Classical Mechanics II

The course is part of a set devoted to Classical Mechanics, which is the branch of physics that studies the movement of material bodies and particles that includes the theory initiated by Galileo and Newton and developed in the XVIII and XIX century by Lagrange and Hamilton. Einstein's Special Relativity is also included in the course

The contents of this broad field are distributed in the two compulsory courses Classical Mechanics I and II in the 2nd year, and in Classical Mechanics III, in the 3rd year. General objectives include the following:

- To introduce the basic concepts of Newtonian and Lagrangian formulations of mechanics and special relativity, for describing mechanical systems and wave phenomena.

- To describe the most relevant applications understanding how the fundamental principles are involved in solving the equations of motion.

- To familiarize the student with appropriate mathematical techniques to solve problems and provide the ability to manipulate concepts to solve these problems creatively.

- To familiarize the student with the terminology and notations of modern physics that will facilitate the transition to the study of other areas of physics, especially quantum mechanics.

Learning Results

With respect to this couse, the student will show he/she has acquires the following capabilities:

-Resolve problems using the techniques of variational calculation
and understand the relationship between the Lagrange equations and the Hamilton principle.

-Use the Lagrange multipliers method comprising its
physical meaning and the role of symmetries and binding forces.

-Write the Hamiltonian and the canonical equations of a mechanical system.

-Calculate the parameters of the orbits in the central forces problems,
especially in the Keplerian case and obtain effective sections in collision problems.

-Calculate the inertia tensors of rigid bodies and know how to use their properties.

-Resolve and interpret the movement of a rigid body in simple cases.

-To understand the principles of special relativity and to know its description in the
four-dimensional space-time. Apply these principles to problem solving
of relativistic particle dynamics and collisions.

The contents of the course are organized in four topics as follows:

1. VARIATIONAL METHODS
- Techniques of variational calculus.
- Hamilton's principle.
- Constrained systems. Method of Lagrange multipliers.
- Hamiltonian formalism.

2. CENTRAL FORCES
- Two body problem.
- Kepler problem. Planetary movement.
- Collisions. Cross-sections.

3. RIGID BODY
- Rotations and tensors.
- Rigid body kinematics.
- Rigid body dynamics.

4. SPECIAL RELATIVITY
- Postulates and Lorentz transformations.
- Four dimensional formalism. Space-time geometry.
- Relativistic mechanics.

TEXTBOOKS

- John R. Taylor: Mecánica Clásica, Ed. Reverté, 2013 (3-A03-21)
Classical mechanics, University Science Books, 2004 (3-A03-248).
- H. Goldstein: Mecánica clásica. Ed. Reverté, 2000. (3-A03-8)
- J. B. Marion: Dinámica clásica de las partículas y los sistemas. Ed. Reverté, 2000. (3-A03-9)

- K. R. Symon: Mecánica. Ed. Aguilar, 1970. (3-A03-44)
Mechanics. Addison-Wesley, 1971. (3-A03-107)
- L.V. Landau, E.M. Lifshitz: Mecánica, E. Reverté, 1991 (3-A03-S. 1-1)
- Atam P. Arya: Introduction to Classical Mechanics, Prentice Hall, 1998 (3-A03-166)
- Tai L. Chow: Classical mechanics , John Wiley, 1995 (3-A03-144).

PROBLEMS

- O. Ecenarro. Problemas de mecánica resueltos y comentados. Universidad del Pais Vasco, 2000. (3-A03-190)
- O. Ecenarro. Mecánica y Ondas: Problemas de examen resueltos y comentados. Universidad del Pais Vasco, 2000. (3-A03-216)
- V.M. Pérez García, L. Vázquez Martínez y A. Fernández-Rañada: 100 problemas de mecánica, Alianza editorial, 1997. (3-A03-159)
- David Morin: Introduction to Classical Mechanics. With Problems and Solutions, Cambridge University Press, 2008. (3-A03-269)

Resources on-line:

Virtual Classroom: Course noted produced by the professors, list of problems, solutions to the problems, exams from previous years, etc.
Virtual Classroom: Links to online resources.

Basic and General

CB1 - That the students have demonstrated to possess and understand knowledge in a
study area that starts from the base of general secondary education, and is usually
find at a level that, while supported by advanced textbooks, also includes
some aspects that imply knowledge coming from the vanguard of its field of study.

CB2 - That students know how to apply their knowledge to their work or vocation in a way
professional and have the skills that are usually demonstrated through the development and defense
of arguments and the resolution of problems within their area of study.

CB3 - That students have the ability to gather and interpret relevant data (normally
within their study area) to make judgments that include a reflection on relevant issues
of a social, scientific or ethical nature.

CG1 - Possess and understand the most important concepts, methods and results of the different branches of
Physics, with a historical perspective of its development.

CG2 - Have the ability to gather and interpret data, information and relevant results, obtain conclusions
and issue reasoned reports on scientific, technological or other problems that require the use of
knowledge of Physics.

CG3 - Apply both the theoretical and practical knowledge acquired as well as the capacity for analysis and abstraction
in the definition and approach of problems and in the search for their solutions both in academic contexts and
professionals.

Transversal

CT1 - Acquire analysis and synthesis capacity.

CT2 - Have the capacity for organization and planning.

CT5 - Develop critical reasoning.

Specific

CE1 - Have a good understanding of the most important physical theories, locating in their logical and mathematical structure,
its experimental support and the physical phenomenon that can be described through them.

CE2 - Be able to clearly manage orders of magnitude and make appropriate estimates in order to develop
a clear perception of situations that, although physically different, show some analogy, allowing the use of solutions
known to new problems.

CE5 - Being able to do the essentials of a process or situation and establish a working model of it,
as well as perform the required approximations in order to reduce the problem to a manageable level. Will prove to possess
critical thinking to build physical models.

CE6 - Understand and master the use of mathematical and numerical methods most commonly used in Physics

CE8 - Be able to manage, search and use bibliography, as well as any source of relevant information and apply it to
research and technical development projects.

One course will be activated at the platform Moodle of the Campus Virtual, where information of interest for the student will be downloaded, as well as various teaching materlals.

The classes take place at a rate of four hours per week, two of them are lectures in large groups and the other two are seminars in small groups, as specified in schedules, up to a total of 28 hours each of them. There are 4hours of tutoring per student to complete 60 hours of classroom teaching.

The lectures in large groups are usually master classes where the teacher presents the theory. The student will know in advance the contents of each class and the bibliography needed to prepare it. Some advanced aspects of each topic can be proposed to the students to work personally.

The seminars will be in small groups and they will seek the participation of students. Classes may consist in theoretical issues, applications, examples and tutorials where students solve exercises, raise questions or present papers. Typically they will be devoted to discuss the exercises distributed during the development of each topic. Also, voluntary work may be proposed in case a student is interested in studying any topic of the syllabus in greater depth. One hour of class on each topic will be devoted to a control assessment, as specified in the paragraph relating to it.

In addition, each student will have 4 hours of very small group or individual tutoring during the course , comprising 60-minute sessions at the end of each of the four topics, where the teacher will monitor the work of each student, will review the results of the assessment and will solve doubts or difficulties.

The course will have its virtual classroom which will serve as support for classroom teaching. It will contain additional materials, notes, links to relevant web pages, simulations, and all information on schedules, grades, etc., and communication tools with the teacher.

The tutoring sessions will be, either presencial, or telematic, when telematic they will require of previous appointment, waht will be also recommended in the case of telematic presential tutoring sessions.

The general evaluation criteria specified in the report of the degree will be applied.

At the end of each of the four topics a control test will be made which will consist in the resolution of a problem. Continuous assessment will take into account the outcome of these tests, together with the assistance and active participation of students in classes and completion of the tasks proposed. This part of the grade will account for 40% of the final grade and will be carried out topic to topic, ie, at the end of each of the four topics the student can obtain up to 1 point over 10 on the final grade for these concepts.

At the end of the course a final written exam will take place, which will consist of solving a number of problems relating to the topics covered during the course. Examination time will be approximately 4 hours and it will not be allowed to use any book or notes. The weight of the examination result of the final grade will be 60%.

The part of the continuous assessment will be saved until the July session in the case of not passing the subject at the first opportunity. Not so for subsequent years.

In any case, the final mark obtained by the student will not be lower than the one obtained in the final exam.

For the cases of fraudulent realization of problems or other tests, what is contained in the Regulation for the evaluation of the students academic performance and for qualifications revision.

WORK IN THE CLASSROOM

Lectures in large groups (32 hours)
Seminars in small groups (24 hours)
Tutorials in very small groups (4 h)

Total number of hours in the classroom: 60 hours

STUDENT WORK

Individual self-study or study in group (75 hours)
Writing exercises and other works (15 hours)

It is recommended to have completed Física Xeral I and II and Métodos Matemáticos I, II, III and IV. The course is complemented by the laboratory of mechanics in Técnicas Experimentais II.

Regarding the study of the subject, it is recommended to attend and participate actively in the classes, keep up the study of the contents imparted using the proposed bibliography, solve the problems individually or in groups and take advantage of tutorials to solve doubts.