The program of the school includes series of main lectures, workshop and poster session. The following programme is tentative and is updated regularly.

The preliminary timetable of the Scientific Program: PDF

### Courses

**A) Arjan van der Schaft: Geometric modeling and control of multi-physics systems**

Abstract.

In this course we will be concerned with the geometric formulation, analysis and control of finite-dimensional physical systems stemming from different physical domains, with an emphasis on mechanical and electro-mechanical systems. The basic starting point will be the conceptual reticulation of the physical system into energy-storing, energy-routing and energy-dissipating elements. This will lead to the definition of a state space manifold with total energy given by a Hamiltonian function, together with a geometric structure on the state space manifold known as a Dirac structure. The notion of a Dirac structure generalizes the notions of symplectic and Poisson structure known from the geometric theory of Hamiltonian dynamics. Interestingly, while symplectic structures in geometric mechanics normally derive from the canonical symplectic structure on the cotangent bundle of the configuration space, the Dirac structure emphasizes the interconnection structure of the reticulated physical system. In the first part of the course we will deal with the geometric theory of Dirac structures, and the resulting class of dynamical systems known as port-Hamiltonian systems, generalizing classical Hamiltonian dynamics in a number of directions. This will be illustrated by a list of examples such as mechanical systems with kinematic constraints, and electro-mechanical systems including synchronous machines and power networks. In the second part of the course we will further analyze Dirac structures and the dynamical properties of port-Hamiltonian systems, including integrability of Dirac structures, symmetries and Casimirs, Lyapunov function theory, and physical network dynamics. Furthermore, we will use the developed theory as a starting point for control. If time permits we will also discuss extensions of the theory of port-Hamiltonian systems to thermodynamical systems, using contact geometry.

The course will use an introductory set of lecture notes on port-Hamiltonian systems, which will be made available during the course.

**B) Ekkehart Winterroth: Field theory: Variational sequences and bicomplexes**

Abstract.

In 1936 Lepage proposed a formulation of the calculus of variations by decomposing the exterior differential of a Lagrangian into Euler-Lagrange form plus „congruences“ (what we call now contact forms). This can be considered the starting point of the modern cohomological formulations of the calculus of variations with which we will deal in this series of lectures. We will start by presenting the evolution of Lepage's ideas using the modern concepts of contact structures of jet bundles and (jet prolongations of) sections of fibered manifolds (fibre bundles). This leads then naturally to the general construction principle „de Rham complex modulo contact forms“ of variational cohmology theories. Along the way we will also introduce the necessary concepts and results from differential geometry and homological algebra.

Instead of studying immediately explicit realisations of this construction principle, we will simply assume a variational cohmology theory to be given and study the general properties of such a theory. In particular, we will study the local and global inverse problem, Noether theorems and cohomological obstructions to the existence of critical sections.

In the final part, we will introduce two explicit constructions of variational cohmology -Krupka's Variational Sequence (on finite order jet bundles) and the Variational Bicomplex- and study some of their features.

### Workshop

The goal is to present recent results in differential geometry, geometric control theory, and applications. Presentations of posters are also possible. The concrete workshop program of oral contributions will be scheduled with respect to the number of registered talks.

**Workshop talks**

L. Accornero and M. Palese, Higher order variations, conservation laws, and the Jacobi equations for Yang-Mills Lagrangians on a Minkowskian background (poster)

Praveen Agarwal, Study of extended fractional derivative operators in a complex domain and their applications

Rosalio Jr. Artes, Bernstein-Szego Inequality on Hardy Space

Bogdan Balcerzak, Dirac type operators on symmetric tensors

Arindam Bhattacharyya, Homology of a type of Heptahedron

Smail Chemikh, On the existence and nonexistence of global signchanging solutions on riemannian manifolds

Ju Chen, Geometric optimal control of robot manipulators

Santu Dey, Conformal Mappings of Mixed-Generalized Quasi-Einstein Manifolds Admitting Special Vector Fields

Kamel Djeddi, Chaos-based of RLC circuit

Shan Gao, Geometric Optimal Control of Robot Manipulators

Fazia Harrache, L^1 contact metrics in dimension 3 (Poster)

Seoung D. Jung, Basic Dolbeault cohomology on transversely Kahler foliations

Young Ho Kim, Ellipic Linear Werigarten Surfaces of 3-spheres

Demeter Krupka: On variational principles in Finsler geometry

Álvaro P. Llarena, On Extremals of Curvature Energies used in Visual Curve Completion

Jae W. Lee, Seiberg-Witten invariants on 3-manifolds with an orientation-reversing involution

Shixing Liu, The nonlinear dynamics based on the nonstandard Hamiltonians

Michael Orieux, Pi-singularities in minimum time control for space mechanics

Marcella Palese, Higgs fields induced by Yang--Mills type Lagrangians on gauge-natural

prolongations of principal bundles

Dhriti S. Patra, The CPE conjecture on K-contact manifold

Alberto Roncoroni, Domain rigidity for some overdetermined elliptic boundary value problems

Nina Rutten, Flows on the space of Poisson structures

Sarita Singh, Multi-Scale Modelling of Carotid Artery

Zbynek Urban, Jan Brajercik: First-order variational field theory for submanifolds

Ben L. Wang, Motion Planning of Industrial Robots: Geometric Mechanics Method

Dmitry Zenkov, Midpoint Rule as a Structure-Preserving Integrator (Abstract)

### Social program

On Friday (August 25) the organizers plan to arrange the following excursions:

**a)** “Wieliczka“ Salt Mine (http://www.wieliczka-saltmine.

**b)** Hiking in the Tatra Mountains (the hiking boots and warm clothes are recommended to take with you)

**c)** Sightseeing walking tour of Old Town of Krakow

On Monday (August 21) evening, a *conference dinner* will take place in the Old Town.