Classical Mechanics

The Newtonian Approach

Dr. Cristian Giovanny Bernal, IMEF FURG


About the Teacher

He holds a bachelor’s degree in Physics by the Universidad del Valle (Colombia). Master’s degree in Astronomy and PhD in Astrophysics by the Universidad Nacional Autónoma de Mexico (Mexico). In addition, he was a postdoctoral fellow by the Instituto Politécnico Nacional (Mexico), the Instituto de Astrofísica Teórica at UNAM (Mexico) and the Instituto de Física at Universidade Federal Fluminense (Brazil).

He was also a Visiting Professor at the Universidad de Cartagena (Colombia), the Universidad Nacional Autónoma de Mexico (Mexico) and the International Center for Relativistic Astrophysics (ICRANet-Rome).

He is currently Teacher/Researcher at the Instituto de Matemática, Estatística e Física (IMEF) at the Universidade Federal Rio Grande (FURG). He is a member of the PPG-Física and MNPEF programs at IMEF-FURG, and participates in several international research projects, including Mexico, Brazil, Colombia and Italy.

He has experience in high energy astrophysical phenomena and plasma astrophysics, with emphasis on the MHD fluxes present in various astrophysical star environments: stellar winds, binary stars, supernovae, neutron stars, pulsars, and astroparticles. He works building theoretical and numerical astrophysical models, and performing computational simulations of great refinement in parallel computers (Clusters).

Finally, he founded the scientific research group ASTROFURG-DAE at IMEF-FURG. The aim of this group is to promote scientific debate, outreach and research through various information channels. Find more information about our group and research in:


This course is intended to expand your knowledge of how macroscopic objects move: kinematics and dynamics.

Many of the topics covered in the Fundamental Physics courses will be repeated in this course, but the depth and level of mathematical sophistication will be much greater.

In addition, I will try to cover several advanced topics in classical mechanics that you have not seen before such as Fluid-Dynamics and Special Relativity in Astrophysics, as well as Chaos Theory.

To obtain a comprehensive education in classical mechanics at the intermediate level the student is encouraged to take the Analytical Mechanics course immediately after completing this course.


To familiarize you with the mathematical tools of classical mechanics (vectors and vector calculus, simple differential equations, Taylor series, complex numbers, etc.)

To enable you to solve problems in classical mechanics. This includes solving realistic problems (for example, problems involving air resistance) as well as problems in areas of current research interest (such as fluid-dynamics and special relativity in astrophysics, as well as  dissipative chaos.)

To introduce you to computational physics using public precompiled software or free source codes.

To give you experience presenting scientific work, as well as critiquing the scientific work of others.

Why Is This Course Important For You?

After taking this course and its sequel (Analytical Mechanics) you should be well-prepared for a graduate course in classical mechanics.

This course will also help to prepare you for physics and engineering courses in statics and dynamics.

In addition, should also provide you with enough grasp of fluid-dynamics, astrophysics, chaos theory and special relativity so that you can comprehend articles about such topics in popular journals (like Scientific American, National Geographic, Physical Review, Astrophysical Journal or Physics Today.)

Moreover, this course will also expose you to a number of mathematical techniques (Taylor series, complex exponentials, etc.) and computational tools (MatLab, Mathematica, Maple, Maxima, Python, C++, Fortran, etc) and techniques (ODE solving algorithms, root finding algorithms, etc.) that may be useful to you in other courses or in your future work.

Finally, this course will help improve your skill at making scientific presentations.

Methods of Instruction

You will have the opportunity to learn Classical Mechanics from a variety of sources during this course, including:

Assigned textbook readings

Classroom lectures and discussions using the CDF (Computable Document Format), supplemented by occasional computer demonstrations

In-class tutorial examples

In-class exercises

Homework assignments and in-class presentations of solutions

Computational projects

Note that I will lecture very frequently. Nevertheless, most of our class time will be focused on you rather than on me.


At least 5 topics will be well covered in this course. The first topic is a mathematical review. The following three topics are fundamental in classical mechanics. A fifth and/or sixth modern topic will be chosen to complement the course.

1. Vector Calculus

Introduction and Basic Definitions

The Scalar Product

Component Representation of a Vector

The Vector Product (Axial Vector)

The Triple Scalar Product

Differentiation and Integration of Vectors

Coordinate Frames

Vector Differential Operations ( gradient, divergence, and curl)

2. Newtonian Mechanics

Newton’s Axioms

Basic Concepts of Mechanics

The General Linear Motion

The Free Fall


Mechanics of Particle Systems

Special Relativity

3. Oscillations

The Harmonic Oscillator

Mathematical Interlude—Series Expansion, Euler’s Formulas

The Damped Harmonic Oscillator

The Pendulum

Nonlinear Mechanics and Chaos using a DDP

4. Gravitation

General Notions of Astronomy and Astrophysics

Planetary Motions

Special Problems in Central Fields

Open Problems in Stellar and Galactic Dynamics.

5. Fluid-Dynamics

Introduction to Fluids

The Governing Equations of Hydrodynamics

The Gas-dynamics Approach

Steady Adiabatic Flows; Isothermal Flows; Sound Waves

A Special Case: Steady, Spherically Symmetric Accretion

Feedback and Evaluation

In order to provide you with some feedback on your progress and eventually to evaluate your learning I will assign numerical grades on all of your work. Your overall grade will be the average of these individual grades, weighted as follows:

In-Class Tests (5 topics = 5 tests): 50 %

Computational Projects (presentation - 10% + physics & code - 10%): 20 %

In-Class and Take-Home Problems: 30%

TextBooks and References

You will need some of the following books for this course. You may find some of these books helpful during this course, or you may be interested in reading more about classical mechanics, fluid-dynamics, astrophysics and chaos after you have completed this course.

Mathematical Methods for physicists. George B. Arfken & Frank E. Harris

Basic Theoretical Physics. Uwe Krey & Anthony Owen

Classical Mechanics: Theory and Mathematical Modeling. Emmanuele DiBenedetto

Classical Mechanics - An Introduction. Dieter Strauch

Classical Mechanics. John R. Taylor

Classical Mechanics-Point particles and relativity. Walter Greiner

Classical Mechanics. Herbert Goldstein

Classical Mechanics. Tom W.B. Kibble & Frank H. Berkshire

Classical Dynamics of Particles and Systems. Thornton & Marion

An Introduction to Modern Astrophysics. Bradley W. Carroll & Dale A. Ostlie

Astrophysical Hydrodynamics. Steven Shore

An Introduction to Relativity. Jayant  V. Narlikar

Chaotic Dynamics: An introduction. Baker & Gollub

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