Jordi Gaset

This work presents a comprehensive overview of three recently developed geometric frameworks for the study of classical action-dependent field theories. Specifically, we introduce the three underlying geometric structures, namely [ ]-contact, [ ]-cocontact and multicontact, and we use them to develop the Lagrangian and Hamiltonian formalisms for such theories. Finally, we analyze the relationships among these three structures in the case of trivial bundles, and we compare them with alternative definitions of multicontact structures found in the literature.

A method to construct a geometric structure with the same solutions as a given variational principle is presented. The method applies to large families of variational principles. In particular, the known results that assign cosymplectic geometry to Hamilton’s principle and cocontact geometry to Herglotz’s principle for regular Lagrangians are recovered. The unified Lagrangian–Hamiltonian formalism is also recovered via the absorption of the holonomy conditions. The method is applied to singular time-dependent Lagrangians, proving that they can always be described with a (pre)cosymplectic structure, although it is not always given by the Lagrangian [ ]-form. When applied to singular action-dependent Lagrangians, the method does not always lead to (pre)cocontact geometry. In these cases, the resulting geometry associated with Herglotz’s variational principle is new.

A method to construct a geometric structure with the same solutions as a given variational principle is presented. The method applies to large families of variational principles. In particular, the known results that assign cosymplectic geometry to Hamilton's principle and cocontact geometry to Herglotz's principle for regular Lagrangians are recovered. The unified Lagrangian-Hamiltonian formalism is also recovered via the absorption of the holonomy conditions. The method is applied to singular time-dependent Lagrangians, proving that they can always be described with a (pre)cosymplectic structure, although it is not always given by the Lagrangian $2$-form. When applied to singular action-dependent Lagrangians, the method does not always lead to (pre)cocontact geometry. In these cases, the resulting geometry associated with the Herglotz's variational principle is new.

This work presents a comprehensive overview of three recently developed geometric frameworks for the study of classical action-dependent field theories. Specifically, the three underlying geometric structures - namely, k-contact, k-cocontact, and multicontact - are first introduced, and then used to develop the Lagrangian and Hamiltonian formalisms of the aforementioned theories. Finally, the relationship among these three types of structures is analyzed in the case of trivial bundles; as well as the comparison with other alternative definitions of multicontact structure presented in the literature.

In the recent years, with the incorporation of contact geometry, there has been a renewed interest in the study of dissipative or non-conservative systems in physics and other areas of applied mathematics. The equations arising when studying contact Hamiltonian systems can also be obtained via the Herglotz variational principle. The contact Lagrangian and Hamiltonian formalisms for mechanical systems has also been generalized to field theories. The main goal of this paper is to develop a generalization of the Herglotz variational principle for first-order and higher-order field theories. In order to illustrate this, we study three examples: the damped vibrating string, the Korteweg–de Vries equation, and an academic example showing that the non-holonomic and the vakonomic variational principles are not fully equivalent.

We present the covariant multisymplectic formalism for the so-called cubic Horndeski theories and discuss the geometrical and physical interpretation of the constraints that arise in the unified Lagrangian-Hamiltonian approach. We analyse in more detail the covariant Hamiltonian formalism of these theories and we show that there are particular conditions that must be satisfied for the Poincaré-Cartan form of the Lagrangian to project onto J ¹ π. From this result, we study when a formulation using only multimomenta is possible. We further discuss the implications of the general case, in which the projection onto J ¹ π conditions are not met.

In contact Hamiltonian systems, the so‐called dissipated quantities are akin to conserved quantities in classical Hamiltonian systems. In this article, a Noether's theorem for non‐autonomous contact Hamiltonian systems is proved, characterizing a class of symmetries which are in bijection with dissipated quantities. Other classes of symmetries which preserve (up to a conformal factor) additional structures, such as the contact form or the Hamiltonian function, are also studied. Furthermore, making use of the geometric structures of the extended tangent bundle, additional classes of symmetries for time‐dependent contact Lagrangian systems are introduced. The results are illustrated with several examples. In particular, the two‐body problem with time‐dependent friction is presented, which could be interesting in celestial mechanics.

This paper proposes a method for quantitative colour analysis aimed at determining colour variation between different sets of films with a common selection criterion. The process is exemplified by comparing films selected at the Cannes Film Festival and the highest-grossing films in the years 1988 and 2018. The method will try to answer several questions. Firstly, whether there are any differences between the films competing in Cannes Film Festival and those most watched by audiences. Secondly, if these differences exist in both the year 1988 and 2018. Thirdly, if the colour of films exhibited at Cannes and highest-grossing film have evolved during the 30-year difference between samples. To analyse these films, one image every 14 s is extracted and all frames are used to quantify the amount of each colour present. Values for each category undergo a Kolmogorov–Smirnov test to determine whether the distribution of each colour is different for both categories compared. The relative difference between colours is computed to find out if those colours where variations are present have increased or decreased their presence within those sets compared. This methodology enables film researchers to statistically prove that there are colour changes in those films included in their research.

We derive the equations of motion of an action-dependent version of the Einstein-Hilbert Lagrangian, as a specific instance of the Herglotz variational problem. Action-dependent Lagrangians lead to dissipative dynamics, which cannot be obtained with the standard method of Lagrangian field theory. First-order theories of this kind are relatively well understood, but examples of singular or higher-order action-dependent field theories are scarce. This work constitutes an example of such a theory. By casting the problem in clear geometric terms we are able to obtain a Lorentz invariant set of equations, which contrasts with previous attempts.