Despite the fact that, at present, skateboarding is one of the most popular forms of sports activities, the dynamics and stability of the motion of a skateboard have not yet been seriously investigated […]

(Alexander Kuleshov, 2007)

So it exists! The pumping motion is described by these Russian mathematicians (and I must assume, avid slalomers). The math exceeds me by far, but someone, somewhere, sometime might find this helpful, hopefully.

Ispolov, Y. G., & Smolnikov, B. A. (1996). Skateboard dynamics. Computer Methods in Applied Mechanics and Engineering, 131(3), 327–333. https://doi.org/10.1016/0045-7825(95)00932-9

(Ispolov & Smolnikov, 1996)

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Other notable papers:

Hubbard, M. (1979). Lateral Dynamics and Stability of the Skateboard. Journal of Applied Mechanics, 46(4), 931–936. https://doi.org/10.1115/1.3424680 [this is widely cited as the first effort to describe a skateboard mathematically; I’ve tried to explain before that there’s always a point with 0 lateral velocity (0 “turn”) and here this fact is shown in the first page as elementary of the model; he also shows a tilt-turn equation, but it’s wrong]

Österling, A. E. (2004). On the Skateboard, Kinematics and Dynamics (SSRN Scholarly Paper No. ID 1823272). Rochester, NY: Social Science Research Network. Retrieved from https://papers.ssrn.com/abstract=1823272 [this BSc math student says Dan Gesmer personally gave him Hubbard’s paper; he then went on to discover and prove the equation between tilt (lean) and turn of axles; I thought I was the first one in the universe and I could publish it some day and become famous (how presumptuous?)]

Kremnev, A., & Kuleshov, A. (2010). Nonlinear dynamics and stability of the skateboard. Discrete and Continuous Dynamical Systems, 3(1), 85–103. https://doi.org/10.3934/dcdss.2010.3.85 [these two are cited by others as getting the tilt-turn function first – Österling did, though he hadn’t published that in a peer-reviewed journal; the authors, of course, explore and prove other stuff too which I cannot even comprehend]

Rosatello, M., Dion, J.-L., Renaud, F., & Garibaldi, L. (2015). The Skateboard Speed Wobble. 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, 6, 1–10. https://doi.org/10.1115/DETC2015-47326 [this paper is probably what you should read on speed wobbles, instead of Carver’s musings; however, they assume rake to be 0, so it’s perhaps not complete; also, they say that bushings determine “a restoring torque between the wheel-set and the board, proportional to the tilt angle” which is not true, as I showed here]

Varszegi, B., Takacs, D., & Stepan, G. (2016, January 19). Skateboard: A Human Controlled Non-Holonomic System. ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. https://doi.org/10.1115/DETC2015-47512 [this is a conference paper and it deals with stability and speed wobbles]

Kunesch, M., & Usunov, A. (2010). Tic-tac: Accelerating a skateboard from rest without touching an external support. European Journal of Physics, 31(4), S25. https://doi.org/10.1088/0143-0807/31/4/S03 [the title of this one is self-explanatory; they expand on Ispolov & Smolnikov’s (1996) paper]

Endruweit, A., & Ermanni, P. (2002). Experimental and numerical investigations regarding the deformation-adapted design of a composite flex slalom skateboard. Sports Engineering, 5(3), 141–154. https://doi.org/10.1046/j.1460-2687.2002.00104.x [this is for the deck geeks; deals with the properties and behavior of a composite deck under slalom]

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A couple of sociological papers:

O’Connor, P. (2018). Beyond the youth culture: Understanding middle-aged skateboarders through temporal capital. International Review for the Sociology of Sport, 53(8), 924–943. https://doi.org/10.1177/1012690217691780

Willing, I., Bennett, A., Piispa, M., & Green, B. (2019). Skateboarding and the ‘Tired Generation’: Ageing in Youth Cultures and Lifestyle Sports. Sociology, 53(3), 503–518. https://doi.org/10.1177/0038038518776886

Robinson, R., Patterson, I., & Axelsen, M. (2014). The “Loneliness of the Long-Distance Runner” No More. Journal of Leisure Research, 46(4), 375–394. https://doi.org/10.1080/00222216.2014.11950333 (while not related to skateboard, it offers a pretty robust theoretical framework for describing the distance skating scene)