# Book recommendation for classical Newtonian gravity

I’m looking for a good textbook that covers Newtonian Gravity in detail (preferably advanced undergrad/grad level). One that covers important things like

1. Calculating trajectories of satellites around some large mass
2. Methods for solving $$n$$-body problem
3. Continuous masses
4. Derivation of tides

What’s a good textbook that does all these (or at least most of them)?

• Do you want an extended school book, i.e. with more and more detailed examples, or do you want to dive into theoretical mechanics, i.e. moving from Newton to principals of extremes or variations? // What books do you have as reference, and what are you missing there? Mar 18 at 8:16
• More theoretical while still being specifically about Newtonian gravity. I have seen the book astrodynamics by bates, but it seems as if it shows the physics of each kind of orbit, rather than deriving it from the law of gravitation. It also doesn’t derive tides. I’m not sure what you mean by “more detailed examples”. I’m just looking for a book that discusses applying Newtonian gravitational field theory to different kinds of masses, derived orbits—sort of like gravitation by misner thorne for Newtonian gravity. But yes I’d expect variations method to be covered in such a book. Mar 18 at 13:01
• "More detailed examples": School books limit themselves to forces, energy, momentum. Wrt orbits moon and planet may be all they illustrate (!), and tides ares shown for pupils around their teenage years. Multibody problems are not discussed, fluids are scratched at best. // "Newtonian gravitational field theory": So you almost can't avoid the concept of potentials, deriving gravity fields from it by differentiation, deriving Newtons law from conserving the vector of angular moment etc. // "I’d expect variations method" So clearly you head for theoretical mechanis ... leaving out tides etc. -> Mar 18 at 13:32
• With this in mind, your search should target books for students in their 1st and 2nd semesters. Tides you'll probably have to derive by yourself, because these books try teaching basic concepts, from forces to Hamilton's formulations, which pave the way to quantum mechanics. // Multibody problems may not be discussed in detail there: Analytical solutions quickly aren't available the more bodies you consider, AND slight parameter variations lead to unpredictable outcomes (link to chaos theory). // So, may be you won't find all in "one" book. Mar 18 at 13:32
• I mention tides because the usual way they are presented is not correct. So I’m looking for one that describes it correctly not just qualitatively but quantitatively. I’m looking for books dedicated to gravity, instead of general lagrangian mech, Hamiltonian mech, etc books. Mar 18 at 14:11

## 1 Answer

Answering your most-in-one-book request seems to be somewhat difficult. I hesitated a little answering, as it's already "some time ago" that I left university. Let me try paving the road for you again, a little.

Basically, "all" you need to do for your list, is coming up with "the right" $$L(q_i, \dot q_i,t)$$ or $$H(q_i, \dot q_i,t)$$ and run the "mathematical mechanics". Being "right or right enough" in this starting point is your task. And it is a task, if not a burden ...

A word to set the stage: The grand attempt is to explain all effects from the real world with concepts from theoretical physics. Decompress the real world from a set of equations and procedures, and see what's still missing.

## 1. Great books from the past

Basically we used a later version of "Classical Mechanics", by Herbert Goldstein, AW. The preface for the 1st edition was written 1950 in Cambridge, MA. I still like it much today.

It's purpose is to teach and illustrate basic principles of theoretical mechanics. So as a student you are prepared to move into (m)any fields of theoretical physics. BTW, the weekly exercises on applying these subjects and concepts where ... tough.

So, starting with $$\vec F = d/dt (\vec p)$$ he expands on multiparticles, treatment of inner forces (mainly cancellation), forced movement, d'Alembert's principle. So Laplace comes into reach, and representation of enforced movements by using generalized coordinates $$q_i$$ and $$\dot q_i$$. He explains reasons and conditions to introduce transformation, e.g. into Hamilton equations. And you'll acquire a deeper understanding of the various (and many) conservation laws which follow from this matheamatics, should they exist.

You'll learn how to derive systems of differential equations to express movements from any of these. Some simple examples illustrate these concepts, e.g. for the simplified 2-body model of planets. Compared to your list, that's it, almost.

Some chapters from the book, to illustrate his drive: Elementary principles, Rigid bodies, Special relativity, Hamilton-Jacoby theory, with some Maxwell and Quantum mechanics here and there. This is the beauty of this book: to witness, how our predecessors used and adjusted concepts from mechanics for unknown territory.

~ ~ ~ You won't be able to work on your list without a sound background here, because it all starts with the right (at least most appropriate) equations as a starting point ~ ~ ~

Another classic is Landau Lifshitz, who dedicated 10 Volumes on subjects treated and touched by Goldstein. See e.g. the Vol.-list in his book on mechanics (pdf).

It's a similar style as Goldstein uses (no surprise), but in more detail and tried or illustrated on more examples. E.g. the planetary problem is treated in some more details (p. 33). As you can see, Vol 6 is on Fluid Mechanics, something you'll need to deal with mathematically when working on your list.

So if you're comfortable with the kind of mathematics (well, from physicist, mathematitions would "die") from Goldstein or LL, you'll be able to master (m)any other publications you need for your list.

~ ~ ~ I'm not aware more innovative books are available today for students in their 2nd year. Are there? ~ ~ ~

## 2. Theoretical astrophysics ?

I thought, this might be a field which comes closest to what you want. And even there many-in-one-books seem to be a problem. Let's illustrate it.

E.g. here is an "Introduction into Theoretical Astrophysiks" in Munich from 2017. Item 2 is your subject, and it mentions 3 sources. Let's have a look into Padmanabhan: Theoretical Astrophysics, Vol. I (Kapitel 2; only basic theory).

P. 51 ff. mentions the "reduced three body problem" . I'm not sure whether or not it considers planets own rotation. If you know Goldstein or LL, you'll immediately feel at home and can follow (hopefully).

Ok, let's have a look at a script from Zurich, 2021. As you can read from the TOC, of course they want to be able to deal with all the effects you encounter in outer space, beyond your focus. So you may want to have a look into e.g. chapters "Generalized kinetic theory" and "Astrophysical fluid dynamics" for treatment.

Tide isn't a subject there. So probably you'll find answers from studying literature, like papers, thesis etc.

And even without detailed mathematical knowledge: you won't be able to find answers in analytical form, when you look at real cases, not oversimplifications from your list. So you'd acquire a firm knowledge on numerical approaches, i.e. computer simulation.