Analytical Mechanics
Keywords |
Classification |
Keyword |
OFICIAL |
Physics |
Instance: 2020/2021 - 2S
Cycles of Study/Courses
Acronym |
No. of Students |
Study Plan |
Curricular Years |
Credits UCN |
Credits ECTS |
Contact hours |
Total Time |
L:F |
51 |
Official Study Plan |
2 |
- |
6 |
56 |
162 |
Teaching language
Suitable for English-speaking students
Objectives
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)
Prerequisite: Classical Mechanics at introductory level.
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.
9. Generalities about chaotic behaviour of mechanical systems and the KAM theorem.
Mandatory literature
H. Goldstein, C.P. Poole, J.L. Safko; Classical Mechanics (3rd ed.), Addison Wesley, 2001
Complementary Bibliography
Woodhouse, N.; Introduction to Analytical Dynamics, Springer, 2009
Comments from the literature
From the adopted work it will be taught subjects addressed in the sections: 1.3 to 1.6; 2.1 to 2.7; 3.1 to 3.3; 3.7 to 3.9; 4.1 to 4.4; 4.7 to 4.9; 5.1 to 5.8; 8.1 to 8.3; 8.5 to 8.6; 9.1-9.9; 10.1 to 10.7; 11.1 to 11.6; 12.1 to 12.5.
Punctually it will be provided slides and notes of the materials presented in class.
Teaching methods and learning activities
Lectures: exposition.
Practical lectures: problem solving.
Evaluation Type
Evaluation with final exam
Assessment Components
designation |
Weight (%) |
Exame |
100,00 |
Total: |
100,00 |
Amount of time allocated to each course unit
designation |
Time (hours) |
Estudo autónomo |
106,00 |
Frequência das aulas |
56,00 |
Total: |
162,00 |
Eligibility for exams
Students who attend the course for the first time can give a maximum of 4 absences to theoretical-practical classes.
Calculation formula of final grade
Final grade = Final exam classification
Observations
The jury of the curricular unit includes:
-
José Miguel Nunes da Silva
-
Miguel Sousa da Costa
- Orfeu Bertolami Neto