Higher Order Hamiltonian Systems with Generalized Legendre Transformation

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.

1. Introduction

Hamiltonian theory on manifolds has been intensively studied since the 1970s (see e.g., [1,2,3,4,5,6,7,8,9,10]). 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 [3,6,13,14,15]. Some second order Lagrangians have also been discussed in [16,17,18].

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 Legendre transformation. We note that the generalized momenta _{pσij pσij} satisfy _pσij≠_{pσji pσij}≠_pσji. 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 π:Y→X , dimX=n dimX=n , dimY=n+m dimY=n+m . Its r-jet prolongation is _πr:^JrY→X _πr:^JrY→X , r≥1 r≥1 and its canonical jet projections are _πr,k:^JrY→^JkY _πr,k:^JrY→^JkY , 0≤k≤r 0≤k≤r (with the obvious notation ^J0Y=Y ^J0Y=Y ). A fibered chart on Y (respectively associated fibered chart on ^JrY ^JrY ) is denoted by (V,ψ) (V,ψ) , ψ=(^xi,^yσ) ψ=(^xi,^yσ) (respectively (_Vr,_ψr) (_Vr,_ψr) , _ψr=(^xi,^yσ,_yiσ,…,_yi1…irσ) _ψr=(^xi,^yσ,_yiσ,…,_yi1…irσ)).

A vector field ξ ξ on ^JrY ^JrY is called _{πr πr} -vertical (respectively _{πr,k πr,k} -vertical) if it projects onto the zeroth vector field on X (respectively on ^JkY ^JkY).

Recall that every q-form η η on ^JrY ^JrY admits a unique (canonical) decomposition into a sum of q-forms on ^Jr+1Y ^Jr+1Y as follows [7]:

_πr+1,r∗η=hη+∑k=1q_pkη,

where hη hη is a horizontal form, called the horizontal part of η η , and _pkη _pkη , 1≤k≤q, 1≤k≤q, is a k-contact part of η η.

We use the following notations:

_ω0=d^x1∧d^x2∧…∧d^xn,_ωi=_{i∂/∂^xi ω0},_ωij=_{i∂/∂^xj ωi},

and

^ωσ=d^yσ−_yjσd^xj,…,_{ωi1 i2…ikσ}=d_{yi1 i2…ikσ}−_{yi1 i2…ikjσ}d^xj.

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 λ on ^JrY ^JrY.

An n-form ρ ρ is called a Lepagean equivalent of a Lagrangian λ λ if (up to a projection) hρ=λ hρ=λ and _p1dρ _p1dρ is a _{πr+1,0 πr+1,0}-horizontal form.

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

ρ=Θ+μ,

where Θ Θ is a (global) Poincaré–Cartan form associated to λ λ and μ μ is an arbitrary n-form of order of contactness ≥2 ≥2 , i.e., such that hμ=_p1μ=0 hμ=_p1μ=0 . Recall that for a Lagrangian of order 1, Θ=_θλ Θ=_θλ where _{θλ θλ} is the classical Poincaré–Cartan form of λ λ . If r≥2 r≥2 , Θ Θis no longer unique, however there is a non-invariant decomposition

Θ=_θλ+_p1dν,

where

_θλ=L_ω0+∑k=0r−1∑l=0r−k−1^(−1)l _{dp1 dp2} …_dpl ∂L∂_{yj1…jk p1…pliσωj1…jkσ}∧_ωi,

and ν ν is an arbitrary at least 1-contact (n−1) (n−1) -form (see [7,23]).

A closed (n+1) (n+1) -form α α is called a Lepagean equivalent of an Euler–Lagrange form E=_Eσ ^ωσ∧_ω0 E=_Eσ ^ωσ∧_ω0 if _p1α=E _p1α=E.

Recall that the Euler–Lagrange form corresponding to an r-th order λ=L_ω0 λ=L_ω0 is the following (n+1) (n+1) -form of order ≤2r ≤2r:

E=_Eσ^ωσ∧_ω0=∂L∂^yσ−∑l=1r^(−1)l _{dp1 dp2} …_dpl ∂L∂_yp1…plσ^ωσ∧_ω0.

By definition of a Lepagean equivalent of E, one can find Poincaré lemma local forms ρ ρ such that α=dρ α=dρ , where ρ 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] [α]. The corresponding Euler–Lagrange equations now take the form

^Jsγ∗ _i^Jsξα=0foreveryπ-verticalvectorfieldξonY,

where α α is any representative of order s of the class [α] [α] . A (single) Lepagean equivalent α α of E on ^JsY ^JsYis also called a Hamiltonian system of order s and the equations

^δ∗ _iξα=0forevery_πs-verticalvectorfieldξon^JsY

are called Hamilton equations. They represent equations for integral sections δ δ (called Hamilton extremals) of the Hamilton ideal, generated by the system _{Dαs Dαs} of n-forms _iξα _iξα , where ξ ξ runs over _{πs πs} -vertical vector fields on ^JsY ^JsY . Also, considering _{πs+1 πs+1} -vertical vector fields on ^Js+1Y ^Js+1Y , one has the ideal _{Dα^s+1 Dα^s+1} of n-forms _iξα^ _iξα^ on ^Js+1Y ^Js+1Y , where α^ α^ (called principal part of α α ) denotes the at most 2-contact part of α α . Its integral sections, which annihilate all at least 2-contact forms, are called Dedecker–Hamilton extremals. It holds that if γ γ is an extremal then its s-prolongation (respectively (s+1) (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 r0} the minimal order of Lagrangians corresponding to E. A Hamiltonian system α α on ^JsY,s≥1 ^JsY,s≥1 , associated with E is called regular if the system of local generators of _{Dα^s+1 Dα^s+1}contains all the n-forms

^ωσ∧_ωi,_ω(j1σ∧_ωi),…,_{ω(j1…jr0−1σ}∧_ωi),

where (…) (…) denotes symmetrization in the indicated indices. If α α is regular then every Dedecker–Hamilton extremal is holonomic up to the order _{r0 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 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 α is regular and such that α=α^ α=α^ then it is strongly regular. A Lagrangian system is called regular (respectively strongly 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 ^J4Y ^J4Y . We shall consider second order Lagrangians λ λthat satisfy one of the following conditions:

(1) The corresponding Euler–Lagrange form is of order 3, i.e., the Lagrangians satisfy the conditions

_{^∂2L∂yijσ∂yklνSym(ijkl)}=0,

where Sym(ijkl) Sym(ijkl)means symmetrization in the indicated indices.

(2) The Euler–Lagrange expressions _{Eσ Eσ} (4) of λ λare second order and “non-affine” in the second derivatives

^∂2 _Eσ∂_yklν∂_yijκ≠0

for some indices i,j,k,l,σ,ν,κ 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 α of order 3 and α=dρ α=dρ, where

ρ=L_ω0+∂L∂_yjσ−_dk∂L∂_yjkσ^ωσ∧_ωj+∂L∂_yijσωiσ∧_ωj+μ¯+_aσνij ^ωσ∧^ων∧_ωij+_bσνkij ^ωσ∧_ωkν∧_ωij+_cσνklij ^ωσ∧_ωklν∧_ωij,

with an arbitrary at least 3-contact n-form μ¯ μ¯ and functions _{aσνij aσνij} , _{bσνkij bσνkij} , _{cσνklij cσνklij} dependent on variables ^{xk xk} , ^{yκ yκ} , _{ykκ ykκ} , _{yklκ yklκ}and satisfying the conditions

_aσνij=−_aσνji,_aσνij=−_aνσij;_bσνkij=−_bσνkji;_cσνklij=_cσνlkij,_cσνklij=−_cσνklji.

Theorem 1.

Ref. [18] Let dimX≥2 dimX≥2 . Let λ=L_ω0 λ=L_ω0 be a second order Lagrangian with the Euler–Lagrange form (7) or (8), and α=dρ α=dρ with ρ of the form (9), (10), be its Lepagean equivalent. Assume that the matrix

_Pσνijkl=_{^∂2L∂yijν∂yklσ+2cνσklijSym(jkl)},

with mⁿ³ mⁿ³ rows (respectively mn mn columns) labelled by σjkl σjkl (respectively νi νi ) has maximal rank equal to mn mn and the matrix

_Qσνijkl=^∂2L∂_yijσ∂_yklν−2_cσνklij,

with mⁿ² mⁿ² rows (respectively mⁿ² mⁿ² columns) labelled by σij σij (respectively νkl νkl ) has maximal rank equal to mnn+1/2 mnn+1/2 . Then the Hamiltonian system α=dρ α=dρ is regular (i.e. every Dedecker–Hamilton extremal is of the form _π3,2∘_δD=^J2γ _π3,2∘_δD=^J2γ , where γ is an extremal of λ).

Moreover, if μ¯ μ¯ is closed then the Hamiltonian system α=dρ α=dρ is strongly regular (i.e., every Hamilton extremal is of the form _π3,2∘δ=^J2γ _π3,2∘δ=^J2γ , where γ is an extremal of λ).

Proof.

Explicit computation α=dρ α=dρgives:

_π4,3∗α=_Eσ ^ωσ∧_ω0+^∂2L∂_yiσ∂^yν−∂∂^yν_dj∂L∂_yijσ−2_{dj aσνij}^ων∧^ωσ∧_ωi+^∂2L∂_yiσ∂_ykν−^∂2L∂^yσ∂_yikν−∂∂_ykνdj∂L∂_yijσ+4_aνσik−2_{dj bσνkijωkν}∧^ωσ∧_ωi+^∂2L∂_yiσ∂_yklν−∂∂_yklνdj∂L∂_yijσ−2_{(bσνkil)Sym(kl)}−2_{dj cσνklijωklν}∧^ωσ∧_ωi−_{^∂2L∂yijσ∂yklν+2cσνklijSym(jkl) ωjklν}∧^ωσ∧_ωi+^∂2L∂_yijσ∂_ykν−4_{(bσνkij)Alt((σj)(νk))ωkν}∧_ωjσ∧_ωi+^∂2L∂_yijσ∂_yklν−2_{cσνklijωklν}∧_ωjσ∧_ωi+_{∂aσνij∂^yκAlt(κσν)}^ωκ∧^ωσ∧^ων∧_ωij+_{∂aσνij∂ypκ+∂bνκpij∂^yσAlt(σν) ωpκ}∧^ωσ∧^ων∧_ωij+_{∂aσνij∂ypqκSym(pq)}+_{∂cνκpqij∂ypqσAlt(σν)ωpqκ}∧^ωσ∧^ων∧_ωij+_{∂bσνqij∂ypκAlt((κp)(νq))} ^ωσ∧_ωqν∧_ωpκ∧_ωij+_{∂bσνkij∂ypqκ−∂cσκpqij∂ykνSym(pq)}^ωσ∧_ωkν∧_ωpqκ∧_ωij−_{∂cσνklij∂ypqκAlt((κpq)(νkl))} ^ωσ∧_ωpqκ∧_ωklν∧_ωij+dμ¯,

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

In the notation of Equations (11) and (12), the principal part of α α (13) takes the form

α^=_Eσ ^ωσ∧_ω0+^∂2L∂_yiσ∂^yν−∂∂^yν_dj∂L∂_yijσ−2_{dj aσνij}^ων∧^ωσ∧_ωi+^∂2L∂_yiσ∂_ykν−^∂2L∂^yσ∂_yikν−∂∂_ykνdj∂L∂_yijσ+4_aνσik−2_{dj bσνkijωkν}∧^ωσ∧_ωi+^∂2L∂_yiσ∂_yklν−∂∂_yklνdj∂L∂_yijσ−2_{(bσνkil)Sym(kl)}−2_{dj cσνklijωklν}∧^ωσ∧_ωi+^∂2L∂_yijσ∂_ykν−4_{(bσνkij)Alt((σj)(νk))ωkν}∧_ωjσ∧_ωi−_{Pνσijkl ωjklν}∧^ωσ∧_ωi+_{Qσνijkl ωklν}∧_ωjσ∧_ωi,

Expressing the generators of the ideal _{Dα^4 Dα^4}, we obtain

_i∂∂^yνα^=_{Eν ω0}+2^∂2L∂_yiσ∂^yν−∂∂^yν_dj∂L∂_yijσ−2_{dj aσνij}^ωσ∧_ωi−^∂2L∂_yiν∂_ykσ−^∂2L∂^yν∂_yikσ−∂∂_ykσdj∂L∂_yijν+4_aσνik−2_{dj bνσkijωkσ}∧_ωi−^∂2L∂_yiν∂_yklσ−∂∂_yklσdj∂L∂_yijν−2_{(bνσkil)Sym(kl)}−2_{dj cνσklijωklσ}∧_ωi+_{Pσνijkl ωjklσ}∧_ωi,_i∂∂ykνα^=^∂2L∂_yiσ∂_ykν−^∂2L∂^yσ∂_yikν−∂∂_ykνdj∂L∂_yijσ+4_aνσik−2_{dj bσνkij}^ωσ∧_ωi+2^∂2L∂_yijσ∂_ykν−4_{(bσνkij)Alt((σj)(νk))ωjσ}∧_ωi+_{Qνσikjl ωjlσ}∧_ωi,_i∂∂yklνα^=^∂2L∂_yiσ∂_yklν−∂∂_yklνdj∂L∂_yijσ−2_{(bσνkil)Sym(kl)}−2_{dj cσνklij}^ωσ∧_ωi+_{Qσνijkl ωjσ}∧_ωi,_{i∂∂yjklν}α^=−_Pσνijkl ^ων∧_ωi

Since the ranks of the matrices _{Pνσijkl Pνσijkl} , _{Qσνijkl Qσνijkl} are maximal then the ^ωσ∧_ωi ^ωσ∧_ωi and _ω(jσ∧_{ωi) ω(jσ}∧_ωi) are generators of the ideal _{Dα^4 Dα^4} . For Dedecker–Hamilton extremals, we obtain _{δD δD π3,2}∘_δD=^J2γ _π3,2∘_δD=^J2γ , where γ γ is a section of π π . Substituting this into Equation (5), we get

_{δD∗ i∂∂^yσ}α^=_Eσ∘^J3γ

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

_{δD∗ i∂∂^yσ}α^=_Eσ∘^J2γ

for the 2nd order Euler–Lagrange form (8) and γ γ is an extremal of λ λ.

Let us prove strong regularity. We have to show that under our assumptions, for every section δ δ satisfying the Hamilton equations, _π3,2∘δ=^J2γ _π3,2∘δ=^J2γ , where γ γ is a solution of the Euler–Lagrange equations of the Lagrangian λ λ . Assuming dμ¯=0 dμ¯=0 , we obtain ^δ∗(_{i∂/∂yjklσ}α)=^δ∗(_Pσνijkl ^ων∧_ωi)=0 ^δ∗(_{i∂/∂yjklσ}α)=^δ∗(_Pσνijkl ^ων∧_ωi)=0 , i.e., ^{δ∗ ων}=0 ^{δ∗ ων}=0 by the rank condition on _{Pσνijkl Pσνijkl} , i.e., ∂(^yσ∘δ)/∂^xi=_yiσ∘δ ∂(^yσ∘δ)/∂^xi=_yiσ∘δ . Hence, ^δ∗(_{i∂/∂yklν}α)=^δ∗_{Qσνijkl ωjσ}∧_ωi=0 ^δ∗(_{i∂/∂yklν}α)=^δ∗_{Qσνijkl ωjσ}∧_ωi=0.

Note that the matrix _{Qσνijkl Qσνijkl} is symmetric in indices kl kl and its maximal rank is mn(n+1)/2 mn(n+1)/2 . Due to the rank condition on _{Qσνijkl Qσνijkl} , ^δ∗ _ωjσ=0 ^δ∗ _ωjσ=0 , i.e., _{∂(yjσ∘δ)/∂^xiSym(ij)}=_yijσ∘δ _{∂(yjσ∘δ)/∂^xiSym(ij)}=_yijσ∘δ . The conditions for δ δ obtained above mean that every solution of Hamilton equations is holonomic up to the second order, i.e., we can write _π3,2∘δ=^J2γ _π3,2∘δ=^J2γ , where γ γ is a section of π π . Now, the equations ^{J3 (_π3,0∘δ)∗}(_i∂/∂ykσα)=0 ^{J3 (_π3,0∘δ)∗}(_i∂/∂ykσα)=0 are satisfied identically and the last set of Hamilton equations— ^{J3 (_π3,0∘δ)∗}(_i∂/∂^yσα)=0 ^{J3 (_π3,0∘δ)∗}(_i∂/∂^yσα)=0 —take the form _Eσ∘^J3γ=0 _Eσ∘^J3γ=0 (7) or _Eσ∘^J2γ=0 _Eσ∘^J2γ=0 (8), proving that γ γ is an extremal of λ λ. ☐

In the next propositon we study a weaker condition which the Hamilton extremals satisfy.

Theorem 2.

Let dimX≥2 dimX≥2 . Let λ=L_ω0 λ=L_ω0 be a second order Lagrangian with the Euler–Lagrange form (7) or (8), and α=dρ α=dρ with ρ of the form (9) and (10) be its Lepagean equivalent. Assume that μ¯ μ¯ is closed and the matrix

_Pσνijkl=_{^∂2L∂yijν∂yklσ+2cνσklijSym(jkl)},

with mⁿ³ mⁿ³ rows (respectively mn mn columns) labelled by σ, j, k, l (respectively νi νi ) has rank mn mn .

Then every Hamilton extremal δ:π(U)⊂V→^J2Y δ:π(U)⊂V→^J2Y of the Hamiltonian system α=dρ α=dρ is of the form _π3,1∘δ=^J1γ _π3,1∘δ=^J1γ (i.e., ∂^yσ∂^xi=_yiσ ∂^yσ∂^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 ^J3Y ^J3Y.

Writing the Lepagean equivalent ρ ρ (9), (10) in the form of a noninvariant decomposition, we get

ρ=−H_ω0+_pσjd^yσ∧_ωj+_pσijd_yiσ∧_ωj+2_{cσνklij yjσ}d_yklν∧_ωi+_aσνijd^yσ∧d^yν∧_ωij+_bσνkijd^yσ∧d_ykν∧_ωij+_cσνklijd^yσ∧d_yklν∧_ωij+μ¯,

where

H=−L+∂L∂_yiσ−_dj∂L∂_yijσyiσ+∂L∂_yijσyijσ−2_{aσνij yiσ yjν}−2_{(bσνkij)Sym(ki) yiσ ykjν}−2_{(cσνklij)Sym(klj) yiσ ykljν},_pσj=∂L∂_yjσ−_di∂L∂_yijσ+4_{aσνij yiν}+2_{(bσνkij)Sym(ki) ykiν}+2_{(cσνklij)Sym(kli) ykliν},_pσij=∂L∂_yijσ+2_{bνσijk ykν}.

Moreover, if the matrix

∂_pσi∂_yklν∂_pσi∂_yklmν∂_pσij∂_yklν∂_pσij∂_yklmν

has maximal rank, then

(^xi,^yσ,_yiσ,_pσi,_pσij)

is part of coordinate system.

We note that the functions _{pσij pσij} do not depend on the variables _{yklmν yklmν}. Then the submatrix of the Jacobi matrix of the transformation takes the form

∂_pσi∂_yklν∂_pσi∂_yklmν∂_pσij∂_yklν0.

The above matrix has maximal rank if and only if the matrices ∂_pσi/∂_yklmν ∂_pσi/∂_yklmν and ∂_pσij/∂_yklν ∂_pσij/∂_yklνhave maximal ranks. Explicit computations lead to

∂_pσi∂_yklmν=_{^∂2L∂yimν∂yklσ+2cνσklimSym(klm)},∂_pσij∂_yklν=^∂2L∂_yijσ∂_yklν+2∂_bκσijq∂_yklνyqκ.

Note that in the notation of Equation (11), PσνijklT=∂pσi/∂yjklν PσνijklT=∂pσi/∂yjklν and the maximal rank is equal to mn mn . The matrix ∂pσij/∂yklν ∂pσij/∂yklν is symmetric in the indices kl kl and therefore the maximal rank of the matrix is equal to mnn+1/2 mnn+1/2 , i.e., the number of independent pσij pσij is mnn+1/2 mnn+1/2 . Contrary to the situation in Hamilton–De Donder theory, the functions pσij pσij are not symmetric in the indices ij ij.

If we suppose that the matrix (19) has maximal rank, then

_ψ3=(^xk,^yν,_ykν,_yklν,_yklmν)→(^xi,^yσ,_yiσ,_pσi,_pσij,^zB)=χ

is a coordinate transformation over an open set U⊂_V2 U⊂_V2 , where ^zB,1≤B≤mn(ⁿ²+3n−1)/6 ^zB,1≤B≤mn(ⁿ²+3n−1)/6 are arbitrary coordinate functions. We call it a generalized Legendre transformation and χ χ (22) the generalized Legendre coordinates. Accordingly, H,_pσi,_pσij H,_pσi,_pσijare called generalized Hamiltonian and generalized momenta, respectively.

Writing the Lepagean equivalent ρ ρ (9) and (10) in the generalized Legendre transformation, we get

ρ=−H_ω0+_pσjd^yσ∧_ωj+_pσijd_yiσ∧_ωj+2_{cσνklij yjσ}∂_yklν∂_pβqd_pβq+∂_yklν∂_pβqrd_pβqr+∂_yklν∂^zBd^zB∧_ωi+_aσνijd^yσ∧d^yν∧_ωij+_bσνkijd^yσ∧d_ykν∧_ωij+_cσνklijd^yσ∧∂_yklν∂_pβqd_pβq+∂_yklν∂_pβqrd_pβqr+∂_yklν∂^zBd^zB∧_ωij+μ¯,

where _{yklν yklν} are functions of variables _pσi,_pσij,^zB _pσi,_pσij,^zB.

The Hamilton Equation (5) in these generalized Legendre coordinates take a rather complicated form, see Appendix A.

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

η=2_{cσνklij yjσ}d_yklν∧_ωi+_aσνijd^yσ∧d^yν∧_ωij+_bσνkijd^yσ∧d_ykν∧_ωij+_cσνklijd^yσ∧d_yklν∧_ωij+_dσνklijd_ykσ∧d_ylν∧_ωij

then the Hamilton Equation (5) have the following form

∂H∂^yκ=−∂_pκj∂^xj,∂H∂_yqκ=−∂_pκqj∂^xj,∂H∂_pκq=∂^yκ∂^xq,∂H∂_pκqr=∂_yqκ∂^xr,∂H∂^zM=0.

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=^J1γ _π3,1∘δ=^J1γ , where γ γ is an extremal of λ λ.

Theorem 3.

Let dimX≥2 dimX≥2 . Let λ=L_ω0 λ=L_ω0 be a second order Lagrangian with the Euler–Lagrange form (7) or (8), and α=dρ α=dρ with ρ of the form (9), and Equation (10) be the expression of its Lepagean equivalent in a fiber chart (V,ψ) (V,ψ) , ψ=(^xi,^yσ) ψ=(^xi,^yσ) .

Suppose that μ¯ μ¯ is closed and ρ admits Legendre transformation (22) defined by Equation (18).

Then _π3,1∘δ=^J1γ _π3,1∘δ=^J1γ , where γ is an extremal of λ.

Proof.

The form ρ ρadmits Legendre transformation, so the matrix

∂_pσi∂_yjklν=_{^∂2L∂yijν∂yklσ+2cνσklijSym(jkl)}

has maximal rank equal to mn mn . In the notation of (11), PσνijklT=∂pσi/∂yjklν PσνijklT=∂pσi/∂yjklν . Acordingly, from Proposition 2, we obtain π3,1∘δ=J1γ π3,1∘δ=J1γ , where γ γ is an extremal of λ 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) The Hamiltonian system is strongly regular and the Legendre transformation exists. (See examples of strongly regular systems in [17]).

(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.

Let X=^R2 X=^R2 , Y=^R2×^R2 Y=^R2×^R2 (i.e., n=2 n=2 , m=2 m=2 ). Denote (V,ψ) (V,ψ) , ψ=(^xi,^yσ) ψ=(^xi,^yσ) a fibered chart on ^R2×^{R2 R2}×^R2. Let us consider the following Lagrangian

λ=L_ω0,L=_{y111 y222}−_{y221 y112}

which satisfies (7).

5.1. Example (a)

View of the above considerations, we take a Lepagean equivalent ρ ρ (of the Euler–Lagrange form E of Lagrangian (25)) in the form α=dρ α=dρ , where ρ ρ is (9), (10).

We consider functions _{aσνij aσνij} , _{bσνkij bσνkij} , _{cσνijkl cσνijkl} (see Equation (10)) on an open set U⊂^{J3 R2} U⊂^{J3 R2} with the conditions _y11≠0 _y11≠0 , _y21≠0 _y21≠0 , _y121≠0 _y121≠0 and _y122≠0 _y122≠0.

The functions _{aσνij aσνij} are arbitrary. The functions _{bκσijp bκσijp} are linear in variables _{yklν yklν} . We denote _dκσνijpkl=∂_bκσijp/∂_{yklν dκσνijpkl}=∂_bκσijp/∂_yklν . Suppose that _{dκσνijpkl dκσνijpkl} are constant functions, then we have only eight non-zero constants and we put _d11212112=_d11212121=−_d11211212=−_d11211221=1 _d11212112=_d11212121=−_d11211212=−_d11211221=1 and _d12122112=_d12122121=−_d12121212=−_d12121221=1 _d12122112=_d12122121=−_d12121212=−_d12121221=1 . Similarly, we assume that _{cσνijkl cσνijkl} are constant functions. We have again only eight non-zero constants, and we choose _c111212=_c112112=−_c112121=−_c111221=1 _c111212=_c112112=−_c112121=−_c111221=1 and _c221212=_c222112=−_c222121=−_c221221=1 _c221212=_c222112=−_c222121=−_c221221=1. Then the Lepagean equivalent takes the form

_ρ1=_θλ+_aσνij ^ωσ∧^ων∧_ωij−4_y121 ^ω2∧_ω21∧_ω12−4_y122 ^ω1∧_ω12∧_ω12+4^ω1∧_ω121∧_ω12+4^ω2∧_ω122∧_ω12+μ¯,

where μ¯ μ¯is an arbitrary closed n-form.

The matrices (11), (12), and (21) take the following form

^{(_Pσνijkl)T}=1300004440000011100−4−4−400000−1−1−100000000−1−1−1000004440011100000−4−4−40000,

and

_Qσνijkl=000000010−2−200000022000000000−1000000−1000000000−2−200000022010000000,

and

∂_pσij∂_yklν=00000−_y21−_y21100000_y11y110000000000000−1000000−10000000000000−_y21−_y21000001_y11y1100000,

We can easily see that rank(_Pσνijkl)=4 rank(_Pσνijkl)=4 and rank(_Qσνijkl)=6 rank(_Qσνijkl)=6 . Since _y11≠0 _y11≠0 and _y21≠0 _y21≠0 rank∂_pσij/∂_yklν=6 rank∂_pσij/∂_yklν=6 . The form α=dρ α=dρis strongly regular and a generalized Legendre transformation exists.

The generalized Hamiltonian and momenta (18) take the form

H=−_{y121 y222}+_{y221 y112}−_y11(8_y1221+_y1222)+_y21(8_y1121+_y1222)−_y12(8_y1222+_y1221)+_y22(8_y1122+_y1221)−4_a1212(_{y11 y22}−_{y21 y12}),

_p11=_y1222−8_y1221+4_{a1212 y22},_p21=−_y1221+8_y1222−4_{a1212 y21},_p12=−_y1122−8_y1121−4_{a1212 y12},_p22=_y1121+8_y1122+4_{a1212 y11},

_p111=_y222−4_{y21 y122},_p112=4_{y11 y122},_p122=−_y112,_p222=_y111+4_{y11 y121},_p221=−4_{y21 y121},_p122=−_y221.

We have only six independent generalized momenta _{pσij pσij} . We note that _p121=_p212=0 _p121=_p212=0.

5.2. Example (b)

For the given Lagrangian (25), we consider another Hamiltonian system on an open set U⊂^{J3 R2} U⊂^{J3 R2}

_ρ2=_θλ+_aσνij ^ωσ∧^ων∧_ωij+_bσνkij ^ωσ∧_ωkν∧_ωij+4^ω1∧_ω121∧_ω12+4^ω2∧_ω122∧_ω12+μ¯,

where _{aσνij aσνij} , _{bσνkij bσνkij} are arbitrary constant functions satisfying Equation (10) and μ¯ 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

∂_pσij∂_yklν=0000000100000000000000000000−1000000−10000000000000000000010000000,

and rank∂_pσij/∂_yklν=4 rank∂_pσij/∂_yklν=4. Therefore the generalized Legendre transformation does not exist.

5.3. Example (c)

On an open set U⊂^{J3 R2} U⊂^{J3 R2} where _y11≠0 _y11≠0 , _y21≠0 _y21≠0 , _y121≠0 _y121≠0 and _y122≠0 _y122≠0, the Lepagean equivalent takes the form

_ρ3=_θλ+_aσνij ^ωσ∧^ων∧_ωij−4_y121 ^ω2∧_ω21∧_ω12−4_y122 ^ω1∧_ω12∧_ω12+4^ω1∧_ω121∧_ω12+μ¯,

where μ¯ μ¯ is an arbitrary closed n-form and _{aσνij aσνij} are arbitrary functions satisfying Equation (10).

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

The matrices (11) and (12) take take the form

^{(_Pσνijkl)T}=1300004440000011100−4−4−400000−1−1−100000000−1−1−10000000000111000000000000,

_Qσνijkl=000000010−2−200000022000000000−1000000−10000000000000000000010000000,

and rank(_Pσνijkl)=4 rank(_Pσνijkl)=4 and rank(_Qσνijkl)=5 rank(_Qσνijkl)=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 was studied and the generalized momenta _pσij≠_{pσji pσij}≠_pσjiwere found. The theory was illustrated using examples of Hamilton systems satisfying:

(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.

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.

Funding

This research was funded by the Institute of Technology and Business in České Budějovice (project No. 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 dμ¯=0 (9) in Legendre coordinates take the following explicit form:

∂H∂^yκ=−∂_pκj∂^xj+2∂_cσνklij∂^yκ_yjσ∂_yklν∂_pβq∂_pβq∂^xi+∂_yklν∂_pβqr∂_pβqr∂^xi+∂_yklν∂^zB∂^zB∂^xi+4∂_aκνij∂^xj∂^yν∂^xi+6_{∂aσνij∂^yκAlt(κνσ)}∂^yσ∂^xi∂^yν∂^xj+4∂_aκνij∂_yqσ∂_yqσ∂^xi∂^yν∂^xj+4∂_aκνij∂_pσq∂_pσq∂^xi∂^yν∂^xj+4∂_aκνij∂_pσqr∂_pσqr∂^xi∂^yν∂^xj+4∂_aκνij∂^zM∂^zM∂^xi∂^yν∂^xj+2∂_bκνkij∂^xj∂_ykν∂^xi+4_{∂bσνij∂^yκAlt(κσ)}∂^yσ∂^xi∂_ykν∂^xj+2_{∂bκνkij∂yqσAlt((νk)(σq))}∂_ykν∂^xi∂_yqσ∂^xj+2∂_bκνkij∂_pσq∂_ykν∂^xi∂_pσq∂^xj+2∂_bκνkij∂_pσqr∂_ykν∂^xi∂_pσqr∂^xj+2∂_bκνkij∂^zM∂^yν∂^xi∂^zM∂^xj+2∂_cκνklij∂^xj∂_yklν∂_pβq∂_pβq∂^xi+∂_yklν∂_pβqr∂_pβqr∂^xi+∂_yklν∂^zB∂^zB∂^xi+4_{∂cσνklij∂^yκAlt(κσ)}∂^yσ∂^xi∂_yklν∂_pβq∂_pβq∂^xj+∂_yklν∂_pβqr∂_pβqr∂^xj+∂_yklν∂^zB∂^zB∂^xj+2∂_cκνklji∂_yqσ∂_yqσ∂^xi∂_yklν∂_pβq∂_pβq∂^xj+∂_yklν∂_pβqr∂_pβqr∂^xj+∂_yklν∂^zB∂^zB∂^xj+2∂_cκνklji∂_pσq∂_pσq∂^xi∂_yklν∂_pβq∂_pβq∂^xj+∂_yklν∂_pβqr∂_pβqr∂^xj+∂_yklν∂^zB∂^zB∂^xj+2∂_cκνklji∂_pσq∂_pσq∂^xi∂_yklν∂_pβq∂_pβq∂^xj+∂_yklν∂_pβqr∂_pβqr∂^xj+∂_yklν∂^zB∂^zB∂^xj+2∂_cκνklji∂_pσqr∂_pσqr∂^xi∂_yklν∂_pβq∂_pβq∂^xj+∂_yklν∂_pβqr∂_pβqr∂^xj+∂_yklν∂^zB∂^zB∂^xj+2∂_cκνklji∂^zM∂^zM∂^xi∂_yklν∂_pβq∂_pβq∂^xj+∂_yklν∂_pβqr∂_pβqr∂^xj+∂_yklν∂^zB∂^zB∂^xj

∂H∂_yqκ=−∂_pκqj∂^xj+2_cκνkliq∂_yklν∂_pβq∂_pβq∂^xi+∂_yklν∂_pβqr∂_pβqr∂^xi+∂_yklν∂^zB∂^zB∂^xi+2∂_aσνij∂_yqκ∂^yσ∂^xi∂^yν∂^xj+2∂_bσκqij∂^xj∂^yσ∂^xi+2_{∂bσκqij∂^yνAlt(νσ)}∂^yν∂^xi∂^yσ∂^xj+4_{∂bσνkij∂yqκAlt((κq)(νk))}∂^yσ∂^xi∂_ykν∂^xj+2∂_bσκqij∂_pνk∂_pνk∂^xi∂^yσ∂^xj+2∂_bσκqij∂_pνkl∂_pνkl∂^xi∂^yσ∂^xj+2∂_bσκqij∂^zM∂^zM∂^xi∂^yσ∂^xj+2∂_cσνklji∂_yqκ∂^yσ∂^xi∂_yklν∂_pβq∂_pβq∂^xj+∂_yklν∂_pβqr∂_pβqr∂^xj+2∂_yklν∂^zB∂^zB∂^xj

∂H∂_pκq=∂^yκ∂^xq+2∂_cσνklji∂^xi∂_yklν∂_pκqyjσ+2∂_cσνklji∂^yβ∂_yklν∂_pκq∂^yβ∂^xi_yjσ+2∂_cσνklji∂_yrβ∂_yklν∂_pκq∂_yrβ∂^xi_yjσ+2∂_cσνklji∂_yrβ∂_yklν∂_pκq∂_pβr∂^xi_yjσ++2∂_cσνklji∂_pκqyjσ∂_yklν∂_pβq∂_pβq∂^xi+∂_yklν∂_pβqr∂_pβqr∂^xi+∂_yklν∂^zB∂^zB∂^xi+2∂_cσνklji∂_pβrs∂_yklν∂_pκq∂_pβrs∂^xi_yjσ+2∂_cσνklji∂^zM∂_yklν∂_pκq∂^zM∂^xi_yjσ+2_cσνklji∂_yklν∂_pκq∂_yjσ∂^xi+2∂_aσνij∂_pκq∂^yσ∂^xi∂^yν∂^xj+2∂_bσνkij∂_pκq∂^yσ∂^xi∂_ykν∂^xj+2_{∂cσνklij∂^yβAlt(βν)}∂^yβ∂^xi∂^yσ∂^xj∂_yklν∂_pκq+2∂_cσνklji∂^xj∂_yklν∂_pκq∂^yσ∂^xi+2∂_cσνklij∂_pκq∂^yσ∂^xi∂_yklν∂_pβq∂_pβq∂^xj+∂_yklν∂_pβqr∂_pβqr∂^xj+∂_yklν∂^zB∂^zB∂^xj+2∂_cσνklij∂_yrβ∂_yklν∂_pκq∂_yrβ∂^xi∂^yσ∂^xj+2∂_cσνklij∂_pβr∂_yklν∂_pκq∂_pβr∂^xi∂^yσ∂^xj+2∂_cσνklij∂_pβrs∂_yklν∂_pκq∂_pβrs∂^xi∂^yσ∂^xj+2∂_cσνklij∂^zM∂_yklν∂_pκq∂^zM∂^xi∂^yσ∂^xj

∂H∂_pκqr=∂_yqκ∂^xr+2∂_cσνklji∂^xi∂_yklν∂_pκqryjσ+2∂_cσνklji∂^yβ∂_yklν∂_pκqr∂^yβ∂^xi_yjσ+2∂_cσνklji∂_ysβ∂_yklν∂_pκqr∂^yβ∂^xi_yjσ+2∂_cσνklji∂_pβs∂_yklν∂_pκqr∂_pβs∂^xi_yjσ+2∂_cσνklji∂_pβst∂_yklν∂_pκqr∂_pβst∂^xi_yjσ+2∂_cσνklij∂_pκqr∂^yσ∂^xi∂_yklν∂_pβs∂_pβs∂^xi+∂_yklν∂_pβst∂_pβst∂^xi+∂_yklν∂^zB∂^zB∂^xi_yjσ+2∂_cσνklji∂^zM∂_yklν∂_pκqr∂^zM∂^xi_yjσ+2_cσνklji∂_yjσ∂^xi∂_yklν∂_pκqr+2∂_aσνij∂_pκqr∂^yν∂^xi∂_ykν∂^xj+2∂_bσνkij∂_pκqr∂^yσ∂^xi∂^yν∂^xj+2∂_cσνklji∂^xj∂^yσ∂^xi∂_yklν∂_pκqr+2_{∂cσνklij∂^yβAlt(βσ)}∂^yβ∂^xi∂^yσ∂^xj∂_yklν∂_pκqr+2∂_cσνklij∂_ysβ∂_yklν∂_pκqr∂_ysβ∂^xi∂^yσ∂^xj+2∂_cσνklij∂_pβs∂_pβs∂^xi∂^yσ∂^xj∂_yklν∂_pκqr+2∂_cσνklij∂_pβst∂_pβst∂^xi∂^yσ∂^xj∂_yklν∂_pκqr+2∂_cσνklij∂_pκqr∂^yσ∂^xi∂_yklν∂_pβs∂_pβs∂^xj+∂_yklν∂_pβst∂_pβst∂^xj+∂_yklν∂^zB∂^zB∂^xj+2∂_cσνklij∂^zM∂_yklν∂_pκqr∂^zM∂^xi∂^yσ∂^xj

∂H∂^zM=2∂_cσνklji∂^xi∂_yklν∂^zM_yjσ+2∂_cσνklji∂^yβ∂_yklν∂^zM∂^yβ∂^xi_yjσ+2∂_cσνklji∂_ysβ∂_yklν∂^zM∂^yβ∂^xi_yjσ+2∂_cσνklji∂_pβs∂_yklν∂^zM∂_pβs∂^xi_yjσ+2∂_cσνklji∂_pβst∂_yklν∂^zM∂_pβst∂^xi_yjσ+2∂_cσνklij∂^zM∂_yklν∂_pβs∂_pβs∂^xi+∂_yklν∂_pβst∂_pβst∂^xi+∂_yklν∂^zB∂^zB∂^xi_yjσ

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