Analytical Mechanics

Code:

FIS2021

Acronym:

FIS2021

Keywords
Classification	Keyword
OFICIAL	Physics

Instance: 2024/2025 - 2S

Active?	Yes
Responsible unit:	Department of Physics and Astronomy
Course/CS Responsible:	Bachelor in Physics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:F	48	Official Study Plan	2	-	6	48	162

Teaching language

Portuguese

Objectives

To endow students to master classical formalisms of analytical mechanics and its application to problems of classical mechanics. In this line the student is exposed to the notions of symmetry and algebrization in the description of movement, as an introductory path to the concepts to be developed in chairs on Quantum Mechanics.

Learning outcomes and competences

This course develops skills to solve more advanced problems in more complex mechanical systems, using more sophisticated mathematical techniques.

In addition, the student will learn several concepts that play an important role in modern theoretical physics, including the principles of symmetry and the geometric structure of mechanics.

Upon completion of the course the student should be able to use the formalisms of Lagrange and Hamilton in specific examples, solve a greater variety of problems using methods of Analytical Mechanics, and apply the mathematical tools that were developed during the course.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Introductory Classical Mechanics

Program

1. Review the basic concepts of Newtonian mechanics. Principle of d'Alembert. Generalized forces.

2. Review of variational methods. Variational formulation of the equations of Lagrange. Case of generalized potentials. Case of dissipative forces. Symmetries and conservation laws. Noether's theorem.

3. The problem of two bodies (point like) with a central potential.

4. Movement of a solid. Euler angles. Euler equations. The symmetrical top.

5. The formulation of Hamilton. Hamilton equations.

6. Canonical transformations. The generating function of a canonical transformation. The Poincaré invariants. Poisson brackets.  Infinitesimal canonical transformations, symmetries and constants of motion.

7. Hamilton-Jacobi theory. Separation of variables in Hamilton-Jacobi equation.

8. Action-angle variables. Adiabatic invariants. Brief presentation of the canonical theory of perturbations.

Mandatory literature

H. Goldstein, C.P. Poole, J.L. Safko; Classical Mechanics, Adison Wesley, 2001

Complementary Bibliography

Woodhouse, N.; Introduction to Analytical Dynamics, Springer, 2009

Teaching methods and learning activities

Theory lectures: exposition of theory and concepts. 
Exercise classes: resolution of problems.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	75,00
Teste	25,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	114,00
Frequência das aulas	48,00
Total:	162,00

Eligibility for exams

Students are allowed to miss up to 3 exercise classes.

Calculation formula of final grade

Type of evaluation:
     The assessment of knowledge will be done through two items:

     A. Problem solved individually in TP class - at the end of the 4th week and the 10th week of classes, students will be given a problem that they must solve in 30 minutes. The final classification of this component (PI) is equal to the average of the classifications obtained in each problem submitted for evaluation. This component is considered in all examination periods, and for the purpose of improving the classification.

     B. Final exam (EF).


     Formula for calculating the final classification:
     The classification of the final exam, in any of the exam periods, for approval or improvement purposes, must not be less than 8.0 out of 20 (EF>=8.0)

     The final classification (CF) is determined by the formula:

     CF = 0.25*PI + 0.75*EF

Observations

The jury of the curricular unit includes:
Miguel Nunes da Silva
Joaquim Agostinho Moreira
Miguel Sousa Costa
Orfeu Bertolami Neto