![]() How can this be? Consider, in particular, the prototypical swimmer confined to (or embedded within) the spherical surface depicted in Fig. Less well known is the fact that curved surfaces permit locomotors embedded within them to self-propel via translation without exchanging momentum with an environment ( 7– 9) (as is done in swimming, flying, and running in typical environments). Additionally, of course, gravitational interactions themselves are derived from the fundamental curvature of four-dimensional space-time ( 6), leading to explanations of dynamics, such as the precessing orbit of Mercury and the gravitational lensing of light. Geometric frustrated extended objects in curved space have been found to move due to the stress from the incompatibility ( 5). These geometric (and topological) effects can profoundly alter conventional dynamics, as when the inability to form periodic crystals on spherical surfaces ( 1, 2) recently gave rise to lively dynamics of essential crystalline defects ( 3, 4). For example, on a spherical surface, the square of the hypotenuse is not the sum of the squares of the legs, “parallel” lines meet at the poles, and the sum of the interior angles of a triangle grows with the triangle’s area. We envision that our work will be of use in a broad variety of contexts, such as active matter in curved space and robots navigating real-world environments with curved surfaces.Ĭurved surfaces are ubiquitous in physics, biology, engineering, and mathematics but are defined by features that defy intuitions derived from flat space. ![]() In this way, the robot both swims forward without momentum and becomes fixed in place with a finite momentum that can be released by ceasing the swimming motion. While this simple geometric effect predominates over short time, eventually the dissipative (frictional) and conservative forces, ubiquitous in real systems, couple to it to generate an emergent dynamics in which the swimming motion produces a force that is counter-balanced against residual gravitational forces. ![]() It produces shape changes comparable to the environment’s inverse curvatures and generates movement of 10 − 1 cm per gait. To realize this idea which remained unvalidated in experiments for almost 20 y, we show that a precision robophysical apparatus consisting of motors driven on curved tracks (and thereby confined to a spherical surface without a solid substrate) can self-propel without environmental momentum exchange. However, as first noted in and later in and, the noncommutativity of translations permits translation without momentum exchange in either gravitationally curved spacetime or the curved surfaces encountered by locomotors in real-world environments. Locomotion by shape changes or gas expulsion is assumed to require environmental interaction, due to conservation of momentum. ![]()
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