Hamilton-Jacobi Theory in k-Symplectic Field Theories

In this paper we extend the geometric formalism of Hamilton-Jacobi theory for Mechanics to the case of classical field theories in the k-symplectic framework

Geometric aspects of nonholonomic field theories

A geometric model for nonholonomic Lagrangian field theory is studied. The multisymplectic approach to such a theory as well as the corresponding Cauchy formalism are discussed. It is shown that in both formulations, the relevant equations for the constrained system can be recovered by a suitable projection of the equations for the underlying free (i.e. unconstrained) Lagrangian system.Comment: 29 pages; typos remove

Nonholonomic constraints in kk-symplectic Classical Field Theories

A $k$-symplectic framework for classical field theories subject to nonholonomic constraints is presented. If the constrained problem is regular one can construct a projection operator such that the solutions of the constrained problem are obtained by projecting the solutions of the free problem. Symmetries for the nonholonomic system are introduced and we show that for every such symmetry, there exist a nonholonomic momentum equation. The proposed formalism permits to introduce in a simple way many tools of nonholonomic mechanics to nonholonomic field theories.Comment: 27 page

On the Hamilton-Jacobi Theory for Singular Lagrangian Systems

We develop a Hamilton-Jacobi theory for singular lagrangian systems using the Gotay-Nester-Hinds constraint algorithm. The procedure works even if the system has secondary constraints.Comment: 36 page

Hamiltonian dynamics and constrained variational calculus: continuous and discrete settings

The aim of this paper is to study the relationship between Hamiltonian dynamics and constrained variational calculus. We describe both using the notion of Lagrangian submanifolds of convenient symplectic manifolds and using the so-called Tulczyjew's triples. The results are also extended to the case of discrete dynamics and nonholonomic mechanics. Interesting applications to geometrical integration of Hamiltonian systems are obtained.Comment: 33 page

Symmetries in Classical Field Theory

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.Comment: 70S05; 70H33; 55R10; 58A2

Optimal Control of Underactuated Mechanical Systems: A Geometric Approach

In this paper, we consider a geometric formalism for optimal control of underactuated mechanical systems. Our techniques are an adaptation of the classical Skinner and Rusk approach for the case of Lagrangian dynamics with higher-order constraints. We study a regular case where it is possible to establish a symplectic framework and, as a consequence, to obtain a unique vector field determining the dynamics of the optimal control problem. These developments will allow us to develop a new class of geometric integrators based on discrete variational calculus.Comment: 20 pages, 2 figure

The influence of cell size on the mechanical properties of nanocellular PMMA

Solid-state foaming experiments are conducted on three grades of polymethyl methacrylate (PMMA). Nanocellular PMMA foams are manufactured with an average cell size ranging from 20 nm to 84 nm and a relative density between 0.37 and 0.5. For benchmarking purposes, additional microcellular PMMA foams with an average cell size close to 1 µm and relative density close to that of the nanocellular foams are manufactured. Uniaxial compression tests and single edge notch bend tests are conducted on the PMMA foams. The measured Young’s modulus and yield strength of the PMMA foams are independent of cell size whereas the fracture toughness of the PMMA foam increases with decreasing average cell size from the micron range to the nanometer range.Financial assistance from MINECO, FEDER, UE (MAT2015-69234-R), the Junta of Castile and Leon (VA275P18) and Spanish Ministry of Science, Innovation and Universities (RTI2018-098749-B-I00) are gratefully acknowledged. Financial support from FPU grant FPU14/02050 (V. Bernardo) from the Spanish Ministry of Education and Junta of Castile and Leon grant (J. Martín-de León) are gratefully acknowledged. Financial support from SABIC and the EPSRC award 1611305 (F. Van Loock), and the ERC project MULTILAT (N. A. Fleck) are acknowledged too

Unambiguous Formalism for Higher-Order Lagrangian Field Theories

The aim of this paper is to propose an unambiguous intrinsic formalism for higher-order field theories which avoids the arbitrariness in the generalization of the conventional description of field theories, which implies the existence of different Cartan forms and Legendre transformations. We propose a differential-geometric setting for the dynamics of a higher-order field theory, based on the Skinner and Rusk formalism for mechanics. This approach incorporates aspects of both, the Lagrangian and the Hamiltonian description, since the field equations are formulated using the Lagrangian on a higher-order jet bundle and the canonical multisymplectic form on its dual. As both of these objects are uniquely defined, the Skinner-Rusk approach has the advantage that it does not suffer from the arbitrariness in conventional descriptions. The result is that we obtain a unique and global intrinsic version of the Euler-Lagrange equations for higher-order field theories. Several examples illustrate our construction.Comment: 21 pages; 4 diagrams; (this version) corrected typos; moved paragraphs; publishe

The entangling side of the Unruh-Hawking effect

We show that the Unruh effect can create net quantum entanglement between inertial and accelerated observers depending on the choice of the inertial state. This striking result banishes the extended belief that the Unruh effect can only destroy entanglement and furthermore provides a new and unexpected source for finding experimental evidence of the Unruh and Hawking effects.Comment: 4 pages, 4 figures. Added Journal referenc