Geometric Mechanics

In 1969, Jean-Marie Souriau has introduced a “Lie Groups Thermo-dynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines thanks to cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dy-namic groups of physics (eg, Galileo's group in classical physics). Souriau model is more general if we consider another Souriau theorem, that we can as-sociate to a Lie group, an homogeneous symplectic manifold with a KKS 2-form on their coadjoint orbits. Souriau method could then be applied on Lie Groups to define a covariant maximum entropy density by Kirillov representa-tion theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant.

This paper introduces and studies a class of optimal control problems based on the Clebsch approach to Euler-Poincaré dynamics. This approach unifies and generalizes a wide range of examples appearing in the literature: the symmetric formulation of Ndimensional rigid body and its generalization to other matrix groups; optimal control for ideal flow using the back-to-label map; the double bracket equations associated to symmetric spaces. New examples are provided such as the optimal control formulation for the N -Camassa-Holm equation and a new geodesic interpretation of its singular solutions.

In 1969, Jean-Marie Souriau introduced a "Lie Groups Thermodynamics" in the framework of Symplectic model of Statistical Mechanics. This Souriau's model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non-null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (eg, Galileo's group in classical physics). Souriau method could then be applied on Lie Groups to define a covariant maximum entropy density by Kirillov representation theory. Based on this model, we will introduce a geometric characterization of Entropy as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping. The dual space of the Lie algebra foliates into coadjoint orbits that are also the level sets on the entropy. The information manifold foliates into level sets of the entropy that could be interpreted in the framework of Thermodynamics by the fact that motion remaining on this complex surfaces is non-dissipative, whereas motion transversal to these surfaces is dissipative. We will explain the 2 nd Principle in thermodynamics by definite positiveness of Souriau tensor extending the (Koszul-)Fisher metric from Information Geometry.

This work studies the symmetries, the associated momentum map, and relative equilibria of a mechanical system consisting of a small axisymmetric magnetic body-dipole in an also axisymmetric external magnetic field that additionally exhibits a mirror symmetry; we call this system the "orbitron". We study the nonlinear stability of a branch of equatorial quasiorbital relative equilibria using the energy-momentum method and we provide sufficient conditions for their T 2 -stability that complete partial stability relations already existing in the literature. These stability prescriptions are explicitly written down in terms of the some of the field parameters, which can be used in the design of stable solutions. We propose new linear methods to determine instability regions in the context of relative equilibria that we use to conclude the sharpness of some of the nonlinear stability conditions obtained.

A nonholonomic mechanical system is a pair (L, D ), where L : TQ → R is a mechanical Lagrangian and D ⊂ TQ is a distribution which is non-integrable (in the Frobenius sense). Although such mechanical systems are manifestly not Hamiltonian (their mechanics are described by the Lagrange-d'Alembert principle, not Hamilton's principle), one can nevertheless attempt to formulate the mechanics of certain classes of nonholonomic systems as almost-Hamiltonian. In this dissertation we study various methods of so-called Hamiltonization of nonholonomic systems and discuss their application to optimal control and the quantization of nonholonomic systems. We begin by constructing second-order associated systems for a class of nonholonomic systems and solving the Inverse Problem of the Calculus of Variations to derive Hamiltonians whose canonical equations, when restricted to certain invariant submanifolds, reproduce the original nonholonomic mechanics. We also introduce the idea of conditionally variational nonholonomic systems, which arise from a comparison with the variational nonholonomic equations, and show that these systems give a straightforward Hamiltonization for certain classes of systems. Lastly, we extend a classical theorem of S. A. Chaplygin, which allows a larger class of nonholonomic systems to be Hamiltonized by reparameterizing time, to higher dimensions. Moreover, in some cases we show that the requirement that the original system possess an invariant measure can be removed. The results are then applied to show that under certain conditions the equations of motion of nonholonomic systems can be derived by considering an associated first-order optimal control problem, similar to the situation in holonomic systems. Moreover, the methods are illustrated throughout by various well known examples of nonholonomic systems. Several future directions based on the research presented are also discussed, among them the relatively new problem of quantizing a nonholonomically constrained system. With the advent of nanomachines we expect the importance of subatomic motions in wheeled robots to raise interest in the classical-quantum equations of motion governing these nonholonomic vehicles. Although there is currently no accepted quantum mechanical treatment of nonholonomic mechanics, we discuss the application of the results of the Hamiltonizations obtained herein to the quantization of a well known nonholonomic mechanical system.

In this paper we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau's symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equiv-ariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits , and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework wit...

We present a new multisymplectic framework for second-order classical field theories which is based on an extension of the unified Lagrangian-Hamiltonian formalism to these kinds of systems. Our model allows us to overcome all the ambiguities inherent to these theories. It therefore provides a straightforward and simple way to define the Poincaré-Cartan form and clarifies the construction of the Legendre map (univocally obtained as a consequence of the constraint algorithm). Likewise, it removes the undesirable arbitrariness in the solutions to the field equations, which are analyzed in-depth, and written in terms of holonomic sections and multivector fields. Our treatment therefore completes previous attempt to achieve this aim. The formulation is applied to describing some physical examples; in particular, to giving another alternative multisymplectic description of the Korteweg-de Vries equation.

By studying the Frölicher-Nijenhuis decomposition of cohomology operators (that is, derivations D of the exterior algebra Ω(M) with Z−degree 1 and D 2 = 0), we describe new examples of Lie algebroid structures on the tangent bundle T M (and its complexification T C M) constructed from preexisting geometric ones such as complex, product or tangent structures. We also study a class of cohomology operators induced by Lie algebroids, giving a simple proof that their cohomology ring is the foliated cohomology determined by the image of the anchor map.

In 1969, Jean-Marie Souriau introduced a "Lie Groups Thermody-namics" in the framework of Symplectic model of Statistical Mechanics. This Souriau's model considers the statistical mechanics of dynamic systems in their "space of evolution" associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non-null cohomology (non equivari-ance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (eg, Galileo's group in classical physics). Souriau method could then be applied on Lie Groups to define a covariant maximum entropy density by Kirillov representation theory. Based on this model, we will introduce a geometric characterization of Entropy as a generalized Cas-imir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping. The dual space of the Lie algebra foliates into coadjoint orbits that are also the level sets on the entropy. The information manifold foliates into level sets of the entropy that could be interpreted in the framework of Thermodynamics by the fact that motion remaining on this complex surfaces is non-dissipative, whereas motion transversal to these surfaces is dissipative. We will explain the 2nd Principle in thermody-namics by definite positiveness of Souriau tensor extending the (Koszul-)Fisher metric from Information Geometry.

This work studies the symmetries, the associated momentum map, and relative equilibria of a mechanical system consisting of a small axisymmetric magnetic body-dipole in an also axisymmetric external magnetic field that additionally exhibits a mirror symmetry; we call this system the "orbitron". We study the nonlinear stability of a branch of equatorial quasiorbital relative equilibria using the energy-momentum method and we provide sufficient conditions for their T 2 -stability that complete partial stability relations already existing in the literature. These stability prescriptions are explicitly written down in terms of the some of the field parameters, which can be used in the design of stable solutions. We propose new linear methods to determine instability regions in the context of relative equilibria that we use to conclude the sharpness of some of the nonlinear stability conditions obtained.

Résumé - En 1969, Jean-Marie Souriau a introduit une "Thermodynamique des Groupes de Lie" dans le cadre du modèle Symplectique de la Mécanique Statistique. Sur la base de ce modèle, nous introduirons une caractérisation géométrique de l'entropie en tant que fonction invariante de Casimir en représentation coadjointe, où le cocycle de Souriau est une mesure de l'absence d'équivariance de l'application moment. L'espace dual de l'algèbre de Lie réalise un feuilletage via les orbites coadjointes ; feuilles symplectiques qui sont également les ensembles de niveaux de l'entropie.  Dans le cadre de la thermodynamique, le mouvement restant sur ces feuilletages est non dissipatif, alors que le mouvement transversal à ces feuilles symplectiques est dissipatif. Le modèle fonde également le 2ème principe de la thermodynamique, liée à définie positivité du tenseur associé à la généralisation de la métrique de Koszul-Fisher de la Géométrie de l'Information. Nous introduirons une nouvelle équation géométrique de la chaleur de Fourier qui s’écrit de façon intrinsèque comme une équation d’Euler-Poincaré. En conclusion, l'entropie en tant que fonction de Casimir est caractérisée par la cohomologie de Poisson introduite par Jean-Louis Koszul.
Abstract - In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in the framework of Symplectic model of Statistical Mechanics. Based on this model, we will introduce a geometric characterization of Entropy as a Casimir invariant function in coadjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment map. The dual space of the Lie algebra foliates into coadjoint orbits that are also the level sets on the entropy. In the framework of Thermodynamics, we interpret that motion remaining on these symplectic leaves is non-dissipative, whereas motion transversal to these leaves is dissipative. We will also explain the 2nd Principle in thermodynamics by definite positiveness of Souriau tensor used to extend the Koszul-Fisher metric from Information Geometry. We introduce a new geometric Fourier heat equation that could be written intrinsically as a Euler-Lagrange equation. In conclusion, Entropy as Casimir function is characterized by Poisson Cohomology introduced by Jean-Louis Koszul.

We present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established.

The multisymplectic formalism of field theories developed by many mathematicians over the last fifty years is extended in this work to deal with manifolds that have boundaries. In particular, we develop a multisymplectic framework for first order covariant Hamiltonian field theories on manifolds with boundaries. This work is a geometric fulfillment of Fock's characterization of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [Ca14]. This framework leads to a true geometric understanding of conventional choices for boundary conditions. For example, the boundary condition that the pull-back of the 1-form on the cotangent space of fields at the boundary vanish, i.e. Π * α = 0 , is shown to be a consequence of our finding that the boundary fields of the theory lie in the 0-level set of the moment map of the gauge group of the theory.

The objective of this chapter is to make better known Jean-Marie Souriau works, more particularly his symplectic model of statistical physics, called "Lie groups thermodynamics". This model was initially described in chapter IV "Statistical Mechanics" of his book "Structure of dynamical sys-tems" published in 1969. We have translated in English some parts of three Souriau's publications which provide more details about this geometric model of Thermodynamics. Entropy acquires a geometric foundation as a function pa-rameterized by mean of moment map in dual Lie algebra, and in term of folia-tions. Souriau established the generalized Gibbs laws when the manifold has a symplectic form and a connected Lie group G operates on this manifold by symplectomorphisms. Souriau Entropy is invariant under the action of the group acting on the homogeneous symplectic manifold. As quoted by Souriau, these equations are universal and could be also of great interest in Mathematics.

Through several years the modified linear model has been proposed for fast and accurate prediction of short and long-range trajectories of spin-stabilized bullets. In this paper the modified linear model is compared with a full 6-DOF trajectory model for 7.62 mm bullet. The computational analysis of the flight takes into consideration i) the Mach number and ii) total angle of attack effects by means that the coefficients of the aerodynamic are variables.

After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we state a unified geometrical version of the Variational Principles which allows us to derive the Lagrangian and Hamiltonian equations for these kinds of systems. Then, the standard Lagrangian and Hamiltonian formulations of these principles and the corresponding dynamical equations are recovered from this unified framework.

The Frank tensor plays a crucial role in linear elasticity, and in particular in the presence of dislocation lines, since its curl is exactly the elastic strain incompatibility. Furthermore, the Frank tensor also appears in Cesaro decomposition, and in Volterra theory of dislocations and disclinations, since its jump is the Frank vector around the defect line. The purpose of this paper is to show to which functional space the compatible strain e belongs in order to imply a homogeneous boundary conditions for the induced displacement field on a portion Γ 0 of the boundary. This will allow one to define the homogeneous, or even the mixed problem of linearized elasticity in a variational setting involving the strain e in place of displacement u. With other puposes, this problem was originaly treated by Ph. Ciarlet and C. Mardare, and termed the intrinsic formulation. In this paper we propose alternative conditions on e expressed in terms of e and the Frank tensor Curl t e only, yielding a clear physical understanding and showing as equivalent to Ciarlet-Mardare boundary condition.

We obtain the affine Euler-Poincaré equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation. Contents 1 Introduction 2 2 The affine Euler-Poincaré Equations via Lagrangian reduction 3 3 Continuum spin system 13 4 Elastic filament dynamics and Kirchhoff's theory 18 4.

We consider various notions of strains-quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of [1], is to select a Riemannian metric on GL n , and use its induced geodesic distance to measure the distance of a linear transformation from the set of isometries. We give a short geometric derivation of the formula for the strain measure for the case where the metric is left-GL n-invariant and right-O n-invariant. We proceed to investigate alternative distance functions on GL n , and the properties of their induced strain measures. We start by analyzing Euclidean distances, both intrinsic and extrinsic. Next, we prove that there are no bi-invariant distances on GL n. Lastly, we investigate strain measures induced by inverse-invariant distances.

This paper deals with Jean-Louis Koszul's works related to Geometric and Analytic Mechanics, and to Souriau's Lie Group Thermodynamics that have appeared over time as elementary structures of Information Geometry. The 2 nd Koszul form has been extended by Jean-Marie Souriau in his Symplectic model of Statistical Physics called "Lie Groups Thermodynamics" providing an extension of Fisher metric for homogeneous Symplectic manifolds, associated to KKS (Kirillov-Kostant-Souriau) 2-form in case of non-null cohomology. Jean-Louis Koszul has developed mathematical foundation of Souriau model in the Lecture "Introduction to Symplectic Geometry".

We study a family of approximations to Euler's equation depending on two parameters ε, η ≥ 0. When ε = η = 0 we have Euler's equation and when both are positive we have instances of the class of integro-differential equations called EPDiff in imaging science. These are all geodesic equations on either the full diffeomorphism group Diff H ∞ (R n ) or, if ε = 0, its volume preserving subgroup. They are defined by the right invariant metric induced by the norm on vector fields given by

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of k-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of k-precosymplectic structure, which is a generalization of the k-cosymplectic structure. Next k-precosymplectic Hamiltonian systems are introduced in order to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a submanifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example.

We explicitly compute the semi-global symplectic invariants near the hyperbolic equilibrium point of the Euler top. The Birkhoff normal form at this point is computed using Lie series. The Picard-Fuchs equation for the action near the unstable equilibrium is derived. Using the method of Frobenius on the Pichard-Fuchs equation we show that the Birkhoff normal form can also be found by inverting the Frobenius series of the regular action integral. Composition of the regular action integral with the singular action integral leads to the symplectic invariant. To our knowledge this is the first time that such invariants near a hyperbolic point have been computed explicitly using the Picard-Fuchs equation. Finally we discuss the convergence of these invariants using both analytical and numerical arguments.

A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces.

This paper studies the mechanics of undulatory locomotion. This type of locomotion is generated by a coupling of internal shape changes to external nonholonomic constraints. Employing methods from geometric mechanics, we use the dynamic symmetries and kinematic constraints to develop a specialised form of the dynamic equations which govern undulatory systems. These equations are written in terms of physically meaningful and intuitively appealing variables that show the role of internal shape changes in driving locomotion. the geometry of amoeba swimming through a viscous fluid (see ). However, to date there exists no unifying methodology for analysing or controlling robot locomotion. Ultimately, we seek a mechanics theory and a control theory for robotic locomotion which is uniformly applicable to a broad class of locomotory problems. This paper introduces a unifying mechanics principle for undulatory locomotion.

The use of solar chimneys for energy production was suggested more than 100 years ago. Unfortunately, this technology has not been realized on a commercial scale, in large part due to the high cost of erecting tall towers using traditional methods of construction. Recent works have suggested a radical decrease in tower cost by using an inflatable self-supported tower consisting of stacked toroidal bladders. While the statics deflections of such towers under constant wind have been investigated before, the key for further development of this technology lies in the analysis of dynamics, which is the main point of this paper. Using Lagrangian reduction by symmetry, we develop a fully three-dimensional theory of motion for such towers and study the tower's stability and dynamics. Next, we derive a geometric theory of optimal control for the tower dynamics using variable pressure inside the bladders and perform detailed analytical and numerical studies of the control in two dimensions. Finally, we report on the results of experiments demonstrating the remarkable stability of the tower in real-life conditions, showing good agreement with theoretical results.

In this paper we have obtained some dynamics equations, in the presence of nonlinear nonholonomic constraints and according to a lagrangian and some Chetaev-like conditions. Using some natural regular conditions, a simple form of these equations is given. In the particular cases of linear and affine constraints, one recovers the classical equations in the forms known previously, for example, by Bloch and all [3, 4]. The case of time-dependent constraints is also considered. Examples of linear constraints, time independent and time depenndent nonlinear constraints are considered, as well as their dynamics given by suitable lagrangians. All examples are based on classical ones, such as those given by Appell's machine.

The objective of this work is twofold: First, we analyze the relation between the kcosymplectic and the k-symplectic Hamiltonian and Lagrangian formalisms in classical field theories. In particular, we prove the equivalence between k-symplectic field theories and the so-called autonomous k-cosymplectic field theories, extending in this way the description of the symplectic formalism of autonomous systems as a particular case of the cosymplectic formalism in non-autonomous mechanics. Furthermore, we clarify some aspects of the geometric character of the solutions to the Hamilton-de Donder-Weyl and the Euler-Lagrange equations in these formalisms. Second, we study the equivalence between k-cosymplectic and a particular kind of multisymplectic Hamiltonian and Lagrangian field theories (those where the configuration bundle of the theory is trivial).

On a Poisson foliation equipped with a canonical and cotangential action of a compact Lie group, we describe the averaging method for Poisson connections. In this context, we generalize some previous results on Hannay-Berry connections for Hamiltonian and locally Hamiltonian actions on Poisson fiber bundles. Our main application of the averaging method for connections is the construction of invariant Dirac structures parametrized by the 2-cocycles of the de Rham-Casimir complex of the Poisson foliation.

The concept of superposition rule for second-order differential equations is stated and conditions ensuring the existence of such superposition rules are analysed. In this way, second-order differential equations become formally included within the theory of Lie systems. The theory is illustrated by analysing the properties of a family of second-order differential equations with applications to Physics and we obtain a superposition rule common for all its members. Finally, time-dependent superposition rules for second-order differential equations are defined and we derive a particular instance for a family of second-order Riccati equations by means of the theory of quasi-Lie schemes.

In this paper we look at the question of integrability, or not, of the two natural almost complex structures \begin{document}$ J^{\pm}_\nabla $\end{document} defined on the twistor space \begin{document}$ J(M, g) $\end{document} of an even-dimensional manifold \begin{document}$ M $\end{document} with additional structures \begin{document}$ g $\end{document} and \begin{document}$ \nabla $\end{document} a \begin{document}$ g $\end{document}-connection. We measure their non-integrability by the dimension of the span of the values of \begin{document}$ N^{J^\pm_\nabla} $\end{document}. We also look at the question of the compatibility of \begin{document}$ J^{\pm}_\nabla $\end{document} with a natural closed \begin{document}$ 2 $\end{document}-form \begin{document}$ \omega^{J(M, g, \nabla)} $\end{document} defined on \begin{document}$ J(M, g) $\end{document}. For \begin{document}$ (M, g) $\end{document} we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civi...

The aim of this paper is to write explicit expression in terms of a given principal connection of the Lagrange-d'Alembert-Poincaré equations in several stages. This is obtained by using a reduced Lagrange-d'Alembert's Principle in several stages, extending methods introduced for the case of two stages by one of the authors and collaborators. The case of the Euler's disk is described as an illustrative example.

In this paper we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau's symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equiv-ariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits , and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multi-variate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplec-tic and multisymplectic variational Lie group integration schemes for some of the equations associated to Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

We present a new multisymplectic framework for second-order classical field theories which is based on an extension of the unified Lagrangian-Hamiltonian formalism to these kinds of systems. Our model allows us to overcome all the ambiguities inherent to these theories. It therefore provides a straightforward and simple way to define the Poincaré-Cartan form and clarifies the construction of the Legendre map (univocally obtained as a consequence of the constraint algorithm). Likewise, it removes the undesirable arbitrariness in the solutions to the field equations, which are analyzed in-depth, and written in terms of holonomic sections and multivector fields. Our treatment therefore completes previous attempt to achieve this aim. The formulation is applied to describing some physical examples; in particular, to giving another alternative multisymplectic description of the Korteweg-de Vries equation.