Tumbling Downhill along a Given Curve

• Published in 2024
• Added on 2024-07-24

A cylinder will roll down an inclined plane in a straight line. A cone will roll around a circle on that plane and then will stop rolling. We ask the inverse question: For which curves drawn on the inclined plane $\mathbb{R}^2$ can one carve a shape that will roll downhill following precisely this prescribed curve and its translationally repeated copies? This simple question has a solution essentially always, but it turns out that for most curves, the shape will return to its initial orientation only after crossing a few copies of the curve - most often two copies will suffice, but some curves require an arbitrarily large number of copies.

Links

• http://arxiv.org/abs/2406.16336v1
• http://arxiv.org/pdf/2406.16336v1
• https://www.ams.org/journals/notices/202406/rnoti-p740.pdf

Other information

BibTeX entry

@article{TumblingDownhillalongaGivenCurve,
	key = {TumblingDownhillalongaGivenCurve},
	type = {article},
	title = {Tumbling Downhill along a Given Curve},
	author = {Jean-Pierre Eckmann and Yaroslav I. Sobolev and Tsvi Tlusty},
	abstract = {A cylinder will roll down an inclined plane in a straight line. A cone will
roll around a circle on that plane and then will stop rolling. We ask the
inverse question: For which curves drawn on the inclined plane {\$}\mathbb{\{}R{\}}^2{\$}
can one carve a shape that will roll downhill following precisely this
prescribed curve and its translationally repeated copies? This simple question
has a solution essentially always, but it turns out that for most curves, the
shape will return to its initial orientation only after crossing a few copies
of the curve - most often two copies will suffice, but some curves require an
arbitrarily large number of copies.},
	comment = {},
	date_added = {2024-07-24},
	date_published = {2024-12-07},
	urls = {http://arxiv.org/abs/2406.16336v1,http://arxiv.org/pdf/2406.16336v1,https://www.ams.org/journals/notices/202406/rnoti-p740.pdf},
	collections = {basically-physics,easily-explained,fun-maths-facts,geometry,things-to-make-and-do},
	url = {http://arxiv.org/abs/2406.16336v1 http://arxiv.org/pdf/2406.16336v1 https://www.ams.org/journals/notices/202406/rnoti-p740.pdf},
	urldate = {2024-07-24},
	year = 2024,
	archivePrefix = {arXiv},
	eprint = {2406.16336},
	primaryClass = {math-ph}
}