## Learning objectives

Knowledge and understanding: the student should acquire the knowledge of foundations of Analytical Mechanics viewed as a branch of Mathematical Physics, with a deep understanding of the applications of mathematical methods to the study of physical problems. Moreover, the student should become able to read and understand advanced text of Analytical Mechanics and Mathematical Physics.

Applying knowledge and understanding: the student must become able to produce formal proofs of results of Analytical Mechanics and Mathematical Physics, and to expose, analyze and solve simple problems with a clear mathematical formulation.

Making judgements: the student must become able to construct, develop and apply theoretical reasoning in the context of Analytical Mechanics and Mathematical Physics, with the ability to distinguish correct and wrong assumptions and methods.

Communication skills: the student must acquire the correct terminology and language of Analytical Mechanics and Mathematical Physics and the ability to expose their results and techniques to a qualified audience.

Learning skills: the student must become able to autonomously continue the study of Analytical Mechanics, Mathematical Physics and in general to complete his preparation in Mathematics or in other scientific field with an open minded approach, and must become able to gain knowledge from specialized text and journals.

## Prerequisites

Basic calculus, algebra and geometry of the first and second year courses; mandatory propedeuticities: Rational Mechanics

## Course unit content

PART I

The first part of course aims at providing the students with the coordinate independent formulation of some concepts of Analytical Mechanics, such as dinamical flow, Poincaré-Cartan forms, Poisson brackets and canonical transformations.

PART II

The second part of the course aims at providing the students with the main analytical and numerical tools of the optimal control theory for finite horizon problems. Some applications to biology and population dynamics will be presented and discussed.

## Full programme

PART I

Differentiable manifolds and maps. Tangent and cotangent bundles, vector fields, differential 1-forms and exterior algebra, tensorial bundles, exterior differential of forms, interior product of vector fields and differential forms.

Flow associated to a vector field, 1-parameter groups of diffeomorphisms, congruences of lines and relations between them.

Lie derivative associated to a vector field and its relation between vector fields commutator.

Lagrangian dynamics: configuration space-time and its tangent bundle, Lagrangian function and dynamical flow, time derivative, Lagrange equations and their invariance, Poincaré-Cartan 1 and 2 forms.

Hamiltonian dynamics: cotangent bundle of configuration space-time and phase space, symplectic structure, Poisson brackets, Hamiltonian function, time derivative, Hamilton equations and their invariance, canonical transformation, first integrals, Hamiltonian vector fields.

PART II

Formulation of a basic optimal control problem: control function, objective functional and maximization problem. Necessary conditions: adjoint variable, adjoint equation, transversality condition. Pontryagin maximum principle.

Results about existence and uniqueness of the optimal control. Characterization of the hamiltonian function.

Objective functional with state conditions at fixed points; payoff terms.

Characterization of the optimal control: bang-bang and singular controls.

Numerical technique: forward-backward sweep method. Examples and applications.

## Bibliography

Teacher’s notes and selected parts of the following books:

1. Crampin-Pirani “Applicable Differential Geometry” - London Mathematical Society Lecture Note Series;

2. DeLeon-Rodrigues “Methods of Differential Geometry in Analytical Mechanics” – Elsevier;

3. Goldstein H., Poole C., Safko J. “Meccanica Classica” – Zanichelli Editore.

4. Lenhart S., Workman J.T. "Optimal Control Applied to Biological Models" - Chapman & Hall/CRC

## Teaching methods

The didactic activities are composed of lessons having theoretical character. Lessons will be of frontal type, and integrated with digital material in video or audio format. Theoretical lessons concerns the formal aspects of Analytical Mechanics, with some of its main results, as well as applications of the optimal control theory to mathematical models in Applied Sciences.

## Assessment methods and criteria

The knowledge will be verified through an oral exam based on the whole program of the course. The oral exam consists in questions about the two parts of the course. By prior agreement, for one of the two parts, the student may be asked to present an in-depth study of one of the topics presented in the course.

## Other information

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