First-order Hamiltonian field theory and mechanics

Abstract

This article deals with the geometric analysis of the evolutionary and the polysymplectic approach in first-order Hamiltonian field theory. Based on a variational formulation in the Lagrangian picture, two possible counterparts in a Hamiltonian formulation are discussed. The main difference between these two approaches, which are important for the application, is besides a different bundle construction, the different Legendre transform as well as the analysis of the conserved quantities. Furthermore, the role of the boundary conditions in the Lagrangian and in the Hamiltonian pictures will be addressed. These theoretical investigations will be completed by the analysis of several examples, including the wave equation, a beam equation and a special subclass of continuum mechanics in the presented framework.

Keywords:

• field theory
• polysymplectic structure
• partial differential equations

1. Introduction

The Hamiltonian formalism is a well known method for describing phenomena that can be modelled by ordinary differential equations. The main ingredient of the theory is a representation of the equations in an evolutionary first-order form, and under some regularity assumptions the Legendre transform yields a connection with the well-known Lagrangian description. There exist several approaches that extend this Hamiltonian description to systems that are described by partial differential equations, where the general question arises if the formulation as an evolutionary description should be maintained. A description in an evolutionary form requires us to single out an evolution parameter, for example the time, and this leads to the introduction of the variational derivative with respect to the Hamiltonian formulation. This approach is well known in the literature, see for example [1] or [2] and references therein. Another possibility for describing field theory is an approach that goes back to De Donder [3], which is based on the conservation of the symmetry with respect to all the independent variables, that is, the time and the space, which leads to the introduction of so-called multimomenta, and, of course, the Legendre transform differs from the case of the evolutionary approach. There exists an extensive literature describing the polysymplectic or the multisymplectic formulation, see for example [4–6]. This article represents an extended version of a preliminary work published in [7].

Before we start to analyse the case of partial differential equations, we want to briefly treat the lumped parameter scenario, because these systems have very useful properties in modelling, stability analysis and control. These recapitulations should be utilized mainly as a point of departure and to allow for a comparison of the lumped and the distributed parameter case. Let us consider a manifold that is equipped with coordinates (q ^α). Then standard constructions lead to the tangent bundle and the cotangent bundle , which may be equipped with the induced coordinates and with respect to the holonomic bases ∂_α and dq ^α. Denoting as p _α as it is standard in the literature, a classical Hamiltonian system can be formulated as in [8]

(1)

where is termed the Hamiltonian and possesses the induced bases . The state manifold is clearly and the system (1) describes an autonomous, lossless system and the Hamiltonian is a conserved quantity. One can include inputs/collocated outputs and dissipation to obtain a more general system class, namely, port-controlled Hamiltonian systems with dissipation (PCHD-systems), which in the case of a mechanical system very often takes the form

(2)

where δ denotes the Kronecker symbol and R _αβ meets . This structure (2) has very pleasing properties concerning the control of such systems, see for example [9]. Therefore it is of interest to also analyse field theory in a Hamiltonian fashion, with the desire to exploit as much as possible of the structural properties of this system class with respect to control theoretic problems.

The main geometric ingredients of Equations (1)(1) and (2)(2) are the following: The ordinary differential equations possess dependent coordinates and they are written in an evolutionary manner with respect to an evolution parameter that in most of the applications is the time. The analysis of field theoretic problems requires using the concept of bundles because derivative coordinates have to be introduced. In mechanics besides the time, spatial independent coordinates also appear and the key question is if the time is treated on the same level as the spatial coordinates or if one tries to use the time as an evolution parameter. This then naturally leads to the polysymplectic and the evolutionary approaches, respectively. This article aims to give a description of both of these approaches in a geometric fashion, singling out the differences in the bundle construction and to show how these two approaches differ in the case of first-order field theory in mechanics. As an example we will treat the wave equation, the Timoshenko beam and classical continuum mechanics, where we assume that a stored energy function exists. Field theory for higher dimensions is an actual research topic, see, for example, the recent contributions [10,11] that deal with a generalization of the Skinner–Rusk formalism to higher order field theory.

2. Technical preliminaries

This section includes the main notions of differential geometry and specifies the tensor notation that will be used in the sequel. In this contribution, we use the concept of bundles [4] and [12]. The main motivation for the use of a geometric formulation is based on the desire to obtain an intrinsic description of the systems under investigation. Especially in field theories, the obligation to separate dependent and independent coordinates leads to the choice of geometric objects like bundles quite naturally. We aim not to give a detailed introduction to these concepts at this stage, instead referring to standard literature [4], but we want to introduce some notation important in the sequel. Given a bundle , we are able to derive the first jet manifold , with coordinates and the first jet of a section is written as j ¹ (σ). The tangent bundle possesses the induced coordinates and the vertical subbundle is equipped with the coordinates , whereas the cotangent bundle possesses the induced coordinates . The special vector field , we omit the pullback bundle structure here, that meets the relation with and is called the first-order total derivative. A connection on the bundle is regarded as the map , which can be represented as

(3)

We use the standard notation for tensor bundles as well as for the exterior algebra concerning differential forms, where the interested reader is again referred to [4] for a detailed exposition. For the Lie derivative of a geometric object Δ with respect to a vector field v, we arrange the notation . For a tensor field , we introduce , such that is met, if it exists, with the Kronecker delta δ.

3. The polysymplectic structure

The aim of this part is to describe the main concepts of the polysymplectic formalism for first-order field theory based on a variational problem in a Lagrangian formulation. Most of the material presented here can be found in [4,6], but contrary to the physicists literature where the boundary terms are often neglected, we want to also discuss the boundary conditions because they are important for the applications, especially in engineering. In the latter part of this section, we will then focus on the conserved quantities to give a connection to the evolutionary point of view. Let us consider the bundle possessing the coordinates . In our application, continuum mechanics, the coordinates y ^α will be denoted by q ^α and correspond to the spatial coordinates in the configuration manifold, whereas the independent coordinates xⁱ will be the spatial coordinates in the reference manifold X^j as well as the time t ⁰ = x ⁰. Therefore we have and i = 0, … , s. Let us consider the first jet bundle , which is affine and a first-order Lagrangian L

(4)

with together with

The variational problem for a section is the following

(5)

where the flow ϵ_ε is used to deform sections and whose generator is a vertical vector field . This is a well-known problem and treated for example in [1,4,12] and references therein. To find a solution for the variational problem and especially to be able to discuss boundary conditions, we rewrite Equations (5)(5). It is obvious that Equation (5)(5) is equivalent to

(6)

see for example [4,12] and an adequate strategy to avoid the use of the jet prolongation of v is the choice for a Lepage equivalent for L, for example, the Poincare Cartan form

(7)

Plugging in the Poincare Cartan form in Equation (6)(6) instead of the Lagrangian and using the Theorem of Stokes, one derives the following relations

(8)

which do not depend on derivatives of the field v by construction. Consequently, the partial differential equations for a first-order Lagrangian follow as

(9)

and the boundary term is the second term in Equation (8)(8) and reads as

(10)

Remark 1

It is worth mentioning that the bundle construction leads to the interpretation that all the independent coordinates are treated on the same level. In physical applications often a situation arises where some coordinates are treated special. This can be seen easily when one considers a system where the independent coordinates are the time t ⁰ = x ⁰ and additional spatial coordinates X ^j . Then it is obvious from Equation (8)(8) that the boundary conditions can be satisfied by a condition on the variational field v, that is, no variation on the boundary, or by a condition on the integral itself for an arbitrary variational field. In practice often a combination of these concepts arises, that is, there is no variation on the time boundary but on the spatial one.

We now turn to the Hamiltonian picture. Let us consider the Legendre bundle [4]

with coordinates for Π, which possesses the transitions functions

with respect to a bundle morphism and . With the Legendre bundle at hand we are able to construct the tangent valued Liouville form

(11)

which can be contracted by and one obtains

(12)

see again [4]. Based on Equation (12)(12) the Hamiltonian form associated to a Lagrangian can be constructed and reads as

(13)

which corresponds to Equation (7)(7) by the Legendre map .

The next step is to introduce the polysymplectic form Ω, which is defined such that is met with . We obtain

(14)

The map H_L yields a fibred morphism

(15)

which is the homogeneous Legendre bundle and possesses the coordinates with the additional transition function

(16)

as well as the canonical form

(17)

This unique form can be characterized by the fact that . The horizontal projection leads to

(18)

and this expression shows that all affine maps can be expressed in coordinates by . Let us choose a section of the bundle with and this construction leads to the Hamiltonian form

(19)

and it is readily observed that in the case of a regular Legendre map (13) and (19) can be related by a given Lagrangian L.

3.1. The differential equations

To obtain the partial differential equations we consider the relation

(20)

where the first jet manifold of the Legendre bundle possesses the coordinates and from

(21)

we end up with

(22)

and the partial differential equations follow to

(23)

Remark 2

Let us consider the analogy to the lumped parameter case. The Liouville form and the symplectic form reads as

and the differential equations follow from

as

It should be mentioned that the connection γ from above can be replaced by the Hamiltonian vector field v_H in this case.

Let us denote by the inverse of the map , that is, then from

and

together with

it is shown that Equations (9)(9) and (23)(23) correspond in the case of a regular Legendre transformation. Concerning the boundary conditions it is easy to see that the expression stemming from the variational approach in the Lagrangian picture now becomes

Of course the same argumentation as above with regard to the boundary expression when the coordinates are split into time and spatial ones remains valid. The case that no variation on the time boundary is performed leads directly to the observation that the boundary conditions do not include the temporal momenta because on the time boundary (integration over the boundary form ) the variational field is then zero by construction.

Remark 3

Let us choose a non-trivial connection on such that

(24)

is met. Therefore, one has the splitting

and the differential equations follow as

(25)

3.2. The conserved quantities

Let us consider the projectable vector field where we denote its first jet-prolongation with j ¹ (w). The Lie derivative of the Lagrangian evaluated on Equation (9)(9) yields

(26)

see also [4], which can be written also as

(27)

In the case of mechanics we split the coordinates into t ⁰ and X^j with the convention that the indices j = 1, … , s in contrast to the general case where the indices meet j = 0, … , s, as we explicitly label the coordinate x ⁰ = t ⁰. If the Lagrangian is independent of the time, that is, , where we assume a trivial connection Γ, then we obtain with w ⁰ = 1, w^j  = 0 and w ^α = 0 the relation

and consequently

shows that the expression

(28)

is the conserved quantity for this special case. We have shown that in the case where the Lagrangian is time independent, the conserved quantity reads as

which corresponds to the total energy in many applications. It is worth mentioning that the Hamiltonian and the conserved quantity differ in the expressions containing the spatial momenta. Thus, we are motivated to construct a Hamiltonian formulation where the Hamiltonian equals the conserved quantity in the time-invariant case. This will be demonstrated in the following section.

4. Hamiltonian evolution equations

A different view of Hamiltonian field theory is obtained when the equations are formulated with respect to the evolution of time only. Let us denote the special Hamiltonian, which contains only momenta with respect to the time coordinate as

(29)

then it is verified that the evolution equations can be written as

(30)

To show this let us denote by the inverse of the map , that is, , then from

and

as well as from

the equivalence is shown.

4.1. The geometric background

To explain the geometric motivation behind this approach, let us consider a different bundle structure with where in contrast to the last section the manifold with only consists of the spatial variables that are denoted as Xⁱ with i = 1, … , s and the manifold is equipped with the coordinates (Xⁱ , w ^α). The volume form is now denoted by

and in the Lagrangian picture we consider the bundle structure with , here w ^α is decomposed in positions and velocities. It is worth mentioning that we have the identification and when the flow parameter of the semigroup corresponds to the time t ⁰ (together with the equation ), but the bundle construction remains different of course. Let us consider a section together with the total time change of the Hamiltonian functional, which is given as

(31)

for the first-order Hamiltonians, and we consider an evolutionary vertical vector field together with the first prolongation . Let us inspect the expression

and integration by parts leads to

(32)

Using the variational derivative δ and the horizontal derivative the equation can be rewritten as

where we have the coordinate expression

It is easily seen that the total derivative d splits into the variational derivative δ and an exact form. Furthermore, the additional map δ^∂ can be introduced with

Therefore, we can conclude that in first-order mechanics we obtain

which shows that again the spatial momenta appear in the boundary map. As stated before the main difference in the bundle construction of the evolutionary and the polysymplectic approach is the role of the time. In the evolutionary approach, the role of the time is simply to be an evolution parameter and therefore one need not distinguish between spatial and temporal boundaries and only the spatial momenta appear due to the bundle construction, whereas in the polysymplectic case it depends on the variational field on the boundary.

An extension of the PCHD-system as in Equation (2)(2) to the field theoretic case, also in an evolutionary manner, can be stated for example in [13]

(33)

where in general can be differential operators, and in the case of mechanical systems the state w is decomposed in positions and momenta. The map is a skew symmetric operator and the map is positive semi-definite. Additionally, the input and the output bundles are dual vector bundles and denotes the adjoint map.

Before we want to treat examples, let us shortly summarize the main ideas and concepts presented so far. We have discussed a variational problem based on a Lagrangian of first order and derived the field equations as well as the boundary conditions. In the Lagrangian setting, all the independent variables (in mechanics, the time and additional spatial ones) are treated on the same level, which is reflected in the chosen bundle structure. Nevertheless, also in the physical application in the Lagrangian picture a splitting of these variables might be of interest, especially with respect to the boundary conditions. On the Hamiltonian side, we have analysed two completely different counterparts to the Lagrangian picture, namely, the polysymplectic and the evolutionary formulations. The polysymplectic approach is a straightforward generalization based on the Lagrangian formulation, which preserves the symmetry with respect to all the independent coordinates. Consequently, the equations are similar to the one in the case of ordinary differential equations. This similarity is achieved because of a construction of the Hamiltonian, which need not be the conserved quantity. This is the key motivation for the evolutionary approach, which is based on singling out the time as an evolution parameter and with a formulation of the partial differential equations based on a conserved quantity. Finally, let us comment on the possible extensions of the presented strategies in the context of control theory in the spirit of the presented lumped parameter example in the introduction. The formulation as in Equation (33)(33) is the counterpart to the case of PCHD-systems as in Equation (2)(2), which also allows to introduce ports in a classical sense, but it should be mentioned that the separation of the pairing in inputs and outputs is no longer unique, see [13] for more details on this fact. Initial conditions arise in the evolutionary picture because of the special treatment of the time coordinate as an evolution parameter. Nevertheless, the inclusion of distributed control inputs and/or boundary inputs can be performed in all discussed approaches in a straightforward manner. In the Lagrangian or the polysymplectic picture, the boundary conditions do not split into temporal and spatial ones in a natural way, and the concept of ports is of course more intuitive with regard to an energy-like storage function where the ports affect its change along solutions. This is mainly due to a different interpretation of the Hamiltonian evolutionary vector field and the variational field in the Lagrangian and the polysymplectic scenario.

5. Examples

This section studies the polysymplectic and the evolutionary approach on concrete first-order examples. The first example is a very simple one, the wave equation that was selected to motivate the different bundle structures and to show how the Lagrangian picture and the Hamiltonian ones are linked. A more complex example is the Timoshenko beam, and finally, we present the class of continuum mechanics, where we also focus on a geometric representation of this system class. In what follows t ⁰ will always denote the time, xⁱ denotes the spatial variable(s) in the polysymplectic approach and Xⁱ denotes the spatial variable(s) in the evolutionary approach. We distinguish the spatial variables because the bundle structures are different.

5.1. The wave equation

The wave equation can be modelled on a bundle with coordinates for and coordinates (t ⁰, x ¹) for . The Lagrangian is given as

with together with the coordinates for . Applying the variational derivative in Equation (9)(9) we obtain

(34)

and the partial differential equation y ₀₀ = y ₁₁. The boundary conditions follow from

(35)

for , since no variation takes place on the time boundaries. The multimomenta follow to

and the Legendre bundle Π possesses the coordinates . It is readily observed that in the boundary expression (35) only the spatial momenta appear. The Hamiltonian in the polysymplectic case reads as

The partial differential equations follow as

(36)

The evolutionary approach is based on a different bundle structure, namely, with coordinates for and coordinate (X ¹) for The Lagrangian then reads as

with and the temporal momentum is given as . The Hamiltonian in the evolutionary scenario follows to

The partial differential equations are then given as

and the identification gives the same result as in Equation (36)(36).

5.2. Timoshenko beam

A model of the Timoshenko beam can be derived by considering the bundle , and the relation

where w denotes the deflection, ψ is the angle of rotation and β is the angle of distortion due to shear. The Lagrangian then reads as

where A is the cross-sectional area, I and J denote the moments of inertia, G is the shear modulus and k denotes a numerical factor depending on the shape of the cross section. Here, the Lagrangian meets again where the coordinates for read as . The variational derivatives consequently can be written as

and Equation (9)(9) leads to

The boundary conditions follow from

where with slight abuse of notation is met and again no variation takes place on the time boundaries.

5.2.1. Polysymplectic approach

The polysymplectic approach is again based on the Legendre bundle Π with coordinates and the multimomenta in this scenario read as

The inverse for the spatial momenta can be computed as

The Hamiltonian can be derived from the formula

(37)

and reads as

(38)

The partial differential equations follow as

and using the Hamiltonian (38) we obtain

Rewriting these equations again by plugging in the multimomenta we obtain the result already derived by the Lagrange formalism

5.2.2. Evolutionary approach

The evolutionary approach then consequently makes use of the Lagrangian

and the bundle structure . The temporal momenta follow to and the Hamiltonian reads as

Consequently, the partial differential equations read as

and again the identifications and lead to the same results as above.

5.3. Continuum mechanics

In this part, we want to discuss the Lagrangian description of continuum mechanics in an intrinsic form, see also [14], by considering the formulations presented so far in this article. We start with some geometric preliminaries necessary for an intrinsic description of mechanics.

5.3.1. The geometry of continuum mechanics

The configuration bundle possesses the coordinates and the reference bundle is introduced with coordinates (t ⁰, Xⁱ ) for . We construct the bundle , with

(39)

and coordinates for . Furthermore, we introduce the following geometric objects. A symmetric vertical metric g on the fibres of , a trivial reference frame and a connection splitting the vertical tangent bundle , is denoted by Λ. In coordinates we obtain for the metric

and for the connection

with and the volume form reads as

On the reference bundle we have a metric on the fibres of

as well as a volume form

For Λ, we choose a linear connection

and for simplicity we only discuss the case .

A motion in the Lagrangian setting is a map with

and the tangent map of is given as

5.3.2. Stress forms

Let us consider the Cauchy stress form

(40)

together with the map that allows to pull back the form part of Equation (40)(40). This leads to the first Piola stress tensor

(41)

The second Piola stress tensor is given as

(42)

and the relation is met. The Cauchy–Green tensor is obtained by pulling back the metric g by the map Therefore, the following quantities are adopted, which do not require the knowledge of Φ. We have

(43)

which means that Then the relation reads .

5.3.3. The Euler–Lagrange equations

We investigate only the first-order Lagrangians, and the variational derivative for this setting looks in coordinates as

Let us consider the density of the kinetic energy

(44)

with the mass density , together with the balance of mass , and the stored energy function which meets

(45)

We use the variational principle

where the body-force density has been added, with

(46)

which consequently leads to

(47)

Using the spatial picture we obtain

(48)

which are partial differential equations in the unknown functions Φ^α and we have and . The boundary expression follows from Equation (10)(10) and includes the expression

where the index i regards only the spatial coordinates. It is easily concluded that the boundary conditions have to be fulfilled by a suitable choice of the second Piola tensor , the deformation gradient and/or conditions on the variational field.

5.3.4. The polysymplectic point of view

Again we have coordinates for that read as and we choose the trivial connection

From the Lagrangian (46), we derive the spatial momenta

(49)

as well as the temporal momenta

and the Hamiltonian can be computed when the relation (49) can be solved with respect to the deformation variables from the expression

It is important to remark that the case of a degenerate Lagrangian can be handled using advanced tools, which can be found for example in [15] where it is shown that the problem of regularity can be treated using a more general equivalence of Lagrangian and Hamiltonian systems not necessarily having the same order of the Lagrangians and the associated Hamiltonians. A different approach where the authors introduce a Lagrangian constrained space, which is the image of the Legendre map and clearly a subset of Π, can be found in [4]. This problem of regularity will not be discussed here and we proceed with the assumption of a non-degenerate Lagrangian such that

(50)

5.3.5. The evolutionary point of view

Let us start by introducing the momentum in the spatial description, which can be given as

(51)

In the material description we obtain

where the symbol of the momentum P should not be confused with the one of the Piola tensor. The total energy is the sum of the kinetic and the stored energy function and consequently the Hamiltonian and reads as

The equations of motion can be written as

and

which are the counterpart to the relations (48). In the set of equations the variational derivatives read as

It is worth mentioning that in the case where the Legendre map only consists of the temporal momenta the problem of regularity is much easier because it depends only on the mass metric tensor.

6. Conclusion

In this contribution, we discussed possible Hamiltonian formulations for field theoretic problems of first order, especially in mechanics. Based on the variational problem in a Lagrangian fashion we motivated the use of two Hamiltonian counterparts, the evolutionary and the polysymplectic approach. We focused on a solid geometric bedding, that enables us to point out the differences of these two formulations from a differential geometric point of view, where the key role is played by the treatment of the independent coordinates and the role of the conserved quantities. Also the role of the boundary conditions has been discussed on the Lagrangian and the Hamiltonian side.

Further directions of research will include beside an analysis of higher- order problems the consideration of the problem of regularity and it is of great interest to apply the proposed methods [4,15] to continuum mechanics.

Acknowledgement

The first author has been supported by the Austrian Center of Competence in Mechatronics (ACCM).