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In the novel Gravity's Rainbow by Thomas Pynchon, there is a chapter in which Slothrop (the main character) thinks about this equation:

$$\theta \frac{d^2\phi}{dt^2} + \delta^* \frac{d\phi}{dt} + \frac{\partial L}{\partial\alpha}(s_1-s_2)\alpha = -\frac{\partial R}{\partial\beta} s_3\beta\,.$$

It seems to be something to do with the V2 rocket, and yaw control. Does anyone recognise this equation from a book or research paper? What do the various variables stand for? He may have invented the equation, I suppose. How would one go about solving an equation like this? I find it strange that it contains both partial derivatives and total derivatives. However, there are equations in Hamiltonian mechanics like this.

Any help would be much appreciated. Thank you.

Russell Borogove
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    I took the liberty of (a) including an image of the equation, and (b) specializing the title to indicate the Pynchon-source of the equation. – Joseph O'Rourke Mar 19 '17 at 02:53
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    Googling, I stumbled on a paper by Magueijo and Smolin (2004, Classical and Quantum Gravity, Volume 21) called "Gravity's rainbow" - in context of relativity theory, they consider energy-dependent families of metrics and call these rainbow metrics. –  Mar 19 '17 at 09:46
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    Maybe somebody on the Physics or Space Exploration site could shed some light. – Jason C Mar 19 '17 at 16:11
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    How can I re-post this on the Space Exploration site? –  Mar 19 '17 at 18:43
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    @user947185 Flag it for moderator attention and ask them to do it. –  Mar 22 '17 at 16:12
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    The partial derivatives are likely constants, where $\alpha$ and $\beta$ deviate small amounts from zero. $\alpha$ and $\beta$ are likely the components of the angle of attack, which are zero when the rocket is pointed into the relative air flow. $\phi$ is usually roll, but I suppose it could be yaw here. $\delta^*$ is a damping coefficient for the oscillation of $\phi$. $\theta$ represents the response of $\phi$ to the forces, though it seems like there should be a $\phi$ term to represent a simple harmonic oscillator. The asymmetry in $\alpha$ and $\beta$ suggests an aircraft. – Mark Adler Mar 26 '17 at 03:36
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    Reading the text preceding the equation in the book, "which describes motion under the aspect of yaw control" (which should be included in the question here), I'm guessing by yaw he meant side-slip, which is $\beta$. Then the terms on the left are responding to the controlled "yaw" on the right. $\phi$ may be roll or some other parameter. – Mark Adler Mar 26 '17 at 03:51
  • L could be lift (it seems related to angle of attack) and R could be side-force (it seems related to side-slip). – Organic Marble Mar 26 '17 at 14:25
  • Looking at some stability and control papers, I suspect ϕ is the bank angle. There should be a control surface deflection angle in here somewhere - maybe θ ? – Organic Marble Mar 26 '17 at 15:55

2 Answers2

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Schachterle, Lance, and P. K. Aravind. "The three equations in Gravity's Rainbow." Pynchon Notes 46-49 (2000): 157-170. Journal Link.

"In our view, Pynchon inscribes these equations into Gravity's Rainbow to challenge readers with yet another form of authority within the text."

Sorry—hit a pay wall...
Later: Got through the pay wall:

          Pynchon1
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Joseph O'Rourke
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    I myself don't accept their conclusion that this is "not a genuine mathematical expression." It seems unlikely Pynchon would concoct this without a basis in some real document. But I defer to "rocket scientists"... – Joseph O'Rourke Mar 19 '17 at 01:17
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    Are there any rocket scientists out there who can help us? Is Pynchon's equation genuine or bogus? –  Mar 19 '17 at 01:19
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    There's a journal devoted entirely to the study of the works of Thomas Pynchon? And here I thought math journals were highly specialized... – Nate Eldredge Mar 19 '17 at 02:48
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    @NateEldredge: "Pynchon Notes was a journal devoted to studying the works of Thomas Pynchon. Running from 1979 to 2009." 30 yrs. – Joseph O'Rourke Mar 19 '17 at 02:50
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    It also seems strange to me that Pynchon would put in a meaningless differential equation, except maybe as a joke. He studied physics at Cornell. –  Mar 19 '17 at 04:34
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    I believe $L$ and $R$ must be known (since otherwise the equation would be underdetermined); in that case, it is just a general ordinary second order equation with solutions of the form$$\phi(t)=c_1+\int_1^te^{-c_0u}\left(c_2+\int_1^ue^{c_0v}F(v)dv\right)du$$with some known $F$ –  Mar 19 '17 at 08:23
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    Hmm. By the criterion of "two literary critics tried to pattern-match the equation against two textbooks and failed, so the equation is fake", I've a feeling that almost my entire career is fake. – David Richerby Mar 19 '17 at 09:07
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    @DavidRicherby, "pattern-match" may be a slightly unfair way of putting it. Note that one of the authors, P.K. Aravind is a professor of physics https://users.wpi.edu/~paravind/ (though, not, so far as I can tell, an expert on rocket science), –  Mar 19 '17 at 15:56
  • @j.c. The way they describe it seems like pattern-matching to me. Knowing that one of the authors is a physicist gives me more confidence in the process but, still, I doubt they'd have spotted, e.g., $a+b-c=d$ as being a genuine equation if the book actually presented $a-x=k$ and $b=d-k$ and Pynchon had combined the two and renamed $x$ as $c$. – David Richerby Mar 20 '17 at 16:08
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    I've added a link to the journal article. @DavidRicherby I find it more likely that, due to the likely nontechnical readership of the journal, their description of what they did in that particular excerpt might be written in a way that sounds like "pattern-matching" on a first reading. The possibility that a professor of physics would be fooled by the simple transformation you describe seems more remote to me. I believe the full article text speaks in favor of my more charitable explanation. –  Mar 22 '17 at 13:19
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    @DavidRicherby Lance Schachterle is also scientifically literate. You're embarrassing yourself by talking about him and Aravind in this fashion. –  Mar 22 '17 at 15:32
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    I am interested in what @j.c. says. It looks as if Pynchon has taken an ODE and a PDE and added them together! The variables are distinct. It may be a real equation, but it could be from an out-of-print book from about 1972. Then again, Pynchon working at Boeing for 2 years, and he may even have invented his own mathematical models! –  Mar 22 '17 at 21:41
  • @NoahStein Why not. It's a satirical novel after all. It's perfectly normal to expect something nonsensical in there, which would fit the theme perfectly. – xji Apr 01 '17 at 14:23
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Engelhardt, N. & Engelhardt, H., (2018) “The Momentum of Pynchon's Secret Formula: Gravity’s Rainbow’s Second Equation between Archival Sources and Fiction”, Orbit: A Journal of American Literature 6(1). Journal link here

The source of the equation was found in a Control System book for the V-2 rockets. The reason the equation wasn't readily understandable was because the coefficients used were part of a larger set of equations that couldn't be known without context.

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SomethingCool
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