## Final Report Summary - GEOMECH (Geometric Mechanics)

The IRSES project GEOMECH is concerned with fundamental, mathematical research in a broad field, called Geometrical Mechanics, which consists of a mixture of (applied) differential geometry, the theory of dynamical systems and theoretical mechanics, with applications in the physical and engineering sciences. The project was centered around the following four general items, called Work Packages (WP’s), among which there are inevitable points of overlap: (WP 1) Geometric structures in mechanics and field theory; (WP 2) Nonholonomic mechanics and geometric control theory; (WP 3) Geometry of second-order differential equations and the calculus of variations; (WP 4) Geometric integration techniques.

For this project, a close collaboration has been set up between the following partners: UGent (Ghent University, Belgium), ICMAT/CSIC (Spain), OU (University of Ostrava, Czech Republic), UNS (Universidad Nacional del Sur, Argentina), Waseda University (Japan), UCSD (University of California at San Diego, USA), La Trobe University (Bundoora, Australia). The public website of GEOMECH is

http://www.geomechnetwork.ugent.be/

We now give a comprehensive description of the work that has been carried out and the main results that have been obtained, by which most of the project’s objectives have been achieved.

WP 1: Geometric structures in mechanics and field theory

The collaboration between various partners (UGent, CSIC, UCSD, UNS and Waseda) has led to results on the following topics: Lagrangian and Hamitonian systems on Lie algebroids; poly-Poisson structures; the geometry of Dirac structures and the study of symmetry and reduction of dynamical systems in terms of these structures; the introduction of a notion of multi-Dirac structure and its relevance in classical field theories; the geometry of continuous and discrete systems with constraints; Routh reduction by stages; study of a multisymplectic formulation of Hamilton-Poincaré field equations and its application to the theory of minimal surfaces; Hamilton-Jacobi equations in the framework of contact structures appearing in thermodynamical processes, and conditions guaranteeing complete integrability of the corresponding Hamilton equations. Progress has been made and some preliminary results are obtained with respect to: an extension of Routh reduction for Lagrangian systems without a priori regularity conditions, leading to implicit Lagrange-Routh equations interpreted in terms of an associated Dirac structure; an interpretation of higher-order field theories in terms of Lagrangian submanifolds of multisymplectic manifolds; the role of Dirac structures in connection with Morse families; study of a reduction scheme for field theories, based on Tulczyjew triples.

WP 2: Nonholonomic mechanics and geometric control theory

Also with respect to this work package there has been a growing interaction between several partners, in particular between UGent, CSIC, UNS, UCSD. Among the main achievements we mention the following: a description of the hydrodynamical behavior of a sleigh immersed in a potential flow with circulation; a noholonomic version of the classical Hamilton-Jacobi theorem, allowing the study of the integrability of some concrete nonholonomic systems, and an extension of the technique so developed to nonholonomic control systems; the development of a method for stabilizing underactuated mechanical systems by imposing kinematic constraints; a characterization of accessibility of affine connection control systems using differential geometric constraints; a geometrical description of optimal control problems in terms of Morse families in the Hamiltonian framework. In addition, contributions by the OU partner have led to results concerning the inverse variational problem for nonholonomic systems; some aspects of the search for conservation laws of mechanical systems with nonholonomic constraints; the study of nonconservative nonholonomic systems and the treatment of relativistic mechanics in a nonholonomic setting.

WP 3: Geometry of 2nd order differential equations and the calculus of variations

This work package has mainly benefited from a collaboration between UGent, OU and La Trobe. Subclasses have been found of non-autonomous second-order ODEs, of general dimension n, for which the inverse problem has a non-unique solution. Considerable progress has been made with respect to the projective Finsler metirizability problem and several related issues have been addressed, such as Jacobi fields and the equivalence of path geometries and projective equivalence classes of sprays. G-metrizability, which refers to the construction of a G-invariant metric field for a symmetric affine connection which is invariant with respect to a group action, has been adequately resolved for the case that G = SO(3). A study has been devoted to the double vector bundle structure of the manifold of double velocities, which has been shown to be mirrored by a structure of holonomic and semi-holonomic subgroups in the principal prolongation of the first jet group. Results were also obtained regarding, among others, the decoupling of systems of second-order ODE’s, providing an intrinsic geometrical description of so-called cofactor systems, the tangent bundle decomposition for systems of second-order PDEs and the study of `shape maps’ for PDEs. The CSIC and UNS partners, on the other hand, have developed a generalized variational calculus for systems with and without constraints, and its extension to the setting of Lie groupoids for the treatment of discrete systems.

WP 4: Geometric integration techniques

Most of the results related to this work package were realized by the CSIC, UNS and UCSD partners. Some of the main results are the following: the construction of a geometric numerical integrator for point vortex dynamics on the sphere, which is second-order accurate and symplectic; a discretization of nonholonomic systems using a discrete version of Hamel’s principle; development of a geometric nonholonomic integrator; construction of exact discrete Lagrangians for higher-order Lagrangian systems, appearing e.g. in applications of trajectory planning and optimal control problems; the design of symplectic-momentum geometric integrators for higher-order variational systems. Finally, it has also been shown that Lagrangian submanifolds of symplectic groupoids may give rise to discrete dynamical systems.

The project has resulted in 68 papers which have already been published in journals with peer reviewing system (and are uploaded in ECAS under “publications”), and approximately 30 papers which appeared in conference proceedings or are already accepted (or at least submitted) for publication. For the complete list: see the public website of GEOMECH.

For this project, a close collaboration has been set up between the following partners: UGent (Ghent University, Belgium), ICMAT/CSIC (Spain), OU (University of Ostrava, Czech Republic), UNS (Universidad Nacional del Sur, Argentina), Waseda University (Japan), UCSD (University of California at San Diego, USA), La Trobe University (Bundoora, Australia). The public website of GEOMECH is

http://www.geomechnetwork.ugent.be/

We now give a comprehensive description of the work that has been carried out and the main results that have been obtained, by which most of the project’s objectives have been achieved.

WP 1: Geometric structures in mechanics and field theory

The collaboration between various partners (UGent, CSIC, UCSD, UNS and Waseda) has led to results on the following topics: Lagrangian and Hamitonian systems on Lie algebroids; poly-Poisson structures; the geometry of Dirac structures and the study of symmetry and reduction of dynamical systems in terms of these structures; the introduction of a notion of multi-Dirac structure and its relevance in classical field theories; the geometry of continuous and discrete systems with constraints; Routh reduction by stages; study of a multisymplectic formulation of Hamilton-Poincaré field equations and its application to the theory of minimal surfaces; Hamilton-Jacobi equations in the framework of contact structures appearing in thermodynamical processes, and conditions guaranteeing complete integrability of the corresponding Hamilton equations. Progress has been made and some preliminary results are obtained with respect to: an extension of Routh reduction for Lagrangian systems without a priori regularity conditions, leading to implicit Lagrange-Routh equations interpreted in terms of an associated Dirac structure; an interpretation of higher-order field theories in terms of Lagrangian submanifolds of multisymplectic manifolds; the role of Dirac structures in connection with Morse families; study of a reduction scheme for field theories, based on Tulczyjew triples.

WP 2: Nonholonomic mechanics and geometric control theory

Also with respect to this work package there has been a growing interaction between several partners, in particular between UGent, CSIC, UNS, UCSD. Among the main achievements we mention the following: a description of the hydrodynamical behavior of a sleigh immersed in a potential flow with circulation; a noholonomic version of the classical Hamilton-Jacobi theorem, allowing the study of the integrability of some concrete nonholonomic systems, and an extension of the technique so developed to nonholonomic control systems; the development of a method for stabilizing underactuated mechanical systems by imposing kinematic constraints; a characterization of accessibility of affine connection control systems using differential geometric constraints; a geometrical description of optimal control problems in terms of Morse families in the Hamiltonian framework. In addition, contributions by the OU partner have led to results concerning the inverse variational problem for nonholonomic systems; some aspects of the search for conservation laws of mechanical systems with nonholonomic constraints; the study of nonconservative nonholonomic systems and the treatment of relativistic mechanics in a nonholonomic setting.

WP 3: Geometry of 2nd order differential equations and the calculus of variations

This work package has mainly benefited from a collaboration between UGent, OU and La Trobe. Subclasses have been found of non-autonomous second-order ODEs, of general dimension n, for which the inverse problem has a non-unique solution. Considerable progress has been made with respect to the projective Finsler metirizability problem and several related issues have been addressed, such as Jacobi fields and the equivalence of path geometries and projective equivalence classes of sprays. G-metrizability, which refers to the construction of a G-invariant metric field for a symmetric affine connection which is invariant with respect to a group action, has been adequately resolved for the case that G = SO(3). A study has been devoted to the double vector bundle structure of the manifold of double velocities, which has been shown to be mirrored by a structure of holonomic and semi-holonomic subgroups in the principal prolongation of the first jet group. Results were also obtained regarding, among others, the decoupling of systems of second-order ODE’s, providing an intrinsic geometrical description of so-called cofactor systems, the tangent bundle decomposition for systems of second-order PDEs and the study of `shape maps’ for PDEs. The CSIC and UNS partners, on the other hand, have developed a generalized variational calculus for systems with and without constraints, and its extension to the setting of Lie groupoids for the treatment of discrete systems.

WP 4: Geometric integration techniques

Most of the results related to this work package were realized by the CSIC, UNS and UCSD partners. Some of the main results are the following: the construction of a geometric numerical integrator for point vortex dynamics on the sphere, which is second-order accurate and symplectic; a discretization of nonholonomic systems using a discrete version of Hamel’s principle; development of a geometric nonholonomic integrator; construction of exact discrete Lagrangians for higher-order Lagrangian systems, appearing e.g. in applications of trajectory planning and optimal control problems; the design of symplectic-momentum geometric integrators for higher-order variational systems. Finally, it has also been shown that Lagrangian submanifolds of symplectic groupoids may give rise to discrete dynamical systems.

The project has resulted in 68 papers which have already been published in journals with peer reviewing system (and are uploaded in ECAS under “publications”), and approximately 30 papers which appeared in conference proceedings or are already accepted (or at least submitted) for publication. For the complete list: see the public website of GEOMECH.