I'm looking for a textbook or online notes or a review article etc on a rigerous formulation of Lagrangian mechanics. I'm well aware of the book by Arnold but I would like something to accompany it. Perhaps something a bit more recent. There are a few things I would like it to have.    At least as much differential geometry as Arnold. Even better would be a dedicated section on DG at the start to cover everything needed to read the book.  It must use local coordinate systems and not try to do everything in a coordinate free way.    The book Foundations of Mechanics by Abraham and Marsden doesn't use local coordinates (and focuses on Hamiltonian mechanics more so than Lagrangian). The book by Spivak is nice too and but there's something about it that doesn't quite suit me.
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I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think of the fields that arise in physics as sections of vector bundles (or maybe principal bundles) and would love an approach to classical mechanics or what have you that took advantage of this.  Now for the questions:   Is there a text book you would recommend that phrases the constructions in classical mechanics via bundles without an appeal to transition functions?   What are the drawbacks to this approach other than the fact that it makes computations less doable? (if it does that)   Are there benefits to thinking about things this way, ie would it be of benefit to someone attempting to learn this material to do it this way?
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I am taking a course in Mathematical Methods for physicsPhysics (Junior Level).   We are working from   > Mathematical Methods for physicists, George B.Arfken .    I just need online resources covering ( Matrices, Determinants, Vector Analysis, Tensors and differential forms and Vector spaces...etc) to study and understand from.
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I'm interested in learning physics. I do realize that the subject is large and that it would be easier if I had a specific area of interest. However, I do not. I suppose I want to learn about the fundamentals of it all; the axioms that combine all physics fields. Or, in other words, a high school physics class.   Specifically, a book or series of videos would be helpful. I looked over MIT and unfortunately the material wasn't for me. I don't mean to be "picky" so I am not completely ruling out any resource just yet.   Thanks in advance.
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This question has been posted before, but I need book with specific qualifications. I do not need books for engineers, book that is centered around calculations and stuff. I need to find a book that is theoretical, proves the statements and has good presentation of the theoretical structure. I have had the book by Tenenbaum, I did not like it. I would be very very thankful if someone shared their knowledge with us about this matter.
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What exactly are  (and )?   I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry only confused me more. I don't know any differential equations yet, so maybe that's why.    But what's the difference between Hamiltonian (&amp; Lagrangian mechanics) and Newtonian mechanics?  And why is Hamiltonian mechanics used for QM instead of Newtonian?  Also, what would the prereqs for studying Hamiltonian mechanics be?
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I'm looking for good book recommendations for preparing for the International Physics Olympiad. As stated on the , the topics covered are about the same as those in the first year or two of a physics degree:        Mechanics: kinematics, dynamics, celestial mechanics, hydrodynamics   Electromagnetism: Maxwell's equations, circuits, matter in EM fields   Waves: damped/driven harmonic oscillators, waves, interference and diffraction, geometrical optics   Modern: special relativity, matter waves, particle and nuclear physics   Thermodynamics: laws of thermodynamics, heat engines, phase transitions      However, there's less of an emphasis on complicated calculations, and more of an emphasis on problem solving and insight. For example, multivariable calculus is almost never used; instead many questions can be elegantly solved by symmetry. I'm looking for textbooks or problem books to bring me to this level, starting from the level of high school physics.
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I am following the first volume of the course of theoretical physics by Landau. So, whatever I say below mainly talks regarding the first 2 chapters of Landau and the approach of deriving Newton's laws from Lagrangian principle supposing Hamilton's principle of extremum action. Please keep this view in mind while reading and answering my queries and kindly neglect the systems to which Action Principle is not applicable:  If we use homogeneity of space in Euler-Lagrange equations, we obtain a remarkable result i.e. the conservation of momentum for a closed system.  Now, this result, using the form of Lagrange for a closed system of particles, transforms into $ \Sigma F = 0 $ . Now, how from this can we conclude that the internal forces that particles exert come in equal and opposite pairs?  Is it because for 2 particles this comes out as $ F_{1} + F_{2} = 0 $ and we take the forces exerted by particles on one other to be independent of other particles (i.e. Superposition Principle) as an experimental fact?  I doubt it as whole of Newtonian Mechanics is derivable from Lagrangian Mechanics and supposed Symmetries. So, according to me, a fact like Newton's Third Law should be derivable from it without using an additional experimental fact.  I have an idea to prove it rigorously. Consider two particles $i$ and $j$. Let the force on $i$ by $j$ be $F_{ij}$ and on $j$ by $i$ be $k_{ij}F_{ij}$. Now the condition becomes $\Sigma (1+k_{ij})F_{ij}=0$ where the terms to be included and rejected in summation understood. As this must be true for any value of $F_{ij}$, we get $k_{ij}=-1$. I don't know if this argument or refinement of such an argument holds or not. I can see many questions arising in this argument and it's not very convincing to me.  I would like to hear from you people as to if it is an experimental result used or not? If not, then is the method given above right or wrong? If wrong, how can we prove it?  Addendum  My method of proof uses the fact of superposition of forces itself, so it is flawed. I have assumed that the coefficients $k_{ij}$ are constants and don't change in the influence of all other particles which is exactly what superposition principle says.  As the superposition of Forces can be derived by superposition of potential energies at a point in space and potential energy being more fundamental in Lagrangian Mechanics, I restate my question as follows:  Is the principle of superposition of potential energies by different sources at a point in space derivable from the inside of Lagrangian Mechanics or is it an experimental fact used in Lagrangian Mechanics?  I, now, doubt this to be derivable as the fundamental assumption about potential energy is only that it is a function of coordinates of particles and this function may or may not respect superposition.
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Basically, I just want to know the advantages of Lagrangian and Hamiltonian mechanics over Newtonian mechanics, that made it much more preferable and widely used!
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I find differential equations in physics to be quite challenging so I'm looking for a book to help me master them.  I'm familiar with solving ordinary differential equations via seperation of variables but haven't really gone much further than that.  I was thinking about buying this:   However I'm open to recommendations on books that are specifically targeted to physics, or will help me in general to solve any differential equation.
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I am a second year undergraduate and studying quantum mechanics from sakurai's 'Modern Quantum Mechanics'. Is it a good idea to solve problems from sakurai, which are mostly mathematical in nature? I need a textbook that has physically relevant problems, maybe going even into condensed matter, or field theory in its exercises. This would probably help me to appreciate and understand qm better. Sorry if this question is too localised but I just had to post it.
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Does a similar easy to read introduction like David Mc Mohan's Demystified books exist for LQG? I'd like something easier to start with than a paper but deeper than a popular book.
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I found Sean Carroll's "A No Nonsense Introduction to General Relativity" (about page .  pdf ), a 24-page overview of the topic, very helpful for beginning study.  It all got me over the hump of learning the meaning of various terms associated with GR, most of which I had heard before without understanding.  It also outlined the most important examples.  Is there a similar document for quantum field theory, which presents the main equations, briefly describes the main ideas, and summarizes the most important applications and results so that the reader can feel the lay of the land before studying in depth?
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I'm looking for a textbook in quantum mechanics that relies heavily on Green functions and the path integral formalism to supplement my QM books. I want to do some calculations using alternative methods (path integrals, green functions, Lagrangians etc).
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I am a 4th-year undergraduate student and I have fully read R. Shankar's book on Quantum Mechanics and Griffiths book Quantum Mechanics. I have also done a bit of the Application of QM on multielectron systems,molecules, etc.  Not going ahead in the application part, I want to focus on the foundations of QM part which I find really interesting. But I have very limited knowledge of mathematics. I have read Curtis "An introduction to Linear Algebra", Gallian "Group Theory"(1-10 chapters only) and have a little bit of knowledge on differential geometry.  With this as my Mathematical background, which would be the best book (a bit mathematical) for introduction to Quantum Mechanics which includes an introduction to Hilbert Space and functional analysis?  PS: My question is specifically for the ones who have limited mathematical knowledge in contrast to answers which assumes you have done graduate level mathematics.
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What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the explanation of the Lagrangian formulation of mechanics you'd give to someone who just finished a semester of college physics.  Things I'm hoping to have explained to me:   What's the overall difference in layman's terms? From what I've read so far, it sounds like Newtonian mechanics takes a more local "cause-and-effect"/"apply a force, get a reaction" view, while Lagrangian mechanics takes a more global "minimize this quantity" view. Or, to put it more axiomatically, Newtonian mechanics starts with Newton's three laws of motion, while Lagrangian mechanics starts with the Principle of Least Action. How do the approaches differ mathematically/when you're trying to solve a problem? Kind of similar to above, I'm guessing that Newtonian solutions start with drawing a bunch of force vectors, while Lagrangian solutions start with defining some function (calculating the Lagrangian...?) you want to minimize, but I really have no idea. What are the pros/cons of each approach? What questions are more naturally solved in each? For example, I believe Fermat's Principle of Least Time is something that's very naturally explained in Lagrangian mechanics ("minimize the time it takes to get between these two points"), but more difficult to explain in Newtonian mechanics since it requires knowing your endpoint.
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What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations?    If someone on the street would flat out ask me that question, then I'd probably mumble something about physicists wanting to be able to use the Lagrangian approach. And to allow a positive rotation and translation invariant energy term, which allows for local propagation, you need something like $-\phi\Delta\phi$.  I assume the answer goes in this direction, but I can't really justify why more complex terms in the Lagrangian are not allowed or why higher orders are a physical problem. Even if these require more initial data, I don't see the a priori problem.  Furthermore you could come up with quantities in the spirit of $F\wedge F$ and $F \wedge *F$ and okay yes... maybe any made up scalar just doesn't describe physics or misses valuable symmetries. On there other hand in the whole renormalization business, they seem to be allowed to use lots and lots of terms in their Lagrangians. And if I understand correctly, supersymmetry theory is basically a method of introducing new Lagrangian densities too.   Do we know the limit for making up these objects? What is the fundamental justification for order two?
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I have always been puzzled by how do you arrive at Lagrangians?  That is, how do you know that the functional you need to get Newton's equations is  $$L  =  T-V(x)~?$$ Do you derive the Lagrangian first somehow or do you just guess the one which would satisfy the action to give equation's of motion? (because then you would need to know the equations of motion first.) Also, is the Lagrangian always equivalent to something resembling $T-V$, or there are other ways to determine it? More generally, is the action known before you know the equations of motion, or vice versa?  What I really wish to know is, was the Einstein Hilbert action discovered before field equations were? If so, how? My major doubt is how do you come across the Lagrangian if you don't know the equations of motion, and you can't guess $L$ then?