Guide in PDF format
Logo

Classical Mechanics

Code: 107618
Credits: 6
2026/2027
Degree programme Type Course
Physics OB 2

Contact lecturer

Name :
Cosimo Nigro
Email :
cosimo.nigro@uab.cat

Group languages

You can consult this information at the end of the document.

Prerequisites

There are no requirements, but it is recommended that students have attended the following subjects:

  • Algebra (107597) - basic knowledge of vector spaces and matrices is required (how to operate with them and diagonalize them);
  • Calculus in One Variable (107596) - it is essential to have ease with elementary derivatives and integrals of functions of one variable;
  • Vector and Multivariable Calculus (107598) - especially for analytical mechanics, it is necessary to be familiar with the partial derivative, the chain rule and Taylor series expansion of functions of several variables;
  • Differential Equations and Complex Variables (107611) - it is necessary to be familiar with linear ordinary differential equations of first and especially second order;
  • Mechanics, Waves and Relativity (107591) - it is essential to have acquired the concepts of Newtonian mechanics presented in this course.

Objectives

The main objective of this subject is to develop in students a new general attitude to solving physical problems, based on the capacity for abstraction and supported by the analytical instruments of calculus. Methodologically, therefore, the geometric character of mechanics will be abandoned to arrive at a purely analytical formulation of the problems.

Fundamental concepts and techniques will be shown (for example: Taylor expansions of terms of equations or their solutions, study of equilibrium points) that have a wide range of applications.

The analytical mechanics section will introduce the terminology and tools that will serve as a basis for other disciplines (for example, statistical and quantum mechanics).

Learning outcomes

  • CM15 (Correctly formulate the problems of classical mechanics in terms of compact analytic expressions.) Correctly formulate the problems of classical mechanics in terms of compact analytic expressions.
  • CM16 (Work in groups to solve problems in the field of classical mechanics using the laws of motion in 3D and the tools of analytical mechanics.) Work in groups to solve problems in the field of classical mechanics using the laws of motion in 3D and the tools of analytical mechanics.
  • KM18 (Describe the fundamentals of classical mechanics.) Describe the fundamentals of classical mechanics.
  • SM14 (Apply mathematics in the description of classical mechanics, selecting the appropriate tools, building appropriate models, interpreting results and critically comparing with experimentation and observation.) Apply mathematics in the description of classical mechanics, selecting the appropriate tools, building appropriate models, interpreting results and critically comparing with experimentation and observation.
  • SM15 (Express problems of classical mechanics, identifying the most relevant principles by using approximations, if necessary, to arrive at a solution that must be presented by making explicit hypotheses and approximations.) Express problems of classical mechanics, identifying the most relevant principles by using approximations, if necessary, to arrive at a solution that must be presented by making explicit hypotheses and approximations.

Contents

1. Motion under a Central Force

  • Two-body problem, change of variables (relative position, position of the center of mass);
  • Conservation of angular momentum, the concept of effective potential;
  • Binet's equation, parameterisation of orbits in Kepler's problem;
  • Stability of an orbit (study of the effective potential);
  • Concept of cross-section and Rutherford scattering.


2. Oscillations

  • Simple harmonic oscillator, damped oscillator, forced oscillator.


3. Coupled Oscillators

  • Two masses and three springs, algebraic form of the problem, normal modes.


4. Systems of Particles

  • Review of conservation laws (linear momentum, angular momentum, and energy of a system of particles);
  • Collisions: elastic collision in two dimensions.


5. Non-Inertial Frames of Reference

  • Time derivatives in a rotating reference frame;
  • Newton's second law in a rotating reference frame;
  • centrifugal force;
  • Coriolis force.


6. Rotational Motion of Rigid Bodies

  • Rotation about any axis; inertia tensor;
  • diagonalisation of the inertia tensor: principal axes of inertia, properties of the inertia tensor;
  • Euler angles;
  • Euler equations;
  • free rotation (without gravity) of a symmetrical top, or free precession;
  • rotation of a symmetrical top under the action of gravity.


7. Analytical Mechanics

  • Introduction to Analytical Mechanics (comparison with Newtonian Mechanics);
  • generalised coordinates and constraints, generalised velocities and expression of kinetic energy;
  • introduction to the Calculus of Variations;
  • Hamilton's principle and Lagrange's equations;
  • conservation laws;
  • Hamiltonian formulation.

Learning activities and methodology

Title Hours ECTS Learning outcomes
Theory Classes 28 1.12 CM15, KM18, SM14, SM15
Seminars 8 0.32 CM15, CM16, KM18, SM14, SM15
Self Study 87 3.48 CM15, CM16, KM18, SM14, SM15
Problems' Classes 14 0.56 CM15, CM16, SM14, SM15

Description of the Seminar Activities

We will offer two types of seminars:

  1. One type will focus on problem-solving using the flipped classroom method. Students will be provided with a reading list to study before the seminar. During the seminar, the instructors will present problems that students will solve collaboratively, interacting and working with the rest of the class and the instructors.
  2. The other type will focus on oral presentations by students on topics related to the course and suggested by the instructors.


Seminar 1 (Problem-solving, 2 hours, evaluated). Harmonic Oscillator: Variations on the Theme

  • Problem-solving using the flipped classroom method. Topic: the different types of oscillators (damped, forced).
  • The solutions to the problems will be assessed.


Seminar 2 (Problem-solving, 2 hours). Coupled Oscillators

  • Problem-solving using the flipped classroom method. Topic: Coupled Oscillators


Seminar 3 (Problem-solving, 2 hours). Non-inertial Systems

  • Problem-solving using the flipped classroom method. Topic: Non-inertial Systems


Seminar 4 (Oral Presentations, 2 hours, graded). Genealogy of Classical Mechanics

  • The historical context in which the ideas of classical mechanics developed is often overlooked, as they are presented as a mere succession of chapters in a book. Guided by the instructors, students will reconstruct a "genealogy" or "timeline" of classical mechanics. From Galileo and Newton, through Leibniz and the debate on vis viva, Bernoulli and the brachistochrone problem, and ending with Hamilton, we will attempt to understand how the ideas of these figures relate to and influence one another. The students will be divided into groups, each assigned a group of historical figures and a specific idea they have developed. The students will research and prepare a presentation on their assigned topic for the seminar.
  • This activity will be assessed.
Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Second partial exam 45% 3 0.12 CM15, KM18, SM14, SM15
Students' presentations 5% 2 0.08 CM15, CM16, KM18, SM14, SM15
Recovery exam 90% 3 0.12 CM15, KM18, SM14, SM15
First partial exam 45% 3 0.12 CM15, KM18, SM14, SM15
Problems solved in the seminar with the inverted-class method 5% 2 0.08 CM15, CM16, KM18, SM14, SM15

Continuous evaluation

The methodology section explains which seminar activities will be assessed. The main part of the assessment will consist of two 3-hour midterm exams, each comprising a series of theory questions and problems.

Continuous evaluation: calculation of the Final Grade

Let:

  • N_CI be the grade for the problem-solving exercise in the flipped classroom format;
  • N_PR be the grade for the presentation;
  • N_1 be the grade for the first midterm;
  • N_2 be the grade for the second midterm;

The final grade for the course, N_F, will be calculated as follows:

N_F = 0.05 * N_CI + 0.05 * N_PR + 0.45 * N_1 + 0.45 * N_2 (1)

Students with a final grade equal to or lower than 2 (N_F <= 2) have failed the course and will not be able to take part in the reexamination.

To pass the course, the final grade must be 5 or higher (N_F >= 5). If the final grade is lower than 5 (N_F < 5), the student must take a reexamination. The reexamination will cover topics from the entire semester. The grade on this exam, when submitted, will replace the grades of the two midterm exams. That is, if N_RE is the reexamination grade:

N_F = 0.05 * N_CI + 0.05 * N_PR + 0.90 * N_RE (2)

The reexamination is open to all students who have passed and wish to improve their grades. In this case, however, the rule for replacing the midterm grades remains the same. That is, the reexamination grade is the final grade, and equation 2 will be used to calculate the final grade. In case of having a final note N_F < 5 after the reexamination, the subject will be considered failed.


Single evaluation

The students who have chosen the single evaluation must solve a written exam with theory questions and problems that cover the entire program of the subject. This test will be held on the same day as the second part of the continuous evaluation. The grade of this exam will be the final grade of the subject (that is, the exam has a weight of 100%). If the student obtains a grade lower than 5, he/she may repeat the same test on the day of the reexamination.


Not assessable

In the case of choosing the continuous assessment, students who have not taken any of the mid-term exams will be considered not assessable.

If the single assessment is chosen, students who do not take the single exam will be considered not assessable.

Bibliography

J. R. Taylor. Classical Mechanics. University Science Books, 2005. isbn: 1-891389-22-X.

S. T. Thornton & J. B. Marion. Classical Dynamics of Particles and Systems, Fith Edition. Thomson Brooks/Cole, 2003. isbn: 0-534-40896-6.

K. R. Symon. Mechanics, Third Edition. Addison Wesley Publishing Company, 1971. isbn: 0-201-07392-7.

H. Goldstein. Mecánica Clásica, Segunda Edición. Editorial Reverté, S. A., 1990. isbn: 84-291-4306-8.

J. L. Bohn. A student’s Guide to Analytical Mechanics. Cambridge Univeristy Press, 2018. isbn: 978-1-316-50907-4.

Software

No specific software is required to follow the course.

In case any simulation or numerical calculation needs to be performed, the Google Colab platform will be used, which allows programming in Python directly from the browser, without the need to install any local software. This tool is free, accessible from any device with an internet connection and very useful for viewing results interactively.


Use of AI

Model 1 - Prohibited Use: In this subject, the use of Artificial Intelligence (AI) technologies is not permitted in any of its phases. Any work that includes fragments generated with AI will be considered a lack of academic honesty and may lead to a partial or total penalty in the grade of the activity, or greater sanctions in serious cases.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan/Spanish first semester morning-mixed
(PAUL) Classroom practices 1 Catalan/Spanish first semester morning-mixed
(TE) Theory 2 Catalan/Spanish first semester morning-mixed
(PAUL) Classroom practices 2 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 11 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 12 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 21 Catalan/Spanish first semester morning-mixed
(SEM) Seminars 22 Catalan/Spanish first semester morning-mixed