Analytical mechanics (2020/2021)

Course code
Name of lecturer
Nicola Sansonetto
Nicola Sansonetto
Number of ECTS credits allocated
Academic sector
Language of instruction
II semestre dal Mar 1, 2021 al Jun 11, 2021.

Lesson timetable

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Learning outcomes

The class is devoted to a modern study of classical mechanics from a mathematical point of view. The aim of the class is to introduce the tools and techniques of global and numerical analysis, differential geometry and dynamical systems to formalise a model of classical mechanics. At the end of the class a student should be able to construct a model of physical phenomena of mechanical type, write the equations of motion in Lagrangian and Hamiltonian form and analyse the dynamical aspects of the problem.


• Introduction. At the beginning of the course we will quickly review the basic aspects of Newtonian mechanics. The structure of the Galilean space-time and the axioms of mechanics. Systems of particles: cardinal equations. Conservative force fields. Mass particle in a central field force and the problem of two bodies.

• Lagrangian and Hamiltonian mechanics on Rn. Equivalence of Euler-Lagrange, Hamilton and Newton equations in the mechanical case. Hamilton's principle, conservation of generalised energy and invariance of Euler-Lagrange equation with respect to lifted change of coordinates. Legendre transformation. Cyclic variables and reduction in the Hamiltonian contest. Poisson brackets and first integrals.

• Review of dynamical systems and differential geometry. Vector fields on a manifolds, flow and conjugation of vector fields. First integrals, foliation of the phase space and reduction of order for a ODE.

• Lagrangian mechanics on manifolds. Constrained systems: d’Alembert principle and Lagrange equations. Models of constraints and their equivalence. Invariance of Lagrange equations for change of coordinates. Jacobi integral. Stability theory for Lagrangian systems and small oscillations. Noether’s Theorem, conserved quantities and Routh’s reduction.

Applications: the Foucault pendulum, the magnetic stabilisation and others.

• Rigid bodies. Orthonormal basis, orthogonal and skew-symmetric matrices. Space and body frame: angular velocities. Cardinal equations in different reference frames. A model for rigid bodies. Euler’s equations.

• Introduction to Lie groups and algebras. Group actions, trivializations and Euler-Poincare' theory.

Reference books
Author Title Publisher Year ISBN Note
R. Abraham and J.E. Marsden Foundations of mechanics. Second Edition. (Edizione 2) Addison-Wesley 1987 080530102X Freely available at
D.D. Holm, T. Schmah and C. Stoica Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions (Edizione 1) Oxford University Press 2009
J.E. Marsden and T.S. Ratiu Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems (Edizione 2) Springer-Verlag 1999

Assessment methods and criteria

To success in the exam, students must show that:
- they know and understand the fundamental concepts of Newtonian, Lagrangian and Hamiltonian mechanics;
- they have abilities in solving problems in mechanics, both from the abstract and the computational point of view
- they support their argumentation with mathematical rigor.

The exam is divided in two part: a written test based on the solution of open-form problems and an oral
test in which the student is required to discuss the written test and to answer some questions proposed
in open form. Only students who have passed the written exam will be admitted to the oral examination.
If positive, the mark obtained in the written test will be valid until the last session of the present
academic year (February 2022).

A student must obtain a mark of at least 18/30 (best) in both the written and oral part to pass the exam,
and the final grade will be given by the arithmetic average of the marks of the written and of the oral part.