Instance: 2018/2019 - 2S
Cycles of Study/Courses
||No. of Students
||Syllabus since 2009/2010
Teaching Staff - Responsibilities
Teaching - Hours
Suitable for English-speaking students
Physics is the foundation of any Engineering field. With the advent of computers, the kind of physical problems that can be solved in an introductory course has expanded significantly. Computational physics and simulation techniques allow students to get a wider view of physical phenomena without requiring complicated analytical methods. The computational techniques developed to solve mechanics problems have been applied to other areas outside physics, giving rise to the general theory of dynamical systems.
This course aims to give the student basic background knowledge on dynamics and the computational techniques used to solve dynamical systems. A Computer Algebra System (CAS) is used, in order to allow the student to solve practical problems in mechanics and dynamical systems, rather than spending their time learning abstract analytical techniques. The knowledge on dynamics and computer modeling of physical systems acquired will be very important in other courses of their curriculum: computer graphics and visualization, games theory, simulation and scientific computing.
Learning outcomes and competences
In order to pass this course students must prove to be able to:
- Solve simple equations of motion analytically, using the separation of variables method.
- Identify the forces and torques acting on a mechanical system and write down the equations of motion.
- Analyze a dynamical system and identify its state variables, evolution equations and type of system.
- Find the equilibrium points of a dynamical system and explain their features.
- Solve the evolution equations of a system numerically and interpret the obtained solutions.
Pre-requirements (prior knowledge) and co-requirements (common knowledge)
High-school physics. Introductory College Calculus. Linear Algebra.
- Kinematics. Solution of simple motion equations using the method of separation of variables.
- Vector kinematics. Identification of the degrees of freedom of a mechanical system and resolution of systems of equations of motion.
- Curvilinear motion. Tangent and normal vectors. Centripetal acceleration. Trajectory curvature. Kinematics of rigid bodies.
- Vector Mechanics. Newton's laws. Types of forces found in mechanical systems.
- Rigid body dynamics. Addition of forces. Torque. Moment of inertia. Equations for the plane motion of rigid bodies.
- Work and energy. Relations between work and energy. Conservative and dissipative forces.
- Dynamical systems. Phase space. Stable and unstable equilibrium. Phase portraits. Conservative systems.
- Lagrangian mechanics. Generalized coordinates, velocities and forces. Lagrange equations. Lagrange multipliers.
- Linear systems. Harmonic oscillators. Classification of the equilibrium points. Eigenvalues and eigenvectors. Stability analysis.
- Non linear systems. Pendulums. Linear approximations. Jacobian matrix. Phase spaces with many dimensions.
- Limit cycles and two-species systems. Van der Pol oscillator. Predator-prey systems. Evolution and coexistence of two species.
- Chaotic systems. Asymptotic behavior, strange attractors and chaotic systems.
Jaime E. Villate; Dinâmica e Sistemas Dinâmicos
, Edição do autor, 2015. ISBN: 978-972-99396-1-7 (Available at http://def.fe.up.pt/dynamics)
S. Targ ; trad. de Albano Pinheiro e Melo; Curso teórico-prático de mecânica
Steven H. Strogatz; Nonlinear dynamics and chaos
. ISBN: 0-7382-0453-6
Lawrence Perko; Differential equations and dynamical systems
. ISBN: 0-387-95116-4
Comments from the literature
The book can be read and freely copied from http://def.fe.up.pt/dynamics
Teaching methods and learning activities
This is a practical course, with an active teaching methodology using computer tools for e-learning, computer algebra system (CAS) and simulations. The practical sessions are conducted in the Physics Studio of the Department of Engineering Physics (room B233). During those sessions students work in groups of two at one of the computers in the room, which has access to the support material including some practical activities or simulations, lecture notes, multiple-choice questions and proposed problems. Students should answer the multiple-choice questions among and solve some of the problems in the chapter for that week. The remaining problems in the chapter are left as homework.
The lectures are used to make experimental demonstrations and simulations, as well as giving further explanations for the textbook material and the computer algebra system used. The support for this course, including lecture notes, teaching materials, quizzes results, and communication among students and teachers, is done using the e-learning server (http://def.fe.up.pt/eic0010) which has public access, except for the sections related to evaluation.
Physical sciences > Physics > Classical mechanics
Physical sciences > Mathematics > Chaos theory
Physical sciences > Mathematics > Computational mathematics
Physical sciences > Mathematics > Mathematical analysis > Differential equations
Distributed evaluation with final exam
Amount of time allocated to each course unit
|Frequência das aulas
Eligibility for exams
The only requirement for a student-worker to pass the course is to obtain a final great of at least 10. In order to pass, all other students must fulfill three requirements, in the following order: 1st. Minimal attendance. 2nd. Distributed-component grade of at least 5. 3rd. Final grade of at least 10. The first two requirements may have already been fulfilled in any previous year.
To fulfill the minimal attedance, a student must be registered in one of the recitation groups and not to be absent to more than 25% of the recitation sessions for that group. For instance, if the student's recitation group has 11 sessions during the semester, he can be absent to two of those sessions, but not to three, because 3 is greater than 25% of 11. The number of recitation sessions for students with late acceptance to the program is counted from the date of their regular registration; for instance, if the student is registered when there are only 7 recitation sessions remaining, one absence is admitted but not two, because 2 is greater than 25% of 7.
The distributed-component grade (with a weight of 40% in the final grade) is obtained as the average of the grades of two quizzes. A student who has already obtained that grade in a previous year may attempt to improve it by fulfilling again the minimal attendance requirement and taking the quizzes this year. If the new grade is lower that the one previously obtained, the previous grade prevails.
An absence can be justified by submitting a written prove to the MIEIC secretary within one week of the date in question.
Calculation formula of final grade
If D denotes the grade for the distributed component and E the exam grade, the final grade is calculated with the following equation:
Maximum ( E; 0.4*D + 0.6*E )
Namely, if the grade of the distributed component is higher than the exam grade, the distributed component will have a weight of 40% and the exam 60%. But if the exam grade is higher, the distributed component will be ignored and the final grade will be the exam grade. There is no minimum grade required in the exam and the exam grade will have one decimal digit. The final grade will be rounded to an integer (9.5 is rounded to 10 but 9.4999 is rounded to 9).
Examinations or Special Assignments
Special assessment (TE, DA, ...)
Students who are not required to attend classes and obtain a grade for the distributed component do not need to make any additional tests or assignments before the exam. The final grade will be equal to the exam grade rounded to an integer.
Students can attempt to improve the grade obtained in an exam, only once, up to the remedial exam of the following year in which the course was passed. The final grade to the course is the highest between that previously obtained and the one resulting from the new exam taken. The distributed-component grade can only be improved in subsequent years (see the section "Distributed-component grade").
It is recommended a period of off-class independent work of at least 3 hours per week, in order to keep off with the subjects introduced every week. Independently of their attendance status, it is expected from all enrolled students to preview at home the chapter of the textbook which will be covered in the following practical session. It is also recommended to periodically check the announcements and forum messages posted in the e-learning server.