as follows. The equations $i_{X_E}{\omega }_L(\partial /\partial x^j)=dE(\partial /\partial x^j),j=1,\dots ,n,$ are equivalent to

$$\sum _{i=1}^n{y}^i\frac{\partial {\alpha }_i^0}{\partial x^j}-X_E\left({\alpha }_j^0\right)=\frac{\partial E}{\partial x^j},j=1,\dots ,n.$$

Considering this equation along the lift ${\gamma }_1$ to $TM$ of the curve $\gamma$, we obtain the equations

$$\frac{\partial L}{\partial x^j}-\frac{d}{dt}\left({\alpha }_j^0\right)=0,j=1,\dots ,n.$$

(17)

Along ${\gamma }_1$, we have

$$\frac{\partial L}{\partial x^j}=〈\frac{d\gamma }{dt},{\nabla }_{\frac{d\gamma }{dt}}\frac{\partial }{\partial x^j}〉$$

and

$$\frac{d}{dt}\left({\alpha }_j^0\right)=〈\frac{d\gamma }{dt},{\nabla }_{\frac{d\gamma }{dt}}\frac{\partial }{\partial x^j}〉-〈\frac{D^2\gamma }{d{t}^2},\frac{\partial }{\partial x^j}〉,j=1,\dots ,n.$$

Consequently, the Equation (17) takes the form

$$〈\frac{D^2\gamma }{d{t}^2},\frac{\partial }{\partial x^j}〉=0,j=1,\dots ,n.,$$

and the geodesic equation follows.

Observe that Equation (17) can also be rewritten as follows:

$$\frac{\partial L}{\partial x^i}-\frac{d}{dt}\left(\frac{\partial L}{\partial y^i}\right)=0,i=1,\dots ,n.$$

4.3. The Cogeodesic Flow

The Legendre transformation $Leg$,

coincides with the isomorphism ♭, with inverse ♯, which are the musical isomorphisms with respect to the Riemannian metric. That is, $Leg\left(y\right)=y^{♭}$, with

$$y^{♭}\left(z\right)=〈y,z〉\iff p\left(z\right)=〈p^{♯},z〉,y,z\in T_{x}M,p\in T_x^{*}M,x\in M.$$

Locally, we have

$$Leg(x^i;y^i)=\left(x^i;\sum _{j=1}^n{g}_{ij}{y}^j\right),$$

with inverse

$$Le{g}^{-1}(x^i;p_i)=\left(x^i;\sum _{j=1}^n{g}^{ij}{p}_j\right).$$

We have a symplectomorphism

$$Leg:(TM,{\omega }_L)\to (T^{*}M,{\omega }_1),$$

where ${\omega }_1$ is the symplectic canonical form on $T^{*}M$. This means that

$${\omega }_L=Le{g}^{*}{\omega }_1.$$

The Hamiltonian function $H_L:T^{*}M\to \mathbb{R}$ is given by

$$H_L\left(p\right)=\frac{1}{2}p\left(p^{♯}\right).$$

Locally, $H_L$ can be written as follows.

$$H_L(x^i;p_i)=\frac{1}{2}\sum _{i,j=1}^n{g}^{ij}{p}_i{p}_j.$$

The dynamics of the Hamiltonian system $(T^{*}M,{\omega }_1,H_L)$ is described by (15) and is locally expressed by the following equations.

$$\begin{array}{cc}{\displaystyle {\dot{x}}^i}\hfill & {\displaystyle =\sum _{j=1}^n{g}^{ij}{p}_j}\hfill \\ {\displaystyle {\dot{p}}_i}\hfill & {\displaystyle =-\frac{1}{2}\sum _{j,k=1}^n\frac{\partial }{\partial x^i}{g}^{jk}{p}_j{p}_k,}\hfill \end{array}$$

for $i=1,\dots ,n.$ The corresponding Hamiltonian flow is called the cogeodesic flow.

5. Second-Order Variational Problem

In this section, the attention goes to the tangent bundles $T^{2}M$ and $T^{3}M$. The canonical local coordinates on $T^{2}M$ and $T^{3}M$ will be denoted by $(x^i;y^i;u^i)$ and $(x^i;y^i;u^i;v^i)$, respectively.

5.1. The Bienergy Functional

We see now the second-order variational problem associated with the 2-energy functional

$${\displaystyle J_2\left(\gamma \right)=\frac{1}{2}{\int }_0^T〈\frac{D^2\gamma }{d{t}^2},\frac{D^2\gamma }{d{t}^2}〉dt.}$$

The Euler–Lagrange equation for this problem is the fourth-order differential equation

$$\frac{D^4\gamma }{d{t}^4}+R\left(\frac{D^2\gamma }{d{t}^2},\frac{d\gamma }{dt}\right)\frac{d\gamma }{dt}=0.$$

(18)

The solutions of the above equation are biharmonic curves, better known as Riemannian cubic polynomials on M.

The lift ${\gamma }_2$ of a curve to $T^{2}M$ $\gamma$ is locally represented by $(x^i;{\dot{x}}^i;{\ddot{x}}^i)$. The value of the 2-energy functional $J_2$ at $\gamma$ can be written as

$$J_2\left(\gamma \right)={\int }_0^T{\gamma }_2^{*}\left(L\right)dt={\int }_0^{T}L(x^i;{\dot{x}}^i;{\ddot{x}}^i)dt,$$

with L being the Lagrangian of order 2 associated with the problem. Therefore, the Lagrangian $L:T^{2}M\to \mathbb{R}$ is defined, for each ${\left[\gamma \right]}_0^2\in T^{2}M$, by

$$L\left({\left[\gamma \right]}_0^2\right)=\frac{1}{2}〈A_2\left({\left[\gamma \right]}_0^2\right),A_2\left({\left[\gamma \right]}_0^2\right)〉,$$

(19)

where $A_2=K\circ j_2,$ with $j_2:T^{2}M\to TTM$ defined according to (6), locally given by

$${\displaystyle j_2(x^i;y^i;u^i)=\sum _{i=1}^n\left(y^i\frac{\partial }{\partial x^i}+u^i\frac{\partial }{\partial y^i}\right),}$$

and $K:TTM\to TM$ is the connection application induced by the Levi–Civita connection, locally given by

$${\displaystyle K\left(\sum _{i=1}^n{X}_i\frac{\partial }{\partial x^i}+\sum _{i=1}^n{X}_{i+n}\frac{\partial }{\partial y^i}\right)=\sum _{i=1}^n\left(X_{n+i}+\sum _{j,k=1}^n{\mathrm{\Gamma }}_{jk}^i{y}^j{X}_k\right)\frac{\partial }{\partial x^i}.}$$

It is important to note that (19) may be locally expressed by

$$L(x^i;y^i;u^i)=\frac{1}{2}〈\overline{u},\overline{u}〉,$$

(20)

where

$$\overline{u}=\sum _{k=1}^n\left(u^k+\sum _{i,j=1}^n{\mathrm{\Gamma }}_{ij}^k{y}^i{y}^j\right){\left.\frac{\partial }{\partial x^k}\right|}_x.$$

(21)

For the sake of simplicity, we should use ${\overline{u}}^k$ to represent $u^k+{\sum }_{i,j=1}^n{\mathrm{\Gamma }}_{ij}^k{y}^i{y}^j$.

5.2. Intrinsic Version of the Biharmonic Equation

Consider now the Lagrangian (20) of the second-order variational problem. Differentiating L, we obtain

$$\begin{array}{cc}\hfill \frac{\partial L}{\partial u^i}& =\sum _{k=1}^n{g}_{ik}\left(u^k+\sum _{j,l=1}^n{\mathrm{\Gamma }}_{jl}^k{y}^j{y}^l\right)=〈\overline{u},\frac{\partial }{\partial x^i}〉\hfill \end{array}$$

(22)

where $\overline{u}\in T_{x}M$ is defined by (21). As observed in Section 3, ${\left[\left({\partial }^{2}L/\partial u^i\partial u^j\right)\right]}_{1\le i,j\le n}={\left[g_{ij}\right]}_{1\le i,j\le n}$ is non-singular, and so we have the guarantee that the Lagrangian L is regular.

We may remark that we can associate with a Lagrangian $L:T^{2}M\to \mathbb{R}$ the following structures:

• The Poincaré–Cartan 2-form on $T^{3}M$ given by

  $${\omega }_L=-d{d}_{J_1}L+\frac{1}{2}{d}_{T}d{d}_{J_2}L,$$

  where d, $d_T$, and $d_{J_i}$ represent, on the exterior algebra of differentiable forms on $T^{2}M$, the usual exterior differentiation, the total derivation operator defined according to (10), and the vertical differentiation of order i (i = 1, 2) defined by (9), respectively. In other words, ${\omega }_L=-d{\alpha }_L$, where ${\alpha }_L$ is the Jacobi–Ostrogradsky form on $T^{3}M$ associated with L defined on $T^{3}M$ by

  $${\alpha }_L=d_{J_1}L-\frac{1}{2}{d}_T{d}_{J_2}L,$$

  (23)

  with

  $${\displaystyle d_{J_1}L=\sum _{i=1}^n\left(\frac{\partial L}{\partial y^i}d{x}^i+2\frac{\partial L}{\partial u^i}d{y}^i\right)\mathrm{and}{d}_{J_2}L=2\sum _{i=1}^n\frac{\partial L}{\partial u^i}d{x}^i.}$$
• The energy function $E:T^{3}M\to \mathbb{R}$ defined by

  $$E=C_{1}L-\frac{1}{2}{d}_T\left(C_{2}L\right)-L,$$

  (24)

  with

  $${\displaystyle C_{1}L=\sum _{i=1}^n\left(\frac{\partial L}{\partial y^i}{y}^i+2\frac{\partial L}{\partial u^i}{u}^i\right)\mathrm{and}{C}_{2}L=2\sum _{i=1}^n\frac{\partial L}{\partial u^i}{y}^i,}$$

  where we have used (8).

The following result allows us to obtain an expression for the Poincaré–Cartan 2-form ${\omega }_L$ associated with (20).

Proposition 7.

The Jacobi–Ostrogradsky form ${\alpha }_L$ associated with the Lagrangian (20) is given by

$${\alpha }_L=\sum _{i=1}^n\left({\alpha }_i^{0}d{x}^i+{\alpha }_i^{1}d{y}^i\right),$$

where ${\alpha }_i^0$ and ${\alpha }_i^1$ are the real functions defined by

$$\begin{array}{cc}\hfill {\displaystyle {\alpha }_i^0(x^i;y^i;u^i;v^i)}& =〈\overline{u},{\nabla }_y\frac{\partial }{\partial x^i}〉-〈\overline{v},\frac{\partial }{\partial x^i}〉\hfill \\ \hfill {\displaystyle {\alpha }_i^1(x^i;y^i;u^i;v^i)}& =〈\overline{u},\frac{\partial }{\partial x^i}〉,\hfill \end{array}$$

(25)

with the vectors $\overline{u}$, y, and $\overline{v}\in T_{x}M$ being, respectively, (21),

$$y=\sum _{k=1}^n{y}^k{\left.\frac{\partial }{\partial x^k}\right|}_{x}and\overline{v}={\left.\sum _{k=1}^n{\overline{v}}^k\frac{\partial }{\partial x^k}\right|}_x,$$

(26)

where

$${\displaystyle {\overline{v}}^k=v^k+3\sum _{i,j=1}^n{\mathrm{\Gamma }}_{ij}^k{y}^i{u}^j+\sum _{i,j,l=1}^n{A}_{ijl}^k{y}^i{y}^j{y}^l,with{A}_{ijl}^k=\frac{\partial {\mathrm{\Gamma }}_{ij}^k}{\partial x^l}+\sum _{r=1}^n{\mathrm{\Gamma }}_{jr}^k{\mathrm{\Gamma }}_{il}^r.}$$

Proof. 

From

$$\begin{array}{cc}\hfill \frac{\partial L}{\partial y^i}& =2\sum _{k,m,s=1}^n{g}_{mk}{\mathrm{\Gamma }}_{is}^m{y}^s\left(u^k+\sum _{j,l=1}^n{\mathrm{\Gamma }}_{jl}^k{y}^j{y}^l\right)=2〈\overline{u},{\nabla }_y\frac{\partial }{\partial x^i}〉,\hfill \end{array}$$

(27)

and (22), we obtain

$$\begin{array}{cc}\hfill {\displaystyle d_{J_1}L}& =2〈\overline{u},{\nabla }_y\frac{\partial }{\partial x^i}〉d{x}^i+2〈\overline{u},\frac{\partial }{\partial x^i}〉d{y}^i\hfill \\ \hfill {\displaystyle d_{J_2}L}& =2〈\overline{u},\frac{\partial }{\partial x^i}〉d{x}^i\hfill \\ \hfill {\displaystyle d_T{d}_{J_2}L}& =2\left(〈\overline{v},\frac{\partial }{\partial x^i}〉+〈\overline{u},{\nabla }_y\frac{\partial }{\partial x^i}〉\right)d{x}^i+2〈\overline{u},\frac{\partial }{\partial x^i}〉d{y}^i.\hfill \end{array}$$

Now substituting these in (23), the expression of ${\alpha }_L$ appears immediately. □

Corollary 2.

The Poincaré–Cartan 2-form ${\omega }_L$ associated with the Lagrangian (20) is given by

$${\omega }_L=\sum _{i=1}^n\left(d{x}^i\wedge d{\alpha }_i^0+d{y}^i\wedge d{\alpha }_i^1\right)$$

where ${\alpha }_i^0$ and ${\alpha }_i^1$ are the real functions (25).

The expression for the energy function E associated with (20) is given in the following result.

Proposition 8.

The energy function E: $T^{3}M\to \mathbb{R}$ associated with the Lagrangian (20) is expressed by

$$E(x^i;y^i;u^i;v^i)=\frac{1}{2}〈\overline{u},\overline{u}〉-〈\overline{v},y〉,$$

with the vectors $\overline{u}$, y, and $\overline{v}\in T_{x}M$ defined, respectively, by (21) and (26).

Proof. 

Using (22) and (27), we obtain the expressions $C_{1}L=2〈\overline{u},\overline{u}〉$, $C_{2}L=2〈y,\overline{u}〉$ and $d_T\left(C_{2}L\right)=2〈\overline{u},\overline{u}〉+2〈\overline{v},y〉$. Consequently, from (24), the results follow. □

Remark 7.

The energy function E associated with the Lagrangian (20) is written, along the lift of ${\gamma }_3$ to $T^{3}M$ for the curve γ, as

$${\gamma }_3^{*}\left(E\right)=\frac{1}{2}〈\frac{D^2\gamma }{d{t}^2},\frac{D^2\gamma }{d{t}^2}〉-〈\frac{D^3\gamma }{d{t}^3},\frac{d\gamma }{dt}〉.$$

The Euler–Lagrange Equation (13) uniquely defines the vector field $X_E$ on $T^{3}M$ since, due to the regularity of the Lagrangian L, ${\omega }_L$ is sympletic (see [18]). Moreover, since $J_1{X}_E=C_1$, $X_E$ is a semispray on $T^{3}M$ of type 1 that represents the fourth-order differential Equation (18). Indeed, as we shall see below, the fourth-order differential Equation (18) is deduced straight from the vector field $X_E$ defined by (13). The vector field $X_E$ can be written as

$$X_E(x^i;y^i;u^i;v^i)=\sum _{i=1}^n\left(y^i\frac{\partial }{\partial x^i}+u^i\frac{\partial }{\partial y^i}+v^i\frac{\partial }{\partial u^i}+G^i\frac{\partial }{\partial v^i}\right),$$

for some functions $G^i$ defined on domains of induced local charts. Thus, the equations $i_{X_E}{\omega }_L(\partial /\partial x^j)=dE(\partial /\partial x^j)$ are equivalent to

$$\sum _{i=1}^n\left(y^i\frac{\partial {\alpha }_i^0}{\partial x^j}+u^i\frac{\partial {\alpha }_i^1}{\partial x^j}\right)-X_E\left({\alpha }_j^0\right)=\frac{\partial E}{\partial x^j},j=1,\dots ,n,$$

where we use Corollary 2. Considering this equation along the lift of ${\gamma }_3$ to $T^{3}M$ for the curve $\gamma$, we obtain the equations

$$\frac{\partial L}{\partial x^j}-\frac{d}{dt}\left({\alpha }_j^0\right)=0,j=1,\dots ,n.$$

(28)

Along the canonical prolongation ${\gamma }_3$ of the curve $\gamma$, we have

$$\frac{\partial L}{\partial x^j}=〈\frac{D^2\gamma }{d{t}^2},{\nabla }_{\frac{d\gamma }{dt}}^2\frac{\partial }{\partial x^j}〉+〈R\left(\frac{D^2\gamma }{d{t}^2},\frac{d\gamma }{dt}\right)\frac{d\gamma }{dt},\frac{\partial }{\partial x^j}〉$$

and

$$\frac{d}{dt}\left({\alpha }_j^0\right)=〈\frac{D^2\gamma }{d{t}^2},{\nabla }_{\frac{d\gamma }{dt}}^2\frac{\partial }{\partial x^j}〉-〈\frac{D^4\gamma }{d{t}^4},\frac{\partial }{\partial x^j}〉,j=1,\dots ,n.$$

Consequently, Equation (28) takes the form

$$〈\frac{D^4\gamma }{d{t}^4}+R\left(\frac{D^2\gamma }{d{t}^2},\frac{d\gamma }{dt}\right)\frac{d\gamma }{dt},\frac{\partial }{\partial x^j}〉=0,j=1,\dots ,n.$$

and the Euler–Lagrange Equation (18) follows.

Thus, the Euler–Lagrange Equation (18) can be interpreted as the semispray of order 3 on $T^{3}M$, $X_E$, whose integral curves project onto the solutions of (18). The flow of the semispray $X_E$ is called the biharmonic flow.

5.3. The Cobiharmonic Flow

The Legendre transformation $Leg$ can be represented as follows:

where $\ll .,.\gg$ represents the pairing duality of vector and covectors on $TM$. and ${\pi }_{T^{3}M}:T{T}^{3}M\to T^{3}M,{\tau }_3^1:T^{3}M\to TM\mathrm{and}{\tau }_{TM}:T^{*}TM\to TM$ are the natural projections. Locally, we have

$$Leg(x^i;y^i;u^i;v^i)=(x^i;y^i;{\alpha }_i^0;{\alpha }_i^1),$$

where ${\alpha }_i^0$ and ${\alpha }_i^1$ are the real functions defined by (25). In this context, $Leg$ is a global diffeomorphism, since its restriction to each fiber of ${\tau }_3^1:T^{3}M\to TM$ is an isomorphism. We have a symplectomorphism

$$Leg:(T^{3}M,{\omega }_L)\to (T^{*}TM,{\omega }_1),$$

where ${\omega }_1$ is the symplectic canonical form on $T^{*}TM$. The inverse of the Legendre map is defined by

$$Le{g}^{-1}(x^i;y^i;p_i;q_i)=(x^i;y^i;u^i;v^i),$$

where

$$\begin{array}{cc}\hfill u^i& =\sum _{j=1}^n{g}^{ij}{q}_j-\sum _{j,k=1}^n{\tau }_{jk}^i{y}^j{y}^k\hfill \\ \hfill v^i& =\sum _{j=1}^n{g}^{ij}{p}_j-\sum _{j,l,r=1}^n{g}^{ij}{\mathrm{\Gamma }}_{jl}^r{y}^l{q}_r-3\sum _{j,l,r=1}^n{\mathrm{\Gamma }}_{rl}^i{g}^{rj}{q}_j{y}^l-\sum _{j,l,m=1}^n{A}_{jlm}^i{y}^j{y}^l{y}^m.\hfill \end{array}$$

Consider the Hamiltonian function associated with L; that is,

$$H_L=E\circ Le{g}^{-1}.$$

The function $H_L:T^{*}TM\to \mathbb{R}$ is locally given by

$$H_L(x^i;y^i;p_i;q_i)=\sum _{i=1}^n{p}_i{y}^i+\sum _{i=1}^n{q}_i\left(\frac{1}{2}\sum _{j=1}^n{g}^{ij}{q}_j-\sum _{j,k=1}^n{\mathrm{\Gamma }}_{jk}^i{y}^j{y}^k\right).$$

The dynamics of the Hamiltonian system $(T^{*}TM,{\omega }_1,H_L)$ is described by (15) and locally expressed by the following equations:

$$\begin{array}{c}{\displaystyle {\dot{x}}^i=y^i}\hfill \\ {\displaystyle {\dot{y}}^i=\sum _{j=1}^n{g}^{ij}{q}_j-\sum _{j,k=1}^n{\mathrm{\Gamma }}_{jk}^i{y}^j{y}^k}\hfill \\ {\displaystyle {\dot{p}}_i=\sum _{j=1}^n{q}_j\left(\sum _{k,l=1}^n\frac{\partial {\mathrm{\Gamma }}_{lk}^j}{\partial x^i}{y}^l{y}^k-\frac{1}{2}\sum _{k=1}^n\frac{\partial g^{jk}}{\partial x^i}{q}_k\right)}\hfill \\ {\displaystyle {\dot{q}}_i=2\sum _{j,k=1}^n{q}_j{\mathrm{\Gamma }}_{ik}^j{y}^k-p_i,}\hfill \end{array}$$

for $i=1,\dots ,n$. We call these Hamiltonian equations the cobiharmonic equations, and we call the flow of the Hamiltonian vector field $X_{H_L}$ the cobiharmonic flow.

Note that the Hamiltonian system $(T^{*}TM,{\omega }_1,H_L)$ coincides with the Hamiltonian system resulting from the study of the following optimal control problem:

$$\begin{array}{ccc}{\displaystyle \underset{u}{min}\frac{1}{2}\sum _{i,j=1}^n{g}_{ij}{u}^i{u}^j}\hfill & \mathrm{subject}\mathrm{to}\hfill & \begin{array}{c}{\displaystyle {\dot{x}}^i=y^i}\hfill \\ {\displaystyle {\dot{y}}^i=u^i-\sum _{j,k=1}^n{\mathrm{\Gamma }}_{jk}^i{y}^j{y}^k,}\hfill \end{array}\hfill \end{array}$$

where $(x^i;y^i)$ are the state variables and $u^i$ represents the control variables, $i=1,\dots ,n$. This optimal control problem was proposed in 2000 by M. Camarinha, P. Crouch, and F. Silva Leite ([11]). The equivalence between the two formulations, the optimal and the variational formulations, has also been explored by L. Abrunheiro, M. Camarinha, and J. Clemente-Gallardo in [20], with special emphasis on the intrinsic approaches.