Higher Order Hamiltonian Systems with Generalized

mathematics Article

Higher Order Hamiltonian Systems with Generalized Legendre Transformation Dana Smetanová Department of Informatics and Natural Sciences, Institute of Technology and Business, Okružní 517/10, ˇ 370 01 Ceské Budˇejovice, Czech Republic; [email protected]; Tel.: +420-387-842-133 Received: 9 August 2018; Accepted: 4 September 2018; Published: 10 September 2018

Abstract: The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition. Keywords: Hamilton equations; Lagrangian; regular and strongly regular systems MSC: 58Z05, 35A15, 58J72

1. Introduction Hamiltonian theory on manifolds has been intensively studied since the 1970s (see e.g., [1–10]). The aim of this paper is to apply an extension of the classical Hamilton–Cartan variational theory on fibered manifolds, recently proposed by Krupková [11,12], to the case of a class of second order Lagrangians and third order Hamiltonian systems. In the generalized Hamiltonian field theory, one can associate different Hamilton equations corresponding to different Lepagean equivalents of the Euler–Lagrange form with a variational problem represented by a Lagrangian. With the help of Lepagean equivalents of a Lagrangian, one obtains an intrinsic formulation of the Euler–Lagrange and Hamilton equations. The arising Hamilton equations and regularity conditions depend not only on a Lagrangian but also on some “free” functions, which correspond to the choice of a concrete Lapagean equivalent. Consequently, one has many different “Hamilton theories” associated to a given variational problem. A regularization of some interesting singular physical fields, the Dirac field, the electromagnetic field, and the Scalar Curvature Lagrangian by various methods has been studied in [3,6,13–15]. Some second order Lagrangians have also been discussed in [16–18]. The multisymplectic approach was proposed in [2,4,8,10]. This approach is not well adapted to study Lagrangians that are singular in the standard sense. Note that an alternative approach to the study of “degenerated” Lagrangians (singular in the standard sense) is the constraint theory from mechanics (see [19,20]) and in the field theory [21]. In this work, we are interested in second order Lagrangians that give rise to Euler–Lagrange equations of the 3rd order or non-affine 2nd order. All these Lagrangians are singular in the standard Hamilton–De Donder theory and do not have a Legendre transformation. Examples of these Lagrangians are afinne (scalar curvature Lagrangians) and many Lagrangians quadratic in second derivatives. However, in the generalized setting, the question on existence of regular Hamilton equations makes sense. For such a Lagrangian, we find the set of Lepagean equivalents (respectively family of Hamilton equations) that are regular in the generalized sense, as well as a generalized Mathematics 2018, 6, 163; doi:10.3390/math6090163

www.mdpi.com/journal/mathematics

Mathematics 2018, 6, 163

2 of 15

ij

ij

ji

Legendre transformation. We note that the generalized momenta pσ satisfy pσ 6= pσ . We study the correspondence between solutions of Euler–Lagrange and Hamilton equations. The regularity conditions are found (ensuring that the Hamilton extremals are holonomic up to the second order). These conditions depend on a choice of a Hamiltonian system (i.e., on a choice of “free” functions). We study the correspondence between the regularity conditions and the existence of the Legendre transformation. Contrary to the classical approach, the regularity conditions do not guarantee the existence of a generalized Legendre transformation. On the other hand, the generalized Legendre conditions do not guarantee regularity. The existence of a generalized Legendre transformation guarantees that the Hamilton extremals are holonomic up to the first order. The regularization procedure and properties of the Legendre transformation are illustrated in three examples. We consider three different Hamiltonian systems for a given Lagrangian. The first system is regular and possesses a generalized Legendre transformation. The second Hamiltonian system is regular and a generalized Legendre transformation does not exist. The last one is not regular but a generalized transformation exists. Throughout the paper, all manifolds and mappings are smooth and the summation convention is used. We consider a fibered manifold (i.e., surjective submersion) π : Y → X, dim X = n, dim Y = n + m. Its r-jet prolongation is πr : J r Y → X, r ≥ 1 and its canonical jet projections are πr,k : J r Y → J k Y, 0 ≤ k ≤ r (with the obvious notation J 0 Y = Y). A fibered chart on Y (respectively associated fibered chart on J r Y) is denoted by (V, ψ), ψ = ( xi , yσ ) (respectively (Vr , ψr ), ψr = ( xi , yσ , yiσ , . . . , yσi ...ir )). 1 A vector field ξ on J r Y is called πr -vertical (respectively πr,k -vertical) if it projects onto the zeroth vector field on X (respectively on J k Y). Recall that every q-form η on J r Y admits a unique (canonical) decomposition into a sum of q-forms r on J +1 Y as follows [7]: πr∗+1,r η = hη +

q

∑ pk η,

k =1

where hη is a horizontal form, called the horizontal part of η, and pk η, 1 ≤ k ≤ q, is a k-contact part of η. We use the following notations: ω0 = dx1 ∧ dx2 ∧ · · · ∧ dx n , ωi = i∂/∂xi ω0 , ωij = i∂/∂x j ωi , and ω σ = dyσ − yσj dx j , . . . , ωiσ1 i2 ...ik = dyiσ1 i2 ...ik − yiσ1 i2 ...ik j dx j . For more details on fibered manifolds and the corresponding geometric structures, we refer to sources such as [22]. 2. Lepagean Equivalents and Hamiltonian Systems In this section we briefly recall the basic concepts on Lepagean equivalents of Lagrangians according to Krupka [7,23], and on Lepagean equivalents of Euler–Lagrange forms and generalized Hamiltonian field theory according to Krupková [11,12]. By an r-th order Lagrangian we shall mean a horizontal n-form λ on J r Y. An n-form ρ is called a Lepagean equivalent of a Lagrangian λ if (up to a projection) hρ = λ and p1 dρ is a πr+1,0 -horizontal form.


3 of 15

For an r-th order Lagrangian we have all its Lepagean equivalents of order (2r − 1) characterized by the following formula ρ = Θ + µ,

(1)

where Θ is a (global) Poincaré–Cartan form associated to λ and µ is an arbitrary n-form of order of contactness ≥ 2, i.e., such that hµ = p1 µ = 0. Recall that for a Lagrangian of order 1, Θ = θλ where θλ is the classical Poincaré–Cartan form of λ. If r ≥ 2, Θ is no longer unique, however there is a non-invariant decomposition Θ = θλ + p1 dν,

(2)

where θλ = Lω0 +

r −1

r − k −1

k =0

l =0

∑

∑

∂L

(−1)l d p1 d p2 . . . d pl

!

∂yσj ...j 1

k p1 ...pl i

ω σj1 ...jk ∧ ωi ,

(3)

and ν is an arbitrary at least 1-contact (n − 1)-form (see [7,23]). A closed (n + 1)-form α is called a Lepagean equivalent of an Euler–Lagrange form E = Eσ ω σ ∧ ω0 if p1 α = E. Recall that the Euler–Lagrange form corresponding to an r-th order λ = Lω0 is the following (n + 1)-form of order ≤ 2r: E = Eσ ω ∧ ω0 = σ

r ∂L ∂L − ∑ (−1)l d p1 d p2 . . . d pl σ σ ∂y ∂y p1 ...pl l =1

! ω σ ∧ ω0 .

By definition of a Lepagean equivalent of E, one can find Poincaré lemma local forms ρ such that α = dρ, where ρ is a Lepagean equivalent of a Lagrangian for E. The family of Lepagean equivalents of E is also called a Lagrangian system and denoted by [α]. The corresponding Euler–Lagrange equations now take the form J s γ∗ i J s ξ α = 0 for every π −vertical vector field ξ on Y,

(4)

where α is any representative of order s of the class [α]. A (single) Lepagean equivalent α of E on J s Y is also called a Hamiltonian system of order s and the equations δ∗ iξ α = 0 for every πs −vertical vector field ξ on Js Y

(5)

are called Hamilton equations. They represent equations for integral sections δ (called Hamilton extremals) of the Hamilton ideal, generated by the system Dαs of n-forms iξ α, where ξ runs over πs -vertical vector fields on J s Y. Also, considering πs+1 -vertical vector fields on J s+1 Y, one has the ideal Dαsˆ +1 of n-forms iξ αˆ on J s+1 Y, where αˆ (called principal part of α) denotes the at most 2-contact part of α. Its integral sections, which annihilate all at least 2-contact forms, are called Dedecker–Hamilton extremals. It holds that if γ is an extremal then its s-prolongation (respectively (s + 1)-prolongation) is a Hamilton (respectively Dedecker–Hamilton) extremal, and (up to projection) every Dedecker–Hamilton extremal is a Hamilton extremal (see [11,12]). Denote by r0 the minimal order of Lagrangians corresponding to E. A Hamiltonian system α on J s Y, s ≥ 1, associated with E is called regular if the system of local generators of Dαsˆ +1 contains all the n-forms ω σ ∧ ωi , ω(σj ∧ ωi) , . . . , ω(σj 1

1 ...jr0 −1

∧ ωi ) ,

(6)


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where (. . . ) denotes symmetrization in the indicated indices. If α is regular then every Dedecker–Hamilton extremal is holonomic up to the order r0 , and its projection is an extremal. (In the case of first order Hamiltonian systems, there is a bijection between extremals and Dedecker–Hamilton extremals). α is called strongly regular if the above correspondence holds between extremals and Hamilton extremals. It can be proved that every strongly regular Hamiltonian system is regular, and it is clear that if α is regular and such that α = αˆ then it is strongly regular. A Lagrangian system is called regular (respectivelystrongly regular) if it has a regular (respectively strongly regular) associated Hamiltonian system [11]. 3. Regular and Strongly Regular 3rd Order Hamiltonian Systems In this section we discuss a part of variational theory which is singular in the standard sense. In general, a second order Lagrangian gives rise to an Euler–Lagrange form on J 4 Y. We shall consider second order Lagrangians λ that satisfy one of the following conditions: (1)

The corresponding Euler–Lagrange form is of order 3, i.e., the Lagrangians satisfy the conditions ∂2 L ∂yijσ ∂yνkl

(2)

!

= 0,

(7)

Sym(ijkl )

where Sym(ijkl ) means symmetrization in the indicated indices. The Euler–Lagrange expressions Eσ (4) of λ are second order and “non-affine” in the second derivatives ∂2 Eσ 6= 0 ∂yνkl ∂yκij

(8)

for some indices i, j, k, l, σ, ν, κ. In what follows, we shall study Hamiltonian systems corresponding to a special choice of a Lepagean equivalent of such Lagrangians, namely α of order 3 and α = dρ, where ρ

=

∂L ∂L − dk σ σ ∂y j ∂y jk

Lω0 + ij

! ωσ ∧ ωj +

∂L σ ω ∧ ω j + µ¯ ∂yijσ i

(9)

kij

+ aσν ω σ ∧ ω ν ∧ ωij + bσν ω σ ∧ ωkν ∧ ωij klij

ν + cσν ω σ ∧ ωkl ∧ ωij , ij

kij

klij

with an arbitrary at least 3-contact n-form µ¯ and functions aσν , bσν , cσν dependent on variables x k , yκ , yκk , yκkl and satisfying the conditions ij

ji

ij

lkij cσν ,

klij cσν

ij

kij

kji

aσν = − aσν , aσν = − aνσ ; bσν = − bσν ; klij cσν

=

=−

(10)

klji cσν .

Theorem 1. Ref. [18] Let dim X ≥ 2. Let λ = Lω0 be a second order Lagrangian with the Euler–Lagrange form (7) or (8), and α = dρ with ρ of the form (9), (10), be its Lepagean equivalent. Assume that the matrix ijkl Pσν

=

∂2 L klij + 2 cνσ ∂yijν ∂yσkl

! , Sym( jkl )

(11)


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with mn3 rows (respectively mn columns) labelled by σjkl (respectively νi) has maximal rank equal to mn and the matrix ! ∂2 L klij ijkl − 2cσν , Qσν = (12) ∂yijσ ∂yνkl with mn2 rows (respectively mn2 columns) labelled by σij (respectively νkl) has maximal rank equal to mn (n + 1) /2. Then the Hamiltonian system α = dρ is regular (i.e. every Dedecker–Hamilton extremal is of the form π3,2 ◦ δD = J 2 γ, where γ is an extremal of λ). Moreover, if µ¯ is closed then the Hamiltonian system α = dρ is strongly regular (i.e., every Hamilton extremal is of the form π3,2 ◦ δ = J 2 γ, where γ is an extremal of λ). Proof. Explicit computation α = dρ gives: ∗ π4,3 α

= Eσ ω ∧ ω0 + σ

+ + −

+ +

+

+

∂2 L ∂ ∂L ij − ν d j σ − 2d j aσν σ ν ∂yi ∂y ∂y ∂yij

! ω ν ∧ ω σ ∧ ωi

! ∂2 L ∂ ∂L ∂2 L kij ν σ − σ ν − ν d j σ + 4aik νσ − 2d j bσν ωk ∧ ω ∧ ωi ∂yiσ ∂yνk ∂y ∂yik ∂yk ∂yij ! ∂ ∂L ∂2 L klij kil ν − ν d j σ − 2(bσν )Sym(kl ) − 2d j cσν ωkl ∧ ω σ ∧ ωi ∂yiσ ∂yνkl ∂ykl ∂yij ! ∂2 L klij ω νjkl ∧ ω σ ∧ ωi + 2cσν ∂yijσ ∂yνkl Sym( jkl ) ! 2 ∂ L kij − 4(bσν ) Alt((σj)(νk)) ωkν ∧ ω σj ∧ ωi ∂yijσ ∂yνk ! ! ij ∂2 L ∂aσν klij ν σ − 2cσν ωkl ∧ ω j ∧ ωi + ∂yijσ ∂yνkl ∂yκ Alt(κσν) ! pij ij ∂b ∂aσν + νκσ ω κ ∧ ω σ ∧ ω ν ∧ ωij + ω κp ∧ ω σ ∧ ω ν ∧ ωij ∂yκp ∂y Alt(σν)   ! ! ij pqij ∂cνκ  ∂aσν  ω κpq ∧ ω σ ∧ ω ν ∧ ωij + κ ∂y pq ∂yσpq Sym( pq) Alt(σν) ! ! qij kij pqij ∂bσν ∂bσν ∂cσκ σ ν κ ω ∧ ωq ∧ ω p ∧ ωij + − ∂yκp ∂yκpq ∂yνk Alt((κ p)(νq)) Sym( pq) ! klij ∂cσν ν ¯ ∧ ωij + dµ, ω σ ∧ ωkν ∧ ω κpq ∧ ωij − ω σ ∧ ω κpq ∧ ωkl ∂yκpq

(13)

Alt((κ pq)(νkl ))

where Alt((. . . ) . . . (. . . )) means alternation in the indicated multi-indices and Sym(. . . ) means symmetrization in the indicated indices.


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In the notation of Equations (11) and (12), the principal part of α (13) takes the form ∂ ∂L ∂2 L ij − ν d j σ − 2d j aσν σ ν ∂yi ∂y ∂y ∂yij

= Eσ ω ∧ ω0 + σ

αˆ

+ + + −

! ω ν ∧ ω σ ∧ ωi

! ∂2 L ∂2 L ∂ ∂L kij ν σ − σ ν − ν d j σ + 4aik νσ − 2d j bσν ωk ∧ ω ∧ ωi ∂yiσ ∂yνk ∂y ∂yik ∂yk ∂yij ! ∂ ∂L ∂2 L klij kil ν − ν d j σ − 2(bσν )Sym(kl ) − 2d j cσν ωkl ∧ ω σ ∧ ωi ∂yiσ ∂yνkl ∂ykl ∂yij ! ∂2 L kij − 4(bσν ) Alt((σj)(νk)) ωkν ∧ ω σj ∧ ωi ∂yijσ ∂yνk ijkl

(14)

ijkl

ν Pνσ ω νjkl ∧ ω σ ∧ ωi + Qσν ωkl ∧ ω σj ∧ ωi ,

Expressing the generators of the ideal Dα4ˆ , we obtain i

∂ ∂yν

αˆ

= Eν ω0 + 2 − − +

i

∂ ∂yν k

αˆ

= +

i

∂ ∂yν kl

αˆ

∂2 L ∂ ∂L ij − ν d j σ − 2d j aσν ∂yiσ ∂yν ∂y ∂yij

=

! ω σ ∧ ωi

! ∂2 L ∂ ∂L ∂2 L kij σ − ν σ − σ d j ν + 4aik σν − 2d j bνσ ωk ∧ ωi ∂yiν ∂yσk ∂y ∂yik ∂yk ∂yij ! ∂2 L ∂ ∂L klij σ kil ∧ ωi − σ d j ν − 2(bνσ )Sym(kl ) − 2d j cνσ ωkl ∂yiν ∂yσkl ∂ykl ∂yij ijkl

Pσν ω σjkl ∧ ωi , ! ∂2 L ∂2 L ∂ ∂L kij σ − σ ν − ν d j σ + 4aik νσ − 2d j bσν ω ∧ ωi ∂yiσ ∂yνk ∂y ∂yik ∂yk ∂yij ! ∂2 L ikjl kij 2 − 4(bσν ) Alt((σj)(νk)) ω σj ∧ ωi + Qνσ ω σjl ∧ ωi , ∂yijσ ∂yνk ! ∂2 L ∂ ∂L klij kil − ν d j σ − 2(bσν )Sym(kl ) − 2d j cσν ω σ ∧ ωi ∂yiσ ∂yνkl ∂ykl ∂yij

(15)

ijkl

+ Qσν ω σj ∧ ωi , i

∂ ∂yν jkl

αˆ

ijkl

= − Pσν ω ν ∧ ωi ijkl

ijkl

Since the ranks of the matrices Pνσ , Qσν are maximal then the ω σ ∧ ωi and ω(σj ∧ ωi) are generators of the ideal Dα4ˆ . For Dedecker–Hamilton extremals, we obtain δD π3,2 ◦ δD = J 2 γ, where γ is a section of π. Substituting this into Equation (5), we get δD∗ i

∂ ∂yσ

αˆ = Eσ ◦ J 3 γ

for the 3rd order Euler–Lagrange form (7) and δD∗ i

∂ ∂yσ

αˆ = Eσ ◦ J 2 γ

for the 2nd order Euler–Lagrange form (8) and γ is an extremal of λ. Let us prove strong regularity. We have to show that under our assumptions, for every section δ satisfying the Hamilton equations, π3,2 ◦ δ = J 2 γ, where γ is a solution of the Euler–Lagrange equations


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ijkl

of the Lagrangian λ. Assuming dµ¯ = 0, we obtain δ∗ (i∂/∂yσ α) = δ∗ ( Pσν ω ν ∧ ωi ) = 0, i.e., δ∗ ω ν = 0 jkl ijkl ijkl by the rank condition on Pσν , i.e., ∂(yσ ◦ δ)/∂xi = yiσ ◦ δ. Hence, δ∗ (i∂/∂yν α) = δ∗ Qσν ω σj ∧ ωi = 0. kl

the

ijkl Note that the matrix Qσν is symmetric inindices kl and its maximal rank is mn(n + 1)/2. Due to ijkl rank condition on Qσν , δ∗ ω σj = 0, i.e., ∂(yσj ◦ δ)/∂xi = yijσ ◦ δ. The conditions for δ Sym(ij)

obtained above mean that every solution of Hamilton equations is holonomic up to the second order, i.e., we can write π3,2 ◦ δ = J 2 γ, where γ is a section of π. Now, the equations J 3 (π3,0 ◦ δ)∗ (i∂/∂yσ α) = 0 k

are satisfied identically and the last set of Hamilton equations—J 3 (π3,0 ◦ δ)∗ (i∂/∂yσ α) = 0—take the form Eσ ◦ J 3 γ = 0 (7) or Eσ ◦ J 2 γ = 0 (8), proving that γ is an extremal of λ. In the next propositon we study a weaker condition which the Hamilton extremals satisfy. Theorem 2. Let dim X ≥ 2. Let λ = Lω0 be a second order Lagrangian with the Euler–Lagrange form (7) or (8), and α = dρ with ρ of the form (9) and (10) be its Lepagean equivalent. Assume that µ¯ is closed and the matrix ! ∂2 L klij ijkl + 2 cνσ , (16) Pσν = ∂yijν ∂yσkl Sym( jkl )

with mn3 rows (respectively mn columns) labelled by σ, j, k, l (respectively νi) has rank mn . Then every Hamilton extremal δ : π (U ) ⊂ V → J 2 Y of the Hamiltonian system α = dρ is of the form ∂yσ π3,1 ◦ δ = J 1 γ (i.e., ∂xi = yiσ ), where γ is an extremal of λ. Proof. The assertion of Theorem 2 follows from the proof of Theorem 1. 4. Legendre Transformation In this section the Hamiltonian systems admitting Legendre transformation are studied. By the Legendre transformation we understand the coordinate transformation onto J 3 Y. Writing the Lepagean equivalent ρ (9), (10) in the form of a noninvariant decomposition, we get ρ

j

ij

klij

= − Hω0 + pσ dyσ ∧ ω j + pσ dyiσ ∧ ω j + 2cσν yσj dyνkl ∧ ωi +

ij aσν dyσ

kij ∧ dyν ∧ ωij + bσν dyσ

(17)

∧ dyνk ∧ ωij

klij

¯ + cσν dyσ ∧ dyνkl ∧ ωij + µ, where H

= −L +

∂L ∂L − dj σ σ ∂yi ∂yij

! yiσ +

kij

∂L σ ij y − 2aσν yiσ yνj ∂yijσ ij

klij

− 2(bσν )Sym(ki) yiσ yνkj − 2(cσν )Sym(klj) yiσ yνklj , pσ

j

=

∂L ∂L ij kij klij − di σ + 4aσν yiν + 2(bσν )Sym(ki) yνki + 2(cσν )Sym(kli) yνkli , ∂yσj ∂yij

ij

=

∂L ijk + 2bνσ yνk . ∂yijσ

pσ

(18)

Moreover, if the matrix  

∂piσ ∂yνkl

∂piσ ∂yνklm

∂pσ ∂yνkl

∂pσ ∂yνklm

ij

ij

 

(19)


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has maximal rank, then ij

( xi , yσ , yiσ , piσ , pσ ) is part of coordinate system. ij We note that the functions pσ do not depend on the variables yνklm . Then the submatrix of the Jacobi matrix of the transformation takes the form  

∂piσ ∂yνkl

∂piσ ∂yνklm

∂pσ ∂yνkl

0

ij

 .

(20)

ij The above matrix has maximal rank if and only if the matrices ∂piσ /∂yνklm and ∂pσ /∂yνkl have maximal ranks. Explicit computations lead to

∂piσ ∂yνklm

ij

!

∂pσ ∂yνkl

=

∂2 L klim ν ∂yσ + 2 cνσ ∂yim kl

,

(21)

Sym(klm)

ijq

∂b ∂2 L + 2 κσ yκ . ∂yijσ ∂yνkl ∂yνkl q

=

ijkl T Note that in the notation of Equation (11), Pσν = ∂piσ /∂yνjkl and the maximal rank is equal ij to mn. The matrix ∂pσ /∂yνkl is symmetric in the indices kl and therefore the maximal rank of the ij

matrix is equal to mn (n + 1) /2, i.e., the number of independent pσ is mn (n + 1) /2. Contrary to the ij situation in Hamilton–De Donder theory, the functions pσ are not symmetric in the indices ij. If we suppose that the matrix (19) has maximal rank, then ij

ψ3 = ( x k , yν , yνk , yνkl , yνklm ) → ( xi , yσ , yiσ , piσ , pσ , z B ) = χ

(22)

is a coordinate transformation over an open set U ⊂ V2 , where z B , 1 ≤ B ≤ mn(n2 + 3n − 1)/6 are arbitrary coordinate functions. We call it a generalized Legendre transformation and χ (22) the ij generalized Legendre coordinates. Accordingly, H, piσ , pσ are called generalized Hamiltonian and generalized momenta, respectively. Writing the Lepagean equivalent ρ (9) and (10) in the generalized Legendre transformation, we get ρ

j

ij

= − Hω0 + pσ dyσ ∧ ω j + pσ dyiσ ∧ ω j +

klij 2cσν yσj

∂yνkl

q q dp ∂p β β

+

∂yνkl

qr qr dp β ∂p β

ij

∂yν B + kl dz ∂z B

!

∧ ωi

kij

+ aσν dyσ ∧ dyν ∧ ωij + bσν dyσ ∧ dyνk ∧ ωij klij

+ cσν dyσ ∧

∂yνkl q ∂p β ij

q

dp β +

∂yνkl

qr qr dp β ∂p β

+

∂yνkl B dz ∂z B

(23) ! ¯ ∧ ωij + µ,

where yνkl are functions of variables piσ , pσ , z B . The Hamilton Equation (5) in these generalized Legendre coordinates take a rather complicated form, see Appendix A.


9 of 15

An interesting case. However, if dη = 0, where η

klij

ij

kij

= 2cσν yσj dyνkl ∧ ωi + aσν dyσ ∧ dyν ∧ ωij + bσν dyσ ∧ dyνk ∧ ωij klij

(24)

klij

+ cσν dyσ ∧ dyνkl ∧ ωij + dσν dyσk ∧ dyνl ∧ ωij then the Hamilton Equation (5) have the following form j qj ∂yκq ∂H ∂H ∂pκ ∂H ∂pκ ∂H ∂yκ ∂H = − , , , = 0. = − = , = q qr ∂yκ ∂x q ∂pκ ∂xr ∂z M ∂x j ∂yκq ∂x j ∂pκ

Contrary to the Hamilton–De Donder theory, the regularity conditions of the Lepagean form (9), (10) and regularity of the generalized Legendre transformation (21) do not coincide. The regularity conditions do not guarantee the existence of the Legendre transformation. On the other hand, the existence of the Legendre transformation does not guarantee the regularity. But we can see that the existence of a Legendre transformation (22) guarantees a weaker relation: π3,1 ◦ δ = J 1 γ, where γ is an extremal of λ. Theorem 3. Let dim X ≥ 2. Let λ = Lω0 be a second order Lagrangian with the Euler–Lagrange form (7) or (8), and α = dρ with ρ of the form (9), and Equation (10) be the expression of its Lepagean equivalent in a fiber chart (V, ψ), ψ = ( xi , yσ ). Suppose that µ¯ is closed and ρ admits Legendre transformation (22) defined by Equation (18). Then π3,1 ◦ δ = J 1 γ, where γ is an extremal of λ. Proof. The form ρ admits Legendre transformation, so the matrix ∂piσ ∂yνjkl

!

=

∂2 L klij + 2 cνσ ν σ ∂yij ∂ykl

has maximal rank equal to mn. In the notation of (11),

! Sym( jkl )

ijkl T

Pσν

=

∂piσ /∂yνjkl . Acordingly,

from Proposition 2, we obtain π3,1 ◦ δ = J 1 γ, where γ is an extremal of λ. 5. Examples The above results (the regularity conditions and the Legendre transformation) can be directly applied to concrete Lagrangians. Let us consider the following examples as an illustration. For a given Lagrangian, we find three different Hamiltonian systems satisfying: (a) (b) (c)

R2

The Hamiltonian system is strongly regular and the Legendre transformation exists. (See examples of strongly regular systems in [17]). The Hamiltonian system is strongly regular and the Legendre transformation does not exist. The Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition. Let X = R2 , Y = R2 × R2 (i.e., n = 2, m = 2). Denote (V, ψ), ψ = ( xi , yσ ) a fibered chart on × R2 . Let us consider the following Lagrangian λ = Lω0 , L = y111 y222 − y122 y211

which satisfies (7).

(25)


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5.1. Example (a) View of the above considerations, we take a Lepagean equivalent ρ (of the Euler–Lagrange form E of Lagrangian (25)) in the form α = dρ, where ρ is (9), (10). ij kij ijkl We consider functions aσν , bσν , cσν (see Equation (10)) on an open set U ⊂ J 3 R2 with the 1 1 1 conditions y1 6= 0, y2 6= 0, y12 6= 0 and y212 6= 0. ij

ijp

ijp

ijpkl

The functions aσν are arbitrary. The functions bκσ are linear in variables yνkl . We denote dκσν = ijpkl

∂bκσ /∂yνkl . Suppose that dκσν are constant functions, then we have only eight non-zero constants and we put d12112 = d12121 = −d11212 = −d11221 = 1 and d22112 = d22121 = −d21212 = −d21221 = 1. 112 112 112 112 121 121 121 121 ijkl

Similarly, we assume that cσν are constant functions. We have again only eight non-zero constants, 2112 2121 1221 1212 2112 2121 1221 and we choose c1212 11 = c11 = − c11 = − c11 = 1 and c22 = c22 = − c22 = − c22 = 1. Then the Lepagean equivalent takes the form ij

θλ + aσν ω σ ∧ ω ν ∧ ωij − 4y112 ω 2 ∧ ω21 ∧ ω12 − 4y212 ω 1 ∧ ω12 ∧ ω12

ρ1 =

1 2 ¯ 4 ω 1 ∧ ω12 ∧ ω12 + 4 ω 2 ∧ ω12 ∧ ω12 + µ,

+

where µ¯ is an arbitrary closed n-form. The matrices (11), (12), and (21) take the following form  ijkl

( Pσν )T =

1   3

0 0 0 0

0 −4 0 1

0 −4 0 1

0 −4 0 1

4 0 −1 0

4 0 −1 0

4 0 0 0 0 0 −1 0 0 0 0 0

0 −1 0 −4

0 −1 0 −4

0 1 1 1 0 −1 0 0 0 0 0 4 4 4 0 −4 0 0 0 0

   , 

and 

ijkl

Qσν

      =      

0 0 0 0 0 0 0 1

0 −2 2 0 0 0 0 0

0 0 0 0 0 0 0 1

0 0 0 0 0 0 −y12 y11

0 −2 2 0 0 0 0 0

0 0 0 0 −1 0 0 0

0 0 0 −1 0 0 0 0

0 0 0 0 0 −2 2 0

0 0 0 0 0 −2 2 0

1 0 0 0 0 0 0 0

       ,      

and     !   ij  ∂pσ = ν  ∂ykl     

ijkl

0 0 0 0 0 0 −y12 y11

0 0 0 0 −1 0 0 0

0 0 0 −1 0 0 0 0

−y12 −y12 y11 y11 0 0 0 0 0 0 0 0 0 0 0 0 ijkl

1 0 0 0 0 0 0 0

       ,      

We see that rank( Pσν ) = 4 and rank( Qσν ) = 6. Since y11 6= 0 and y12 6= 0 can easily ij rank ∂pσ /∂yνkl = 6. The form α = dρ is strongly regular and a generalized Legendre transformation exists.


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The generalized Hamiltonian and momenta (18) take the form H

= −y112 y222 + y122 y211 − y11 (8y1122 + y2122 ) + y12 (8y1112 + y2122 ) 1 2 1 2 − y21 (8y2122 + y1122 ) + y22 (8y2112 + y1122 ) − 4a12 12 ( y1 y2 − y2 y1 ),

p11 p21

2 1 1 2 12 1 = y2122 − 8y1122 + 4a12 12 y2 , p2 = − y122 + 8y122 − 4a12 y2 ,

−y2112

=

− 8y1112

2 − 4a12 12 y1 ,

p22

=

y1112

+ 8y2112

(26)

1 + 4a12 12 y1 ,

p11 1

1 2 22 2 = y222 − 4y12 y212 , p12 1 = 4y1 y12 , p1 = − y11 ,

p22 2

1 1 22 1 = y111 + 4y11 y112 , p21 2 = −4y2 y12 , p1 = − y22 . ij

12 We have only six independent generalized momenta pσ . We note that p21 1 = p2 = 0.

5.2. Example (b) For the given Lagrangian (25), we consider another Hamiltonian system on an open set U ⊂ J 3 R2 ρ2 =

+ ij

ij

kij

θλ + aσν ω σ ∧ ω ν ∧ ωij + bσν ω σ ∧ ωkν ∧ ωij 1 2 ¯ ∧ ω12 + µ, 4 ω 1 ∧ ω12 ∧ ω12 + 4 ω 2 ∧ ω12

kij

where aσν , bσν are arbitrary constant functions satisfying Equation (10) and µ¯ is an arbitrary closed n-form. We can easily see that matrices (11) and (12) have the same form as in Example (a), i.e., the Hamiltonian system is strongly regular. The matrix (21) takes the form      !  ij  ∂pσ = ν  ∂ykl     

0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 −1 0 0 0

0 0 0 −1 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

1 0 0 0 0 0 0 0

       ,      

ij and rank ∂pσ /∂yνkl = 4. Therefore the generalized Legendre transformation does not exist. 5.3. Example (c) On an open set U ⊂ J 3 R2 where y11 6= 0, y12 6= 0, y112 6= 0 and y212 6= 0, the Lepagean equivalent takes the form ρ3 =

+

ij

θλ + aσν ω σ ∧ ω ν ∧ ωij − 4y112 ω 2 ∧ ω21 ∧ ω12 − 4y212 ω 1 ∧ ω12 ∧ ω12 1 ¯ 4 ω 1 ∧ ω12 ∧ ω12 + µ, ij

where µ¯ is an arbitrary closed n-form and aσν are arbitrary functions satisfying Equation (10). ij

It is easy to see that rank ∂pσ /∂yνkl = 6 and the matrix has the same form as in Example (a).


12 of 15

The matrices (11) and (12) take take the form  0 0 0 0 4 4 4 0 0  0 −4 −4 −4 1 0 0 0 0 0 ijkl  ( Pσν )T =  3 0 0 0 0 −1 −1 −1 0 0 0 1 1 1 0 0 0 0 0



ijkl

Qσν

ijkl

      =      

0 0 0 0 0 0 0 1

0 −2 2 0 0 0 0 0

0 −2 2 0 0 0 0 0

0 0 0 0 −1 0 0 0

0 0 0 −1 0 0 0 0

0 −1 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 −1 0 0

1 0 0 0 0 0 0 0

 0 1 1 1 0 −1 0 0 0 0   , 0 0 0 0 0  0 0 0 0 0

       ,      

ijkl

and rank( Pσν ) = 4 and rank( Qσν ) = 5. The Hamiltonian system is not regular but it is holonomic up to first order and the generalized Legendre transformation exists (see Theorem 3). 6. Conclusions This paper presents a generalization of classical Hamiltonian field theory on a fibered manifold. The regularization procedure of the first order Lagrangians proposed by Krupkova and Smetanová is applied to the case of a third order Hamiltonian system satisfying the conditions (7) or (8). Hamilton equations are created from the Lepagean equivalent whose order of contactness is more than 2-contact (contrary to the Hamilton p2-equations in [16]). The generalized Legendre transformation ij ji was studied and the generalized momenta pσ 6= pσ were found. The theory was illustrated using examples of Hamilton systems satisfying: (a) (b) (c)

The Hamiltonian system is strongly regular and the Legendre transformation exists. The Hamiltonian system is strongly regular and the Legendre transformation does not exist. The Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition.

Contrary to the standard approach, where all afinne and many quadratic Lagrangians are singular, we show that these Lagrangians are regularizable, admit Legendre transformation, and provide Hamilton equations that are equivalent to the Euler–Lagrange equations (i.e., they do not contain constraints). Within this setting, a proper choice of a Lepagean equivalent can lead to a “regularization” of a Lagrangian. The method proposed in this article is appropriate for the regularization of 2nd order Lagrangians (e.g., scalar curvature Lagrangians). The proposed procedure is different from [6,13,15] since it does not change order of the Lepagean equivalent . ˇ Budˇejovice (project No. Funding: This research was funded by the Institute of Technology and Business in Ceské IGS201805—Innovation of mathematical part of study programs). Conflicts of Interest: The author declares no conflict of interest.

Appendix A Hamilton Equations (5) with dµ¯ = 0 (9) in Legendre coordinates take the following explicit form:


13 of 15

j

∂H ∂yκ

=

klij

κ − ∂p ∂x j

+ 2 ∂c∂yσνκ ij

yσj

ν

κν ∂y +4 ∂a +6 ∂x j ∂xi q

ij

q

∂yνkl ∂p β q ∂p β ∂xi ij

∂aσν ∂yκ

+

qr

∂pσ σ ∂y +4 ∂aκνq ∂p + 4 ∂aκν qr ∂pσ ∂xi ∂x j ∂pσ ∂xi ij kij ν ∂bσν κν ∂yk +2 ∂b + 4 κ ∂y ∂x j ∂xi

∂yν ∂x j

Alt(κσ)

+2 +2

kij ∂bκν ∂yσq

kij ∂bκν qr ∂pσ klij

+2 ∂c∂xκνj

+4

klji

klji

+2 ∂cκνq

∂pσ

Alt((νk )(σq))

∂yνk ∂pqr σ ∂xi ∂x j

klij

∂cσν ∂yκ

+2 ∂c∂yκνσ q

q

∂yνkl ∂p β q ∂p β ∂xi

Alt(κσ)

∂yσq ∂xi

q

∂pσ ∂xi q

qr

klji σ +2 ∂cκνqr ∂p ∂pσ ∂xi klji

∂H ∂yκq

qj

=

∂z M ∂xi

∂yσ ∂xi

q

∂yνkl ∂p β q ∂p β ∂x j

q kij ∂yν k ∂pσ ∂pσ ∂xi ∂x j

+

∂yνkl ∂z B ∂z B ∂xi

q


+

∂yνkl ∂p β qr ∂p β ∂x j

+


+


+


qr

+

∂yνkl ∂z B ∂z B ∂x j ∂yνkl ∂z B ∂z B ∂x j

+

+

∂yνkl ∂z B ∂z B ∂x j

+

∂yνkl ∂z B ∂z B ∂x j ∂yνkl ∂z B ∂z B ∂x j

+

qr

qr

q

qr

∂yνkl ∂p β ∂yνkl ∂p β ∂yν + qr + kl q i i ∂z B ∂p β ∂x ∂p β ∂x ! qij qij ∂bσκ ∂yσ ∂bσκ ∂yν ∂yσ +2 j +2 +4 ν i ∂y ∂xi ∂x j ∂x ∂x Alt(νσ)

(A1)


qr

∂p kliq − κj + 2cκν ∂x

+

qr

q


∂yν ∂x j

+ 2 ∂bκνq

+

q


M

∂yσq ∂yν ∂xi ∂x j

∂yσ ∂yνk ∂xi ∂x j

qr

q


ij


q


q

∂yν ∂z M ∂xi ∂x j

∂yνkl ∂p β qr ∂p β ∂xi

+

ij

κν + 4 ∂a ∂yσ

κν ∂z + 4 ∂a ∂z M ∂xi

qr

klji σ +2 ∂cκνq ∂p ∂pσ ∂xi

κν +2 ∂c ∂z M

kij

κν + 2 ∂b ∂z M

∂yνk ∂yσq ∂xi ∂x j

∂yνkl ∂z B ∂z B ∂xi

+

∂yσ ∂yν ∂xi ∂x j

Alt(κνσ )

ij

ν

qr

∂yνkl ∂p β qr ∂p β ∂xi

∂z B ∂xi

+

∂yνkl ∂z B ∂z B ∂x j

ij

!

kij

∂bσν ∂yκq

+2

∂aσν ∂yσ ∂yν ∂yκq ∂xi ∂x j

!

qij qij qij σ ∂b ∂pkν ∂yσ ∂bσκ ∂pkl ∂bσκ ∂z M ∂yσ ν ∂y + 2 + 2 +2 σκk i j ∂z M ∂xi ∂x j ∂pν ∂xi ∂x j ∂pkl ν ∂x ∂x ! q qr klji ∂yνkl ∂p β ∂yνkl ∂z B ∂cσν ∂yσ ∂yνkl ∂p β +2 κ + qr +2 B ∂yq ∂xi ∂pqβ ∂x j ∂z ∂x j ∂p β ∂x j

Alt((κq)(νk ))

∂yσ ∂yνk ∂xi ∂x j (A2)


∂H q ∂pκ

=

14 of 15

klji klji klji β ∂cσν ∂yνkl ∂yr σ ∂yκ ∂cσν ∂yνkl σ ∂cσν ∂yνkl ∂y β σ y + 2 y + 2 y + 2 q β i j ∂x q ∂xi ∂pκq j ∂y β ∂pκq ∂xi j ∂yr ∂pκ ∂x r klji klji ∂cσν σ ∂cσν ∂yνkl ∂p β σ y + + 2 +2 β q q yj j i ∂pκ ∂yr ∂pκ ∂x

q

qr

∂yνkl ∂p β ∂yνkl ∂z B ∂yνkl ∂p β + + q qr ∂z B ∂xi ∂p β ∂xi ∂p β ∂xi

rs klji klji ν ∂cσν ∂yνkl ∂z M σ ∂cσν ∂yνkl ∂p β σ klji ∂ykl y + 2 y + 2c +2 rs σν q q q j j ∂p β ∂pκ ∂xi ∂z M ∂pκ ∂xi ∂pκ ! ij kij klij ∂aσν ∂yσ ∂yν ∂bσν ∂yσ ∂yνk ∂cσν +2 q +2 q +2 ∂y β ∂pκ ∂xi ∂x j ∂pκ ∂xi ∂x j

∂H qr ∂pκ

∂yσj ∂xi

∂y β ∂yσ ∂yνkl ∂xi ∂x j ∂pκq Alt( βν) ! q qr ∂yνkl ∂p β ∂yνkl ∂z B ∂yνkl ∂p β + qr + B q ∂z ∂x j ∂p β ∂x j ∂p β ∂x j

+2

klji klij ∂cσν ∂yνkl ∂yσ ∂cσν ∂yσ + 2 q q ∂x j ∂pκ ∂xi ∂pκ ∂xi

+2

r rs β klij klij klij ∂cσν ∂yνkl ∂p β ∂yσ ∂cσν ∂yνkl ∂p β ∂yσ ∂cσν ∂yνkl ∂yr ∂yσ + 2 + 2 q q β i j i j ∂prβ ∂pκq ∂xi ∂x j ∂prs ∂yr ∂pκ ∂x ∂x β ∂pκ ∂x ∂x

+2

klij ∂cσν ∂yνkl ∂z M ∂yσ ∂z M ∂pκq ∂xi ∂x j

=

∂yκq ∂xr

+2

!

klji klji klji ∂cσν ∂yνkl ∂y β σ ∂cσν ∂yνkl σ ∂cσν ∂yνkl ∂y β σ y + 2 y + 2 y qr β i j ∂xi ∂pκqr j ∂y β ∂pκqr ∂xi j ∂ys ∂pκ ∂x

s st klji klji ∂cσν ∂yνkl ∂p β σ ∂cσν ∂yνkl ∂p β σ +2 s y + 2 st y ∂p β ∂pκqr ∂xi j ∂p β ∂pκqr ∂xi j klij

∂c ∂yσ +2 σν qr ∂pκ ∂xi

s st ∂yνkl ∂p β ∂yνkl ∂p β ∂yν ∂z B + + kl s st i i ∂p β ∂x ∂z B ∂xi ∂p β ∂x

! yσj

klji ij ∂yσj ∂yν ∂cσν ∂yνkl ∂z M σ ∂aσν ∂yν ∂yνk klji kl y + 2c + 2 σν qr ∂z M ∂pκqr ∂xi j ∂xi ∂pκqr ∂pκ ∂xi ∂x j ! kij klji klij ∂bσν ∂yσ ∂yν ∂y β ∂yσ ∂yνkl ∂cσν ∂yσ ∂yνkl ∂cσν +2 qr i j + 2 + 2 ∂xi ∂x j ∂pκqr ∂x j ∂xi ∂pκqr ∂y β ∂pκ ∂x ∂x Alt( βσ)

+2

s st klij β klij klij ∂cσν ∂yνkl ∂ys ∂yσ ∂cσν ∂p β ∂yσ ∂yνkl ∂cσν ∂p β ∂yσ ∂yνkl + 2 + 2 qr qr β i j i j ∂psβ ∂xi ∂x j ∂pκqr ∂pst ∂ys ∂pκ ∂x ∂x β ∂x ∂x ∂pκ ! s st klij ∂yνkl ∂p β ∂yνkl ∂z B ∂cσν ∂yσ ∂yνkl ∂p β +2 qr + st + B ∂z ∂x j ∂p β ∂x j ∂pκ ∂xi ∂psβ ∂x j

+2

+2

∂H ∂z M

klij ∂cσν ∂yνkl ∂z M ∂yσ ∂z M ∂pκqr ∂xi ∂x j

= 2

klji klji klji ∂cσν ∂yνkl σ ∂cσν ∂yνkl ∂y β σ ∂cσν ∂yνkl ∂y β σ y + 2 y + 2 y M β i j ∂xi ∂z M j ∂y β ∂z M ∂xi j ∂ys ∂z ∂x

+2

s st klji klji ∂cσν ∂yνkl ∂p β σ ∂cσν ∂yνkl ∂p β σ y + 2 y ∂psβ ∂z M ∂xi j ∂z M ∂xi j ∂pst β klij

∂c +2 σν ∂z M

s st ∂yνkl ∂p β ∂yνkl ∂p β ∂yνkl ∂z B + + i ∂psβ ∂xi ∂z B ∂xi ∂pst β ∂x

! yσj


15 of 15

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13.

14. 15. 16.

17. 18.

19. 20. 21. 22. 23.

Betounes, D.E. Extension of the classical Cartan form. Phys. Rev. D 1984, 29, 599–606. [CrossRef] Cantrijn, F.; Ibort, A.; De León, M. On the geometry of multisymplectic manifolds. J. Aust. Math. Soc. 1999, 66, 303–330. [CrossRef] Dedecker, P. On the generalization of symplectic geometry to multiple integrals in the calculus of variations. In Lecture Notes in Math. 570; Springer: Berlin, Germany, 1977; pp. 395–456. De León, M.; Salgado, M.; Vilariño, S. Methods of Differential Geometry in Classical Field Theories, k-Symplectic and k-Cosymplectic Approaches; World Scientific: Singapore, 2016. Goldschmidt, H.; Sternberg, S. The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier Grenoble 1973, 23, 203–267. [CrossRef] Hoˇrava, P. On a covariant Hamilton-Jacobi framework for the Einstein-Maxwell theory. Class. Quantum Gravity 1991, 8, 2069–2084. [CrossRef] Krupka, D. Some geometric aspects of variational problems in fibered manifolds. Folia Fac. Sci. Nat. UJEP Brun. 1973, 14, 1–65. Prieto-Martínez, P.D.; Roman-Roy, N. A new multisymplectic unified formalism for second order classical field theories. J. Geom. Mech. 2015, 7, 203–253. [CrossRef] Sardanashvili, G.A. Handbook of Integrable Hamiltonian Systems; URSS: Moscow, Russia, 2015. Vey, D. Multisymplectic formulation of vielbein gravity: I. De Donder-Weyl formulation, Hamiltonian (n − 1)-forms. Class. Quantum Gravity 2015, 32, 095005. [CrossRef] Krupková, O. Hamiltonian field theory. J. Geom. Phys. 2002, 43, 93–132. [CrossRef] Krupková, O. Hamiltonian field theory revisited: A geometric approach to regularity. In Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, Debrecen, Hungary, 25–30 July 2000; University of Debrecen: Debrecen, Hungaria, 2001; pp. 187–207. Krupka, D.; Štˇepánková, O. On the Hamilton form in second order calculus of variations. In Geometry and Physics, Proceedings of the International Meeting, Florence, Italy, 12–15 October 1982; Modugno, M., Ed.; Pitagora Ed.: Bologna, Italy, 1983; pp. 85–101. Krupková, O.; Smetanová, D. Legendre transformation for regularizable Lagrangians in field theory. Lett. Math. Phys. 2001, 58, 189–204. [CrossRef] Rossi, O. The Lagrangian Order-Reduction Theorem in Field Theories. Commun. Math. Phys. 2018, 362, 107–128. [CrossRef] Smetanová, D. On Hamilton p2 -equations in second-order field theory. In Steps in Differential Geometry, Proceedings of the Colloquium on Differential Geometry, Debrecen, 25–30 July 2000; Kozma, L., Nagy, T.P., Tamassy, L., Eds.; University of Debrecen: Debrecen, Hungaria, 2001; pp. 329–341. Smetanová, D. The Regularization of the Second Order Lagrangians in Example. Acta Univ. Palacki. Olomuc. Fac. Rerum Nat. Math. 2016, 55, 157–165. Smetanová, D. The second order Lagrangians-regularity problem. In Proceedings of the 14th Conference on Applied Mathematics, APLIMAT, English, Slovak, 3–5 February 2015; STU: Bratislava, Slovakia, 2015; pp. 690–697. Swaczyna, M.; Volný, P. Uniform motions in central fields. J. Geom. Mech. 2017, 9, 91–130. [CrossRef] Swaczyna, M.; Volný, P. Geometric concept of isokinetic constraint for a system of particles. Miskolc Math. Notes 2013, 14, 697–704. Krupková, O.; Volný, P. Euler-Lagrange and Hamilton equations for non-holonomic systems in field theory. J. Phys. A Math. Gen. 2005, 40, 8715–8745. [CrossRef] Saunders, D.J. The Geometry of Jets Bundles; Cambridge University Press: Cambridge, UK, 1989. Krupka, D. Lepagean forms in higher order variational theory. In Modern Developments in Analytical Mechanics I: Geometrical Dynamics, Proceedings of the IUTAM-ISIM Symposia, Torino, Italy, 12–15 October 1982; Benenti, S., Francaviglia, M., Lichnerowitz, A., Eds.; Accad. delle Scienze di Torino: Torino, Italy, 1983; pp. 197–238. c 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access

article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).