On Geometry of Schmidt Legendre Transformation

A geometrization of Schmidt-Legendre transformation of the second order Lagrangians is proposed by building a proper Tulczyjew's triplet. The symplectic relation between Ostrogradsky-Legendre and Schmidt-Legendre transformations is obtained. Several examples are presented.


Introduction
The dynamics of a system can either be formulated by a Lagrangian function on the tangent bundle of a configuration space or by a Hamiltonian function on the cotangent bundle [1,4]. For a physical system, ideally, these two formalisms should be transformed to each other with the Legendre transformations. The transformation is immediate by means of fiber derivative of the Lagrangian function if Hessian of the Lagrangian (or Hamiltonian) function is nondegenerate. If the Hessian is degenerate or/and the system possesses various constraints, then defining the transformation becomes complicated.
To overcome the obstructions due to the existences of singularities and constraints, in the beginnings of the 50's, Dirac proposed an algorithm, nowadays called Dirac-Bergmann algorithm [18,19]. This algorithm was geometrized at the end of 70's by Gotay, Nester and Hinds [26,27,28,29]. At the late 70's, Tulczyjew showed that the dynamics can be represented as a Lagrangian submanifold of certain symplectic manifold on higher order bundles [7,56,57,61,62]. In this setting, Hamiltonian and Lagrangian formulations are, in fact, two different generators of the same Lagrangian submanifold. So that, Tulczyjew redefined the Legendre transformation as a passage between these two different generators.
Although in classical mechanics a Lagrangian density is a function of positions and velocities, it is possible to find theories involving Lagrangian densities depending on the higher order derivatives as well. In such cases, to pass the Hamiltonian picture, it is a tradition to employ the Ostrogradsky-Legendre transformation [44]. The Ostrogradsky approach is based on the idea that consecutive time derivatives of initial coordinates form new coordinates, hence a higher order Lagrangian can be written in a form of a first order Lagrangian on a proper iterated tangent bundle. In some recent studies, the higher order Lagrangians has received attention [8,37,38,39,40,41,50], extending the previous works of Pais and Uhlenbeck [45]. The linear harmonic oscillator is a perennial favorite of quantum theorists besides being the fundamental bedrock of many classical theoretical models in both physics and engineering. In view of its fundamental importance it provides a good reference point on which to build higher-order equations which have possible physical relevance. There are extensive studies in the literature for singular or/and constraint higher order Lagrangians as well, let us give an incomplete list of such studies [6,11,10,23,34,30,31,42,43,47]. For the system whose configuration is a Lie group, we additionally refer [14,24,25], and for field theories, see [63]. We cite a recent study on the geometry of higher order theories in terms of Tulczyjew's triplet [15].
At the middle of 90's, Schmidt proposed an alternative method for the Legendre transformations of higher order Lagrangian systems working both for non-degenerate and degenerate systems [48,49]. Schmidt defined the acceleration as a new coordinate instead of the velocity. Although, Schmidt-Legendre transformation has not been credited as it deserves, it is possible to find some related works in the literature [17,32,33]. Two works [2,3] done on the Schmidt-Legendre transformation have motivational importance for the present work. In these papers, the comparison and the relation between the Legendre transformations in the senses of Ostrogradsky and Schmidt have been studied.
The main concern of this manuscript is the higher order differential equations generated by the higher order Lagrangian functions and their Legendre transformations. The main objectives are to construct a geometric framework, namely a Tulczyjew' triplet, for the Schmidt-Legendre transformation, and construct the symplectic relations between Ostrogradsky-Legendre and Schmidt-Legendre transformations in pure geometric ways.
In order to achieve the goals of the paper, we shall start to the following section by reviewing some basic ingredients of the Tulczyjew's construction of the Legendre transformation, see also [22,21]. In section 3, we will review the geometry of higher order tangent bundles, the theory of higher order Lagrangian formalism, and the Ostrogradsky-Legendre transformation. In section 4, the acceleration bundle will be defined. Tangent and cotangent bundles of the acceleration bundle are presented. Then, the Schmidt-Legendre transformation will be presented for the Lagrangians in terms of the Tulczyjew triplet. I section 5, the symplectic relation between Ostrogradsky-Legendre and Schmidt-Legendre transformations will be constructed. The last section, will be reserved for several examples including Pais-Uhlenberg, Sarıoglu-Tekin and Clemént Lagrangians.

Special symplectic structures
Let P be a symplectic manifold carrying an exact symplectic two form Ω P = dϑ P . A special symplectic structure is a quintuple (P, π P M , M, ϑ P , χ) where π P M : P → M is a fibre bundle and χ : P → T * M is a fiber preserving symplectic diffeomorphism such that χ * θ T * M = ϑ P for θ T * M being the canonical one-form on T * M. χ can be characterized uniquely by the condition χ(p), X M (x) = ϑ P (p), X P (p) for each p ∈ P , π P M (p) = x and for vector fields X M and X P satisfying π P M * X P = X M [35,54,58].
A real valued function F on the base manifold M defines a Lagrangian submanifold of the underlying symplectic manifold (P, Ω P = dϑ P ). The function F together with a special symplectic structure (P, π P M , M, ϑ P , χ) are called a generating family for the Lagrangian submanifold S P . Since χ is a symplectic diffeomorphism, it maps S P to the image space im (dF ) of the exterior derivative of F , which is a Lagrangian submanifold of T * M.

Morse Families
Let P, π P M , M be a fibre bundle. The vertical bundle V P over P is the space of vertical vectors U ∈ T P satisfying T π P M (U) = 0. The conormal bundle of V P is defined by Let E be a real-valued function on P , then the image im (dE) of its exterior derivative is a subspace of T * P . We say that E is a Morse family (or an energy function) if for all z ∈ im (dE) ∩ V 0 P , [7,36,58,59,61,62]. In local coordinates (x a , r i ) on the total space P induced from the coordinates (x a ) on M, the requirement in Eq.(2) reduces to the condition that the rank of the matrix be maximal. A Morse family E on the smooth bundle P, π P M , M generates an immersed Lagrangian submanifold of (T * M, Ω T * M ). Note that, in the definition of S T * M , there is an intrinsic requirement that Consider a Morse family E on a fibre bundle P → M, and the Lagrangian submanifolds S T * M defined in Eq. (4). Assume that there exists a subbundle of P ′ ⊂ P → M of where the induced function E| ′ P satisfies the conditions of being a Morse family presented in (3) and generates the same Lagrangian submanifold S T * M of T * M. This procedure is called the reduction of the Morse family. Note that, in this case, the final structure E| P ′ on P ′ → M is called the reduced Morse family generating S T * M . We refer [7,60] for more elaborated formulations and more precise discussions on this subject.

The Legendre transformation
Let (P, Ω P = dϑ P ) be an exact symplectic manifold, and (P, π P M , M, ϑ P , χ) be a special symplectic structure. A function F on M defines a Lagrangian submanifold S P ⊂ P as described in Eq.(1). If S P = im (Υ) is the image of a section Υ of P, π P M , M then we have χ • Υ = dF . Assume that (P, π P M ′ , M ′ , ϑ ′ P , χ ′ ) is another special symplectic structure associated to the underlying symplectic space (P, Ω P ). Then, from the diagram it follows that the difference ϑ P −ϑ ′ P of one-forms must be closed in order to satisfy Ω P = dϑ P = dϑ ′ P . When the difference is exact, there exists a function ∆ on P satisfying d∆ = ϑ P − ϑ ′ P . If S P is the image of a section Υ ′ of the fibration P, π P M ′ , M ′ , then the function generates the Lagrangian submanifold S P [57,59,61]. This is the Legendre transformation. If, finding a global section Υ ′ of π P M ′ satisfying im(Υ ′ ) = S P is not possible, the Legendre transformation is not immediate. In this case, define the Morse family on a smooth subbundle of P, π P M ′ , M ′ , where E satisfies the requirement (2) of being a Morse family. Then, E generates a Lagrangian submanifold S T * M ′ on T * M ′ as described in Eq.(4). The inverse of χ ′ maps S T * M ′ to S P bijectively, that is S P = (χ ′ ) −1 (S T * M ′ ).
A way to generate the symplectic transformation between T * M and T * M ′ that is generate a symplectic diffeomorphism χ ′−1 e.g. from T * M to T * M ′ is to define a generating function

Higher order tangent bundles
Let Q be an n-dimensional differentiable manifold with coordinates x → (q) = (q 1 , q 2 , ..., q n ). Two differentiable curves γ andγ are called equivalent γ ∼ kγ if they agree at x and agree up to their k−th derivatives, that is if for r = 0, 1, 2, ..., k.
An k-th order tangent vector v k (x) at x is an equivalence class of curves at x. This class will be denoted by t k γ (0). The set of all equivalence classes of curves, that is the set of all k-th order tangent vector at x is the k-th order tangent space x Q of all k-th order tangent spaces T k x Q as x varies on Q is the total space of the k-th order tangent bundle of Q with fibers being T k x Q. The k − th order tangent bundle T k Q of Q is (k + 1) n dimensional manifold. There are hierarchic fibrations defined on each fiber where r > s, and r = 1, ..., k and s = 0, 1, ..., k − 1. In local charts, this looks like projecting an r + 1-tuple (q;q;q; ...; q (r) ) to its first s + 1 components (q;q;q; ...; q (s) ).

Higher order Euler-Lagrange equations
Let Q be an n-dimensional manifold with local coordinates (q) = (q 1 , q 2 , ..., q n ). The k − th order tangent bundle T k Q of Q is (k + 1) n dimensional manifold with local coordinates (q;q;q; ...; q (k) ) induced those from Q. A Lagrangian density L = L(q;q;q; ...; q (k) ) is a real-valued function on T k Q from which we define the action integral S = b a L(q;q;q; ...; q (k) )dt by fixing two points q (a) and q (b) in Q. To find the extremum values, we take the variation of the action integral which results with k-th order Euler-Lagrange equations If the partial derivative ∂L/∂q (k) depends on q (k) then, k-th order Euler-Lagrange equations (10) are a set of differential equation of order 2k.

Ostrogradsky-Legendre transformation
The traditional framework for obtaining the Hamiltonian formulation of the higher order Lagrangian formalisms is due to Ostrogradsky [44]. The Ostrogradsky approach is based on the idea that consecutive time derivatives of initial coordinates form new coordinates, hence a higher order Lagrangian can be written in a form of a first order Lagrangian on iterated tangent bundles. There are several different ways to express a k-th order Lagrangian formalism in a first order form. Most general one is to define (k + 1)n-dimensional configuration space by imposing the set of constraintsq (i) − q (i+1) = 0 for i = 1, ..., k − 1. In this case, starting with a k-th order Lagrangian L = L q;q; ...; q (k) on T k Q, we define the following constraint Lagrangian density In this formulation, the canonical Hamiltonian function is given by where (π 1 , ..., π k ) is set of the conjugate momenta and the local representatives of the cotangent bundle T * N.
This motivates a shorter way of defining momenta without refereing to the Lagrangian multipliers as follows. The Ostrogradsky -Legendry transformation can be realised by the introduction of the momenta where see also for the step by step transformation [5,6,31]. 4 Schmidt-Legendre transformation 4

.1 Acceleration bundle
Let x be a point in a manifold Q, and consider the set C ∞ x (Q) of smooth curves passing through x. We define a subset A x (Q) of C ∞ x (Q) by only considering the curves whose first derivatives are vanishing at It is worthless to say that since the vanishing of the first derivative is asked only at a single point, the curve γ needs not to be a constant.
We now define an equivalence relationship on A x (Q). Take two curves γ, β ∈ A x (Q), that is γ (0) = β (0) = x andγ (0) =β(0) = 0. We call γ and β are equivalent if they are agree up to their second derivatives. In other words, two equivalent curves γ and β satisfy We denote an equivalence class by t 2 γ (0) and denote it by a (x). The set of all equivalence classes of curves at x is the acceleration space A x Q at x ∈ Q. We denote an equivalence class t 2 γ (0) by a x ∈ A x Q. The union AQ = ∪A x Q of all acceleration spaces is a 2n−dimensional manifold with local charts a x → (q, a =q) induced those from the coordinates on Q. This suggests the fiber bundle structure of the acceleration manifold AQ over Q with projection An alternative definition of the acceleration bundle may be stated as follows. First recall the imbedding of the second order tangent bundle In literature, such an element V is called second order. To define an imbedding of the acceleration bundle AQ into T T Q, we additionally require that a second order vector field must satisfy T τ Q (V ) = τ T Q (V ) = 0. By this way, we define AQ as a subbundle of the bundle T T Q → Q. In this respect, we may also understand AQ as a subbundle of T 2 Q → Q as well. Diagrammatically, we summarize the sequence of subbundles as follows.
This gives that, the acceleration bundle AQ can be identified with the intersection of the vertical subbundle V T Q (consisting of vectors on T Q and projecting to the zero vector on Q via the mapping T τ Q ) of T T Q and the second order tangent bundle T 2 Q. Alternatively, we may write a sequence where the first mapping is the inclusion and the second is the projection 2 1 τ Q in (9). When a connection is defined on Q, T 2 Q becomes a vector bundle over Q, and it can be written as the Withney some of two copies of its tangent bundle [20]. For a more general discussion on this, we refer [55]. In such a case, one may identify the acceleration bundle AQ with the tangent bundle T Q.
Let φ be a differential mapping Q → M between two manifolds Q and M. We define the acceleration lift Aφ : AQ → AM of φ as follows. Take a x ∈ A x Q, it can be represented by a curve γ lying in the equivalence class a x = t 2 γ (0). The image space of the curve under φ • γ is a curve in M. Note also that, the velocity of φ • γ at t = 0 vanishes, hence φ • γ is lying in one of the equivalence classes in the acceleration bundle A φ(x) M consisting of the velocity free curves passing through φ(x) ∈ M. Accordingly, we define the acceleration lift by where t 2 (φ • γ) (0) is the equivalence class containing φ • γ. Now, we define a local diffeomorphism linking the tangent and acceleration bundles and second order tangent bundle. Let a curve γ represents an element in T 2 x Q, then one may write γ(t) = x + vt + at 2 + O(t). We define a curveγ(t) = x + at 2 + O(t) which is velocity free. Consider the following mapping In a local coordinate frame, this mapping looks like It is possible to define the mixed iterated tangent and acceleration bundles AT Q and T AQ with local coordinate charts It is evident that, there exist various projections of the iterated bundles given by Consider also the following velocity-acceleration rhombics establishing the double fiber bundle structures of AT Q and T AQ.
These commutative diagrams enable us to define the canonical maps We note that, the third order tangent bundle T 3 Q and the acceleration bundle AT Q are isomorphic and composing this with the canonical map κ Q in (19), we arrive the bundle isomorphism T AQ → T 3 Q via the map S : T AQ → T 3 Q : (q, a;q,ȧ) → (q,q; a,ȧ). (20)

Higher order acceleration bundles
Let us generalize the previous discussions on the acceleration bundle to the bundles higher than 3. To this end, first define curves on a base manifold Q with coordinates (q) and consider the second order acceleration space A 2 q Q at the point q ∈ Q by defining an equivalence relationship on the space of curves with vanishing velocitiesq(t) = 0 and the third derivative ... q (t) = 0 by requiring that their second and fourth derivatives are equal. The collection of these spaces results with the second order acceleration bundle defines a fiber bundle structure over Q. One may define the following iterative picture as well To generalize this to higher order bundles is straight forward. So that, one can define A k Q and the following projections Let us now make the local isomorphism between the tangent bundle of a higher order acceleration bundle with the odd order iterated tangent bundles as follows. For the fifth order tangent bundle T 5 Q, the isomorphism is given by In a similar we may generalized this by identifying T A r Q → T (2r+1) Q : (q, a q , ..., b q ;q,ȧ q , ...,ḃ q ) → (q;q; a q ;ȧ q ; ...; b q ;ḃ q ).

Gauge symmetry of second order Lagrangian formalisms
The essence of the Schmidt's Method presented in [2,3,48,49] is to replace the higher-order Euler-Lagrange equation of motion by the more familiar first-order Euler-Lagrange equation by the introduction of a new Lagrangian. As it is very well known, the symmetry of the Euler-Lagrange equations (10) in any order is the addition of a total derivative to the Lagrangian L. Consider a second order non-degenerate Lagrangian density on T 2 Q. We add the total derivative of a function F = F (q,q,q) in order to arrive a new Lagrangian functionL (q;q;q; ... q ) = L (q;q;q) + d dt F (q;q;q) = L (q;q;q) + ∂F ∂qq of order 3. Note that,L is defined on T 3 Q whereas L and F are defined on T 2 Q. A straightforward computation proves that, the third order Euler-Lagrange equations generated byL are the same with the second order Euler-Lagrangian equations generated by L.
By recalling the surjection S presented in Eq. (20), we pull the LagrangianL in Eq. (22) back to the tangent bundle T AQ of the acceleration bundle AQ which results with a first order Lagrangian on the first order tangent bundle T AQ considering the base manifold as AQ. The first order Euler-Lagrange equation generated by the Lagrangian density (23) takes the particular form We find upon expanding the total time derivative in the second equation in (24) and using the fact that L is independent ofȦ Recall that, we have started with a second order non-degenerate Lagrangian density L presented in (21). The non-degeneracy of L is equivalent to say that the rank of the Hessian matrix [∂ 2 L/∂A 2 ] is full. We additionally assume that the auxiliary function F satisfies the equality ∂L ∂A In this case, taking the partial derivative of this equality (26) with respect to A, we observe that the matrix [∂ 2 F/∂A∂Q] is non-degenerate. In this case, the second set of the Euler-Lagrangian equations presented in (24) reduce to the constraint A −Q = 0. Accordingly, the first set of the Euler-Lagrangian equations in (24) give the second order Euler-Lagrange equations generated by the Lagrangian L in (21). Let us take the partial derivative of the assumption on the auxiliary function F in (26) with respect toQ. This brings an integrability condition on F as follows. The symmetry of the matrix [∂ 2 F/∂Q 2 ] enforces the symmetry of the matrix [∂ 2 L/∂A∂Q]. In other words, to define a proper auxiliary function, the matrix [∂ 2 L/∂A∂Q] has to be symmetric. If not, then there may be no such auxiliary function F establishing the reduction. In the forthcoming subsection, we shall present some examples pointing out this fact.

Schmidt-Legendre transformation for even orders
This construction enables us to interpret the second order Euler-Lagrange equations as Lagrangian submanifolds. To this end we propose the following Tulczyjew's triplet for the case of acceleration bundle where L 2 is the Lagrangian density in (23). Introducing the coordinates (Q, A, P Q , P A ;Q,Ȧ,Ṗ Q ,Ṗ A ) ∈ T T * AQ one finds the symplectomorphisms and the potential one-forms where the difference ϑ 2 − ϑ 1 is the exact one-form d P Q ·Q + P A ·Ȧ on T T * AQ. Let us note that, with the triplet (27), we have achieved to recast the second order Lagrangian formalism as a Lagrangian submanifold given by where T π AQ is the tangent mapping and the superscript denotes the pull-back operation. In the local chart, this Lagrangian submanifold can be written as Note that, by combining the first and third, combining the second and the fourth, we arrive at the Euler-Lagrange equations (24) which covers the second order Euler-Lagrange equations under the assumption that (26) is satisfied and the matrix [∂ 2 F/∂A∂Q] is non degenerate.
Recalling the potential function, we define the energy function as follows satisfies the requirements, given in Eq.
(2), of being a Morse family on the Pontryagin bundle T AQ × T * AQ over the cotangent bundle T * AQ. Hence, E L→H generates a Lagrangian submanifold S T * T * Q of T * T * Q as defined in Eq.(4). In coordinates the submanifold S T * T * Q is given by The inverse musical isomorphism Ω ♯ T * Q maps this Lagrangian submanifold S T * T * Q to the one defined in 32 generated by the Lagrangian L 2 . The last two equations define the associated canonical momenta After the substitution of (38), the Hamiltonian Morse family (33) on the Whitney product space T * AQ × AQ T AQ over T * AQ becomes free ofȦ and turns out to be This is an example of a Morse family reduction. If we assume that the matrix [∂ 2 F/∂A∂Q] is non-degenerate, from the equation (38), we can solveQ in terms of the momenta that iṡ This enables us to arrive the isomorphic copies of the functions on T AQ depending only on the variables Q,Q, A on the space T * AQ by the direct substitution. So that, futher reduction of the Morse family is possible given by can be defined on the cotangent bundle T * AQ. In this case, the Hamiltonian function H on T * AQ determines the Lagrangian submanifold S T * T * Q in (36) by means of the Hamilton's equations where the Hamiltonian function H is the one in (40) and ϑ 1 is in Eq. (31). Locally, the Hamilton's equations arė The first equations are the defining identity (39), the second is identically satisfied after the substitutions of the transformations (37 and 38), whereas the third and fourth ones are the Euler-Lagrange equations.
In order to generalize the present discussion to any even order Lagrangian density, one simply needs to replace AQ with the higher order bundle A k Q. In this case, simple manipulations and presentation some indices will be enough to achieve this.

Schmidt-Legendre transformation for odd orders
For a third order Lagrangian theory L (q;q;q; ... q ), to link the acceleration A with the derivative of the velocityQ, we need to introduce a trivial bundle T AQ × T M over the tangent bundle T AQ. Here, M is an n−dimensional manifold with local coordinates (r). Using the isomorphism (20), we define a Morse family is employed.
In the classical sense, canonical momenta for the cotangent bundle T * (AQ × M) are where (P Q , P A , P r ) is the conjugate fiber coordinates. If the non-degeneracy of [∂ 2 F/∂Q∂r] is assumed, it is possible to solveQ in terms of (Q, A, r,P r ). In this case, we writeQ = W (Q, A, r,P r ). Note that, the third order Lagrangian density L is no need to be nondegenerate in the sense of Ostragradsky. Now we assume particularly that the third order Lagrangian L that we started with depends only on (q,q,q). That is, it actually is a second order Lagrangian. In this case, the Lagrangian L 3 becomes where we assume F = F (Q,Q, r). This time the canonical momenta (45) take the particular form where P A = 0 is a set of primary constraints. The non-degeneracy of [∂ 2 F/∂Q∂r] enough to writeQ andṙ in terms of Q, A, r, P Q ,P r . The total Hamiltonian is given by where λ is the vector of Lagrange multipliers. Geometrically, by employing the constraint Φ 1 = P A = 0, the total Hamiltonian can be reduced to (T * Q × Q AQ) × T * M. But further reduction on the total space is needed for the consistency of the primary constraint Φ 1 . To check the consistency of Φ 1 = P A , we take the Poisson bracket of Φ 1 and H T which gives that, the time derivative must be weakly zero. To check the consistency of Φ 2 , we computė If the Lagrangian is assumed to be non-degenerate then this step determines the Lagrange multipliers λ, and the constraint algorithm is finished up. If the Lagrangian is degenerate further steps may be needed to determine the Lagrange multipliers as well as to close up the Poisson algebra. We refer [2] for the present and further discussions on this. It is immediate to generalize this for the higher order theories.

For even order formalisms
For a second order Lagrangian density on T 2 Q, the corresponding Hamiltonian formulation obtained after performing Ostrogradsky Legendre transformation is on the canonical symplectic space T * T Q, whereas the corresponding Hamiltonian formulation obtained after performing Schmidt Legendre transformation is on T * AQ. Here, AQ is the acceleration bundle presented in the first section. In this section, we establish a purely symplectic transformation between T * T Q and T * AQ. The symplectic transformation (relation) between two spaces can be obtained by following [7,56,57,58]. To this end, we first recall the following coordinates (q 1 , q 2 ; π 1 , π 2 ) on T * T Q and Q, A; P Q , P A on T * AQ. The canonical Liouville one-forms are given by ϑ T * T Q = π 1 · dq 1 + π 2 · dq 2 and ϑ T * AQ = P Q · dQ + P A · dA, whose exterior derivatives dϑ T * T Q and dϑ T * T Q are symplectic forms on T * T Q and T * AQ. Recall that the momenta in Eq.(37 and 38) defines the following isomorphism whereȦ is a smooth function of Q, A; P Q , P A . Composing the canonical mapκ Q in (19) map, and the acceleration bundle projection t T Q presented in (18), we arrive a surjective projection of T * AQ on T Q given by This mapping enables us to define a Whitney product over the tangent bundle T Q, where the projection T * AQ → T Q in the one in Eq.(51) and T * T Q → T Q is the cotangent bundle projection. Hence, we can take coordinates on the Whitney product W as Q, π 1 , π 2 ; A; P Q , P A by identifying q = Q andQ = Z (Q, A, P A ), so that W is 6n-dimensional. By pulling the Liouville forms ϑ T * T Q and ϑ T * AQ back to W, we arrive the following one form The exterior derivative of ω W = dϑ W is a symplectic manifold on W . A symplectic transformation (or a relation) between T * AQ and T * T Q is a Lagrangian submanifolds of W . A Lagrangian submanifold S of W has dimension 3n and the restriction of the symplectic form to S vanishes. The total product space W in (52) can be understood as the cotangent bundle T * T 2 Q of the second order tangent bundle T 2 Q identified with the Whitney product T Q × Q AQ. In this case, the exterior derivative of a function F = F (Q, Z, A) on T 2 Q defines a Lagrangian submanifold S. To arrive the Legendre transformation, we equate the one-form ϑ W with the exterior derivative of F , and this locally looks like For this we arrive the transformation by the following list of relations where we chooseQ = Z (Q, A, P A ). These equations are the Legendre transformation relating the Schmidt and Ostrogradsky transformations [2]. Explicitly, the symplectic diffeomorphism is given by As a summary, we can say that the Gauge invariance of the Lagrangian is the addition of a total derivative dF/dt to the Lagrangian function. For second order theories, F is defined on T 2 Q identified with T Q × Q AQ so that its exterior derivative is a Lagrangian submanifold of T * T 2 Q which enaples us to define the canonical diffeomorphism (55) from T * AQ to T * T Q.

For the odd order formalisms
Define two bundle structures by considering the base manifold as AQ, namely the trivial bundle AQ × M and T 2 Q. Here, the manifold M is the one presented in the subsection (4.5) whereas the bundle structure of T 2 Q is the one described in (15). Accordingly, we take the Whitney product of these two bundle and arrive T 2 Q × AQ (AQ × M) with local coordinates (Q, A,Q, r).
The function F = F (Q, A,Q, r) generating the gauge invariance of the third order Lagrangian function is defined on T 2 Q × AQ (AQ × M) hence the image space of its exterior derivative is a Lagrangian submanifold of the cotangent bundle So that it generates a symplectic diffeomorphism between the cotangent bundles T * T 2 Q and T * (AQ × M)). Observe that, T * T 2 Q is the momentum phase space for the third order Lagrangian formalism when it is transformed in the sense of Ostragradsky, and T * (AQ × M)) is the momentum phase space for the third order Lagrangian formalism when it is transformed in the sense of Schmidt. Let us first introduce the coordinates (q 1 , q 2 , q 3 ; π 1 , π 2 , π 3 ) on T * T 2 Q then the canonical one-forms given by ϑ T * T 2 Q = π 1 · dq 1 + π 2 · dq 2 + π 3 · dq 3 , ϑ T * (AQ×M ) = P Q · dQ + P A · dA + P r · dr.
By pulling the one-forms ϑ T * T Q and ϑ T * AQ back to U we arrive at the following one form ϑ U = ϑ T * AQ ⊖ ϑ T * T Q = P Q · dQ + P A · dA + P r · dr − π 1 dQ − π 2 dW − π 3 dA whose exterior derivative is a symplectic two-form on U. Here, we take q 3 = A and q 1 = Q to verify the definition of the Whitney bundle U in (54). To arrive at the Legendre transformation, we equate the one-form ϑ U with the exterior derivative of F , and this locally reads where we writeQ = W (Q, A, r, P A ) by solving the third equation in (45).

Example 1
Let Q = R and consider the non-degenerate second order Lagrangian given by on T 2 Q with coordinates (q,q,q). The 2nd order Euler-Lagrange equations (??) yield the following fourth-order equation On the coordinates (q, a,q,ȧ) on T AQ, integrating the condition (26), we arrive the auxiliary function F (q, a,q,ȧ) = −qa + W (q, a), for W being an arbitrary function. The Lagrangian L 2 in (23) takes the particular form It is obvious that the total derivative term may be neglected. Following (37)(38), we define the canonical momenta as then the canonical Hamiltonian function (40) on T * AQ with coordinates given by The Hamilton's equations (42) arė The Ostrogradsky Legendre transformation of the second order Lagrangian (56) can be achieved by defining the momenta (13) as where we choose q 1 = q and q 2 =q. The canonical Hamiltonian (12) turns out to be The transformation between these two system is given by whereas the generating function F = −q 2 a.

Example 2
The Pais-Uhlenbeck fourth-order oscillator equation of motion, viz, which is a quantum mechanical prototype of a field theory containing both second and fourth order derivative terms [37,41,45]. Mechanically the Pais-Uhlenbeck oscillator can be realized as a two-degrees of freedom spring-mass system where k is the stiffness constant. This can be expressed by the Pais-Uhlenbeck fourth-order oscillator equation where ω 1 = 3k m and ω 2 = k m . The Lagrangian of the coupled system is given by It is easy to show that the corresponding Hamiltonian H = T +U ≡ E m.e. ( mechanical energy) and satisfies dH dt = 0. The Pais-Uhlenbeck equation (60) is generated by a degenerate fourth order Lagrangian In this paper, we consider the non-degenerate second order Lagrangian of the Pais-Uhlenbeck oscillator given by We consider Q = R with coordinate q, T 2 Q with coordinates (q,q,q), T AQ with coordinates (q, a,q,ȧ), and write the Lagrangian L 2 in (23) for the case of Pais-Uhlenbeck Lagrangian L P U in (64) as follows where we choose F (q,ȧ) = −qa by solving the defining equation (26). the canonical momenta (37)(38) take the particular form whereas the canonical Hamiltonian function (40) turns out to be In this case the motion is generated bẏ It is interesting at this juncture to compare these equations with those following from Ostrogradsky's formulation. For the latter the new coordinates are chosen to be q 1 = q and q 2 =q while the corresponding momenta (13) are The Hamiltonian in (12) has the explicit form H = 1 2 π 2 2 + π 1 q 2 + 1 2 (w 2 1 + w 2 2 )q 2 2 − 1 2 w 2 1 w 2 2 q 2 1 and leads to the following equations: We refer [10]. Comparing these first-order equations with those obtained earlier (66), we observe the following relations which clearly defines a canonical transformation between the phase space variables as a particular case of (55). Here, the generation function is F = −q 2 a.
Using the method for degenerate cases, the Legendre transformation of (69) is possible because the integrability condition (26) is not forced. In order to achieve this goal, we take Q = R 2 with coordinates (x, y), and AQ with induced coordinates (x, y; a, b). The augmentation of AQ is given by the manifold M = R 2 with auxiliary variables (r, s). So that, the tangent bundle of the product manifold T (AQ × M) has the coordinates x, y, a, b, r, s;ẋ,ẏ,ȧ,ḃ,ṙ,ṡ .
Obeying the condition (44), we introduce the auxiliary function F =ẋr + bs, then the canonical momenta, denoted by (p x , p y , p a , p b , p r , p s ) and defined by (47), have the following expressions, Accordingly, the total Hamiltonian H T in (50) takes the particular form where the Lagrange multipliers are (λ 1 , λ 2 ). The consistency check for the primary constraints p a ≈ 0 and p b ≈ 0 result with the secondary constraints The consistency checks of the secondary constraints read the tertiary constraints which defines the Lagrangian multipliers

Example 4
Take Q = R 6 with coordinates (X, Y) both of which are three dimensional vectors. We consider a degenerate second order Lagrangian density introduced in [50] where an investigation on topologically massive gravity was performed. The acceleration bundle AQ = R 12 is equipped with the induced coordinates (X, Y; A, B) and six number of auxilary variables (r, s) as the coordinates of the manifold M are introduced. We take the auxiliary function F =Ẋ · r +Ẏ · s obeying the condition (44). So that, we are ready to define the Lagrangian density (46) in the present case as follows on the tangent bundle T (AQ × M) with the induced fiber coordinates Ẋ ,Ẏ,Ȧ,Ḃ,ṙ,ṡ . The canonical momenta (47) are computed to be P X =Ẋ +ṙ, P Y =Ẏ +ṡ + A, P A = P B = 0, P r =Ẋ, P s =Ẏ, where (P X , P Y , P A , P B ; P r , P s ) are the fiber coordinates of the product cotangent bundle T * AQ × T * M. Accordingly, the primary constraints are P A ≈ 0 and P B ≈ 0. In this case, the total Hamiltonian (50) turns out to be where λ and β are Lagrange multipliers. The consistency of the primary constraints leads to the secondary constraints We check the compatibility of these secondary constraints as well. These give the tertiary constraints Note that, the secondary and tertiary constraints in (73) and (74) define the auxiliary variables (r, s) and their conjugate momenta (P r , P s ) in terms of the coordinates T * AQ. Continuing in this way, we compute the fourtiery constraints So we arrive the Hamiltonian formulation of the dynamics generated by the Lagrangian density (72). The reduced Hamiltonian is given by on the manifold T * AQ with the constraints P A ≈ 0 and P B ≈ 0 are employed.

Example 5
We now consider the degenerate second order Lagrangian density which is similar to the one defined in [12]. Here, the inner product X 2 = T 2 − X 2 − Y 2 is defined by the Lorentzian metric and the triple product is X · (Ẋ ×Ẍ) = ǫ ijk X iẊ jẌ k where ǫ ijk is the completely antisymmetric tensor of rank three. We consider the base manifold Q = R 3 with coordinates (X), the acceleration bundle AQ = R 6 with coordinates (X, A), and we consider the auxiliary variables (r) as a coordinate frame of a manifold M = R 3 . We introduce the auxiliary function F =Ẋ · r which enables us to write the Lagrangian density (46) as L = − 1 2Ẋ 2 + X · Ẋ × A +Ẋ ·ṙ + r · A on T (AQ × M) with coordinates X, A, r;Ẋ,Ȧ,ṙ . The conjugate coordinates on the cotangent bundle T * (AQ × M) are (X, A, r; P X , P A , P r ) and following (47) they are defined by P X = −Ẋ + A × X +ṙ, P A = 0, P r =Ẋ.
This gives that P A ≈ 0 are the primary constraints. The total Hamiltonian (50) becomes where λ are the Lagrange multipliers.
Let us now check the consistency of the primary constraints which determine a set of secondary constriants. The consistency check of these secondary constraints (77) lead to the tertiary constraints We check the compatibility of these constraints as well. These give the fourtiery constriants From these last equations, while solving λ, only two of these three equations are independent, so that two components of the Lagrange multiplier λ can be determined depending on the third one. By taking the dot product of (78) with X we arrive at a new constraint 3A × P r · X − A · X ≈ 0.
The consistency condition for this scalar constraint {3 (A × P r ) · X − A · X, H T } = 3 (λ × P r ) · X − λ · X − A · P r ≈ 0 determines the third and last component of the Lagrange multiplier λ. Substitution of the Lagrange multiplier result with the explicit determination of the total Hamiltonian (76). We refer [16] for the application of the Ostrogradsky-Legendre transformation to the Sarıoglu-Tekin Lagrangian (72) and Clemént Lagrangian (75).

Conclusions & Future work
In this paper, we have constructed Tulczyjew triplet for the case of acceleration bundle. This allowed us to study the higher order Lagrangian formalism in terms of Lagrangian submanifolds and symplectic diffeomorphisms. We have presented the symplectic relation between Ostrogradsky and Schmidt methods of Legendre transformation. Several examples both for the degenerate Lagrangians and the non degenerate Lagrangians have been presented. Some possible future works: • To construct a unified formalism for the Schmidt-Legendre transformation in terms Skinner-Rusk [51,52,53]. This was done for the case of Ostrogradsky-Legendre transformation in [46], and for the Hamilton-Jacobi theory [13].
• To study the Schmidt-Legendre transformation when the configuration space is a Lie group and under the existence of some symmetries. We cite [14,24,25] for the Ostrogradsky-Legendre transformation on Lie groups, and Ostrogradsky-Lie-Poisson reduction.

Acknowledgement
One of us (OE) is grateful to Hasan Gümral for pointing out the examples of Sarıoglu-Tekin and Clemént Lagrangians, and for many discussions on this subject.