# Does any classical mechanics textbook solve Kepler's Problem?

• Will Flannery
In summary, none of the classical physics texts that I have access to solve the Kepler equation, M = E-e*sin(E).
Will Flannery
I have several* classical physics and mechanics texts, and none solve the Kepler problem (as far as I can tell), succinctly, solving the Kepler equation, M = E - e*sin(E), for E given M and e, or more generally determining the equations of motion for an orbiting object. In fact none even mention the Kepler equation.

The mechanics books all derive, from energy equations, Kepler's laws, but none even mention the Kepler equation.

*Goldstine, OpenStax, Thornton/Marion, Kleppner/Kolenkow, Young and Freedman

'Solving' is a big word. But there are books with exactly that in the title, as a simple google search demonstrates:

But I suppose you found that too ?

Wiki said:
It is often claimed that Kepler's equation "cannot be solved analytically"; see for example here. Whether this is true or not depends on whether one considers an infinite series (or one which does not always converge) to be an analytical solution. Other authors make the absurd claim that it cannot be solved at all; see for example Madabushi V. K. Chari; Sheppard Joel Salon; Numerical Methods in Electromagnetism, Academic Press, San Diego, CA, USA, 2000, ISBN 0-12-615760-X, p. 659

And somehow I suspect you don't mean numerical solutions...

PeroK
BvU said:
'Solving' is a big word. But there are books with exactly that in the title, as a simple google search demonstrates:

But I suppose you found that too ?

And somehow I suspect you don't mean numerical solutions...
I have Colwell's book. What I'm wondering is if the, or any, solution is included in any textbook used in the universities. Numerical is fine. None of the textbooks I've looked at even mention the Kepler equation, which is a little surprising to me because of its importance in the history of physics.

I'm also wondering if Kepler's equation plays much of a role in 'celestial mechanics' today. I have Danby's 'Celestial Mechanics' (1988) and it is central to the book. On the other hand I spent many years in aero engineering, computing trajectories, analyzing inertial nav systems, etc., and I'd never heard of it!, we used the computer and Newton's laws for everything.

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Will Flannery said:
I have Colwell's book
I have Danby's 'Celestial Mechanics'
I spent many years in aero engineering, computing trajectories, analyzing inertial nav systems, etc
Means you're the top expert (from my point of view) !
No relativistic corrections either ?

As ar as I know, textbooks treat Kepler's laws & equation, mention that the latter is transcendental, but stay clear of actually solving (rightly so, IMHO).

See John Taylor's Classical Mechanics, chapter 8 ("Two Body Central Force Problems"). I have not studied this so I'm not sure how close it comes to "solving" but it sure discusses at length, and does mention the "Kepler problem" by name.

EDIT: No, let's just say it discusses.

Will Flannery said:
*Goldstine, OpenStax, Thornton/Marion, Kleppner/Kolenkow, Young and Freedman

Goldstine certainly doesn't...

vela, BvU and gmax137
I worked at Stanford Telecom and knew a lot about the incredibly complex and precise GPS system, and there was a relativistic correction to something :), I think it was the clock (on the satellite), but I've forgotten the details.

The surprising thing to me is that the books don't even mention Kepler's equation. Once you have the equation Newton's method for approximating a solution is trivially easy and super accurate. And when you get to analytic solutions you get a theorem by LaGrange for inverting infinite series, so it's pretty neat but too much for me to try to understand. Colwell (I think) has a heuristic inversion that is immediate !

I have Taylor's book (pdf) ... he mentions Kepler's problem but as several others do misidentifies it, he says it's 'sketching the potential energy' for an orbit. He doesn't mention Kepler's equation.

Will Flannery said:
He doesn't mention Kepler's equation.

What in the name of reason are you talking about?
The two most important "forces" in the universe derive from 1/r potential functions. There is no more extensively studied system in all of classical Physics. If you cannot find these studies, perhaps a fellow named Newton, I. would be a good start. There is some literature after that.

!

Many books and papers treat Kepler's problems. I do not know of any that "solve" Kepler's problem if in the sense of "solve" you mean giving and expression for position vector r as a function of time t with a finite number of terms, and familiar (exponential, trigonometric, or even special functions). (Not sure about elliptic functions, Weirstrass, or Jacobi Functions. They seem to solve everything else.) On the other hand, it sometimes seems not a month goes by when a paper does not come out describing "regularization" where Kepler's equation is "solved" by their algorithm effectively.

I confess I misunderstood your earlier posts when you suggested no textbooks solves the equation. To my mind, and in most textbooks, they regard the solution to the inverse square problems as the proof that the orbit is an elliptic path, with Kepler's three laws in consequence.

One area that I am familiar with that seems intriguing is the use of Kustaheimo-Stiefel transformation to change the Kepler's problem to four dimensional harmonic oscillator. (I am sure this is in google) It seems like harmonic oscillator equations would have a simple solution, so the transformation could give you a solution that might be satisfying to you.

One advantage to series solutions to the equation would appear to be numerical integration eror would grow with time when integrating with finite time steps, for example Euler integration. A series solution that converges could be truncated after (for example, the 15th decimal ) which is all the computer precision may allow anyway.

dextercioby, hutchphd and BvU
The 2nd edition of Goldstein certainly has this. I just pulled it off the shelf and see Kepler's equation on page 101, and at the end of the chapter (pages 123-124) there is a sequence of 3 problems (18-20) that ask the reader to derive a few solution methods. I don't know if the other editions have this material, since I'm not familiar with them.

It is a pretty standard book, or at least used to be.

jason

PeroK, hutchphd, Vanadium 50 and 1 other person
I was so convinced Will mentioned Goldstein that I didn't even check !
I have the same edition and on p 509 he even unleashes perturbation theory and brings in general relativity . Needless to say I never got that far ...

##\ ##

BvU said:
I was so convinced Will mentioned Goldstein that I didn't even check !
I have the same edition and on p 509 he even unleashes perturbation theory and brings in general relativity . Needless to say I never got that far ...

##\ ##
Will mentioned Goldstine. Perhaps it wasn’t a typo!

BvU
jasonRF said:
The 2nd edition of Goldstein certainly has this. I just pulled it off the shelf and see Kepler's equation on page 101, and at the end of the chapter (pages 123-124) there is a sequence of 3 problems (18-20) that ask the reader to derive a few solution methods. I don't know if the other editions have this material, since I'm not familiar with them.

It is a pretty standard book, or at least used to be.

jason
I have the first edition, I think, printed in 1965. I downloaded the 3rd edition, Goldstein, Poole, and Safko, and there is a new section that doesn't appear in ed. 1, titled 'The motion in time in the Kepler Problem' where Kepler's equation is covered, and it adds ...
'Indeed, it can be claimed that the practical need to solve Kepler's eq to accuracies of an arc second over the whole range of eccentricities fathered many of the developments in numerical math in the 18th and 19th centuries'
This seems unlikely to me as Newton's method is super accurate with 10 iterations, and I am under the impression that even today it's the gold standard*.
*why - from Colwell, covering the Lie Series soln, .. "to display some numerical results comparing <the Lie series method> to the Newton method for each e, M pair until successive itertions differ by less that 10^-10 ... <table follows>

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At work today I looked at my copy of Marion and Thornton (3rd edition). It presents a derivation of the Kepler equation and mentions that approximation techniques must be use to solve it. There is a footnote indicating that more than 100 techniques have been developed and pointing the reader to a reference, but they don't present any of them. I know the 2nd edition (before Thornton) also has Kepler's equation but I don't know if it present solutions.

You may already know this book, but if not you might be interested in
An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition (AIAA Education): Battin, Richard H, R Battin, Massachusetts Institute of Technology: 9781563473425: Amazon.com: Books
It isn't a mechanics book for a physics class, of course, but it is pretty good. I had it checked out of a library at some point in the past and thought it would be fun to work through ... perhaps when I'm retired!

jason

jasonRF said:
At work today I looked at my copy of Marion and Thornton (3rd edition). It presents a derivation of the Kepler equation and mentions that approximation techniques must be use to solve it. There is a footnote indicating that more than 100 techniques have been developed and pointing the reader to a reference, but they don't present any of them. I know the 2nd edition (before Thornton) also has Kepler's equation but I don't know if it present solutions.

You may already know this book, but if not you might be interested in
An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition (AIAA Education): Battin, Richard H, R Battin, Massachusetts Institute of Technology: 9781563473425: Amazon.com: Books
It isn't a mechanics book for a physics class, of course, but it is pretty good. I had it checked out of a library at some point in the past and thought it would be fun to work through ... perhaps when I'm retired!

jason
Weird ! I have Thornton & Marion, Classical Dynamics of Particles and Systems, 2008, and it doesn't mention Kepler's equation, and in the section 'Planetary Motion - Kepler's Problem' Kepler's laws are derived but there is no mention of the equations of motion.

Apparently Thornton removed it sometime after the 3rd edition, which came out in 1988. It was contained in a section with the words "Kepler's equation" in the title. I don't understand the motivation for dropping a section that was already written - weird indeed! Anyway, according to the preface Jerry Marion passed away in 1981 so the 2nd edition was the last one he worked on. In grad school my officemate had the 2nd edition, and for some reason (I don't recall why) at the time I thought it was superior to the 3rd.

jason

I think the reason you don't see much (if anything) about the Kepler equation in most mechanics textbooks might be because it is not easy to even define the issue in simple terms. This thread and the several predecessor threads piqued my interest so I spent a little time investigating (via wiki). The terms are pretty arcane: mean motion, mean anomaly, eccentric anomaly, true anomaly, aphelion, periapsis, and on and on. It is all very interesting but unless you're actually trying to plot the orbit of a comet from a few observations, it is a lot of baggage to take on mentally. And then, once you do get up to speed on the nomenclature, you find a transcendental equation. As a problem to solve, fascinating. As a pedagogical exercise, maybe not.

Compare that to say, SHM of a pendulum.

jasonRF, vela and BvU
I felt the same way a few weeks ago about the anomalies ... but with a few graphs they're clear, and now I like them, they're kind of fun! And Kepler's eq. has an easy graphical derivation ...
For many years Kepler's eq was the only way to plot an orbit as a function of time.

Oh, I agree, it is interesting. And it turns out, "anomaly" means "how much different from a circular orbit." Figuring this out is on my bucket list.

It is not too surprising that mechanics texts at the upper undergrad/graduate level do not address solutions of Kepler's equation, where they may demonstrate and ask the student to learn or at least to follow the proof that an inverse square field leads to conic section orbits, in the case of closed orbits, elliptic orbits. Perhaps if there were a "closed form" solution, it would be noted by them. In the same manner, few mechanics books show the solution to a heavy spherical top with one point fixed, and present limiting cases rather than general cases or qualitative solutions.

I think Battin shows a few algorithms for Kepler's equation, although I cannot say for sure. I use Battin to examine Lambert's problem. BTW, I also can understand how Kepler's problem is missing or understated in Flannery's background. My work in space sciences examines more Lambert's problem. Sometimes Lambert is more applied, the idea that given two positions and a time of flight, what velocity do you need to get from here to there in a prescribed time in an inverse square field. Isn't this more relevant to am intercept or spacecraft rendezvous. Like Kepler's problem, Lambert's problem is "solved" by several good methods, but (as far as I know), there is not closed form for the required velocity. Most algorithms need iteration, Thorne's method doesn't , but there the velocity is still determined by a method, not an equation.

Back to the main point. Physics curricula and the textbook authors sometimes use historical context to choose the material, and their treatment of Kepler's problem and where their treatment finishes relies on view that every student should know Kepler's laws and Newton's insight of elliptic orbits, and not necessarily on how to solve specific equations, that are not easy to attack.

jasonRF
Will Flannery said:
I worked at Stanford Telecom and knew a lot about the incredibly complex and precise GPS system, and there was a relativistic correction to something :), I think it was the clock (on the satellite), but I've forgotten the details.
See https://www.researchgate.net/publication/26386594_Relativity_in_the_Global_Positioning_System by Neil Ashby which describes in detail the multiple relativistic corrections applied in the GPS.

Orbital Mechanics for Engineering Students
Howard Curtis
Appendix D gives several MATLAB algorithms for solving Kepler's equation (all by Newton's method).

Paragraph before the first algorithm:

These programs are presented solely as an alternative to carrying out otherwise lengthy hand computations and are intended for academic use only. They are all based exclusively on the introductory material presented in this text and therefore do not include the effects of perturbations of any kind.

It was a homework exercise in my "University Physics" textbook at Miami University of Ohio in a calculus based first year physics class in the 1988-1989 academic year. I don't know who the authors were.

## 1. What is Kepler's Problem in classical mechanics?

Kepler's Problem is a fundamental problem in classical mechanics that involves calculating the motion of a planet or satellite orbiting a central body under the influence of gravity.

## 2. Why is solving Kepler's Problem important in classical mechanics?

Solving Kepler's Problem is important because it allows us to accurately predict the motion of celestial bodies, which is essential for space exploration, navigation, and understanding the laws of gravity.

## 3. How do classical mechanics textbooks solve Kepler's Problem?

Classical mechanics textbooks typically use mathematical methods such as Newton's laws of motion, the law of gravitation, and calculus to solve Kepler's Problem.

## 4. Are there any limitations to solving Kepler's Problem in classical mechanics?

Yes, there are limitations to solving Kepler's Problem in classical mechanics. For example, it can only accurately predict the motion of a two-body system, and it does not take into account the effects of relativity.

## 5. Are there any alternative methods for solving Kepler's Problem?

Yes, there are alternative methods for solving Kepler's Problem, such as using computer simulations and numerical methods. These methods can account for more complex factors, such as the influence of multiple bodies and relativistic effects.

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