Modern Developments In Fluid Dynamics Goldstein Pdf

[Do Kieu]

Thepurpose of reading Goldstein's Classical Mechanics as an undergraduate is to gain a deeper understanding of classical mechanics, which is a fundamental branch of physics. It covers topics such as Newton's laws of motion, conservation laws, and the motion of particles and systems. It is also a valuable resource for students pursuing degrees in physics, engineering, or related fields.

Yes, Goldstein's Classical Mechanics is suitable for undergraduate students. While it may be more challenging than other introductory textbooks, it provides a comprehensive and rigorous treatment of classical mechanics. It is often used as a textbook for upper-level undergraduate and graduate courses in classical mechanics.

Goldstein's Classical Mechanics is known for its mathematical rigor and its treatment of advanced topics such as rigid body dynamics, Hamiltonian mechanics, and small oscillations. It also includes many examples and problems for students to work through, making it a valuable resource for those looking to deepen their understanding of classical mechanics.

While a strong background in mathematics is helpful, it is not necessary to read Goldstein's Classical Mechanics. The book includes a review of the necessary mathematical concepts and techniques, making it accessible to students with a basic understanding of calculus and linear algebra.

Yes, Goldstein's Classical Mechanics is still relevant in modern physics. While it may not cover some of the more recent developments in classical mechanics, it provides a solid foundation for understanding the principles and concepts that are still used in modern physics. It also serves as a bridge to more advanced topics in theoretical physics.

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Self-sustained turbulent structures have been observed in a wide range of living fluids, yet no quantitative theory exists to explain their properties. We report experiments on active turbulence in highly concentrated 3D suspensions of Bacillus subtilis and compare them with a minimal fourth-order vector-field theory for incompressible bacterial dynamics. Velocimetry of bacteria and surrounding fluid, determined by imaging cells and tracking colloidal tracers, yields consistent results for velocity statistics and correlations over 2 orders of magnitude in kinetic energy, revealing a decrease of fluid memory with increasing swimming activity and linear scaling between kinetic energy and enstrophy. The best-fit model allows for quantitative agreement with experimental data.

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There have been similar questions on mathoverflow, but the answers always gave some advanced introduction to the mathematics of quantum field theory, or string theory and so forth. While those may be good introduction to the mathematics of those subjects, what I require is different: what provides a soft and readable introduction to the (many) concepts and theories out there, such that the mathematics involved in it is in comfortable generality. What makes this is a "for mathematicians" question, is that a standard soft introduction will also assume that the reader is uncomfortable with the word "manifold" or certainly "sheaf" and "Lie algebra". So I'm looking for the benefit of scope and narrative, together with a presumption of mathematical maturity.

Math Answer: It is the class of cohomology of the action of the group of Galilee, measuring the lack of equivariance of the moment map, on a symplectic manifold representing the isolated dynamical system.

These answers are the mathematical versions of physics classical constructions, but it would be very difficult to appreciate them if you have no pedestrian introduction of physics. You may enjoy also Aristotles' book "Physics", as a first dish, just for tasting the flavor of physics :)

Just before entering in the modern world of physics I would suggest few basic lectures for the winter evenings, near the fireplace (I'm sorry I write them down in french because I read them in french).

" I want to explore the working of elementary physics ... which I have always found so hard to fathom.[...]I have written this work in order to learn the subject myself, in a form that I find comprehensible.[...]By physics I mean ... well, physics, what physicists mean by physics, i.e., the actual study of physical objects ... (rather than the study of symplectic structures on cotangent bundles, for example)."

It tries to touch almost all areas in physics, including the hot ones. Penrose emphasizes the mathematical part (especially the geometric interpretations), and avoids to be superficial (many scientific writers, when trying to make the things easier, use misleading metaphors). One warning is to be careful that sometimes he expresses his personal viewpoint, which is not always mainstream. But it is clear when he does this, and he is very careful to make justice to the mainstream viewpoint, by presenting it very well.

Second, if you have the time I would encourage you to read physics books that are written for physicists, not for mathematicians. There are numerous differences in terminology and worldview between the physics and mathematics community, even when the underlying subject matter is in some sense the same. It's very valuable for a mathematician to be able to read and understand recent physics arxiv postings, and the only way to do this is to go through some (perhaps accelerated) version of physics grad school.

Electricity and Magnetism, Berkeley Physics Course Vol. II by Edward M. Purcell. This book presupposes knowledge of special relativity, but I thought is was really great when I read it as an undergraduate.

The Quantum Theory of Fields, volume 1 by Steven Weinberg. I found this book to be much less impenetrable (from the point of view of a mathematician who foolishly stopped taking physics courses when he was an undergraduate) than the typical QFT textbook.

The Feynman lectures are good, but one of the main things which separates physics from mathematics is the role of experiment and observation. Physics is not just a matter of getting the formulae and models right, but also of testing mathematical models against observations to see whether they stand up or break down in "the real world". Part of the role of mathematical models is to give physicists some guidance on potentially fruitful places to look.

So it rather depends whether you are looking at mathematical/theoretical physics as a mathematician/theoretician would understand it, or whether you are looking to understand the role of mathematics in physics as a discipline.

Edit: The list below fits not that good to the requirements you describe, but the texts there are what I found helpfull. If you can read German books, I would recommend W. Greiner's "Theoretische Physik", which explains basically all the needed mathematics. Usefull too may be J. Baez' "Gauge Fields, Knots and Gravity", which contains a "rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations".

And then, I found F. C.'s recommendation to Nahm's very fascinating "Conformal Field Theory and Torsion Elements of the Bloch Group" very good, e.g. Nahm writes "readable for mathematicians", "much of this article is aimed at mathematicians who want to see quantum field theory in an understandable language .... all computations should be easily reproducible by the reader". Nahm's issue is a strange connection between some quantum field theories and algebraic K-theory and he hopes, his article could stimulate mathematicians to become interested in these exciting topic. A forthcoming article by Zagier on "quantum modular forms" may relate to that too.

Connes/Marcolli's book"Noncommutative Geometry, Quantum Fields and Motives" contains a very readable introduction in quantum field theory, renormalization etc., Marcolli's "Feynman motives" a chapter "Perturbative Quantum Field Theory and Feynman Diagrams".

Here is a list of books I find useful that present some physical topics from a mathematical viewpoint. Sadly I don't know a good reference for electromagnetism, quantum field theory or statistical physics.

Jeffrey Rabin has written a lightning-fast introduction to physics designed for exactly the audience you describe: people with "the mathematical background of a first-year graduate student," but "[no] prior knowledge of physics beyond F = ma."

Hmmm, the first thing that occurs to me is that mathematicians need to learn about "time", because something as fundamental as "conservation of energy" is not directly to be found in mathematics in its physics form. The two are connected by what is usually known as "Noether's theorem" and so this provides a more manageable question: where can mathematicians genuinely learn about the role of symmetry principles in physics? This starts getting us somewhere, but observe what goes on: the traditional route goes through calculus of variations in some form, and that is a theory not in Bourbaki.

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