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Swimming in Spacetime: Motion by Cyclic Changes in Body Shape

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Swimming in Spacetime: Motion by Cyclic Changes in Body Shape RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing ␪ by ⌬␪ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding ␾ fixed, then increasing ␾ by (1). At low Reynolds number, the effects of in space can be accomplished through cyclic, ⌬␾ while holding ␪ fixed, decreasing ␪, then inertia are negligible and, in the absence of engineered changes in the shape of a body. decreasing ␾, which brings the system back external forces, bodies are at rest. Nevertheless, Motion in space can be accomplished without to the original relative configuration. In those as a body changes its shape, its location and thrust or external forces. phases of the cycle in which ␪ is held fixed, orientation generally change. A cyclic change in Swimming with spherical caps. The two- the system does not rotate. The rotation of the the shape of a body can lead to a net translation sphere example can be generalized to two rigid system during the two phases in which ␪ is or rotation. The net translation or rotation does bodies of arbitrary shape with a common fixed changed do not balance because ␾ is differ- not depend on the speed with which the shape point (3). Consider two circular spherical caps ent. For small ␥ (small caps), small ␾ (small changes are carried out; it is a consequence of of angular radius ␥, with uniform surface den- separation), and small ⌬␪ and ⌬␾ (small the geometry of the sequence of shapes, a clas- sity and mass m, on a sphere of radius R. The deformations), the net motion of the system sical example of geometric phase (2). For ex- relative configuration is specified by two angles per cycle of deformation is approximately ample, the cilia of a paramecium effectively (Fig. 1). Let xˆ, yˆ, and zˆ be a right-handed 1 define and allow changes in its shape, and loco- orthonormal basis with origin at the center of ⌬␺ ϭ ␥2 ⌬␪⌬␾ (2) motion is accomplished through cyclic changes the sphere. The caps are symmetrically dis- 2 in its shape through motion of its cilia. placed above and below the x-y plane by the Having made everything small compared Consider two concentric spheres in flat angle ␾, and they are rotated oppositely about with the size of the sphere, it is apparent that space with equal moments of inertia connect- their centers by the angle ␪. The angles ␪ and ␾ the essence of the matter is not that these are ed at their centers (3–5). If one sphere is are deformation coordinates (6). As the relative rigid bodies, but that the system lives on a rotated with respect to the other, then the configuration changes, the orientation of the curved manifold. orientation of the system adjusts appropriate- system adjusts dynamically. Because of the Swimming on curved manifolds. Rigid ly. The adjustment can be determined using symmetry of the system, it will move by rotat- bodies, defined by a large number of redundant the fact that angular momentum is conserved. ing around the zˆ axis. Let ␺ be the longitude of distance constraints between constituents, can- If the total angular momentum is zero, the the system, the angular displacement of the not, in general, move on a manifold of noncon- angular velocities of the spheres are equal but system around the zˆ axis. Assuming zero total stant curvature because the redundant con- opposite. Therefore, if one sphere is rotated angular momentum, the equation of motion of straints cannot remain consistent. Consider three about an axis by an angle ␪ with respect to the ␺ is masses on the vertices of a triangle, with geo- other sphere, then the first sphere rotates in desics of fixed length connecting them. Now D␪͑t͒Csin␾͑t͒ space about that axis by ␪/2 and the other ␺͑ ͒ ϭ place a fourth mass in the middle of the triangle, D t ͑ ␾͑ ͒͒2 ϩ ͑ ␾͑ ͒͒2 rotates by Ϫ␪/2. If the two spheres undergo a A cos t C sin t equidistant from each of the three masses. The sequence of rotations with respect to each (1) distance to those masses will depend on the other and are then brought back to their orig- curvature of the manifold. This system of four inal relative orientation, then their orientation where D is the derivative operator, C is the masses cannot move to a region of different in space can undergo a net rotation. The net moment of inertia of each cap about the radial curvature without some of the constraint dis- rotation of the system does not depend on the line through its center, and A is the moment of tances changing. This is a consequence of the speed with which these relative rotations oc- inertia about any line through the center of the fact that the constraints are redundant—more cur; it depends only on the sequence of rela- sphere perpendicular to this line. The denomina- constraints are specified than are needed to tive orientations. This is another classical tor is half the moment of inertia of the system specify the relative location of the masses. If example of geometric phase. about the zˆ axis; the numerator is half the zˆ instead of specifying that the fourth mass is Generalizing the latter example, I show here component of the angular momentum due to the equidistant from the three masses it is specified that it is possible for quasi-rigid bodies to swim twisting (nonzero D␪) of the cap. Each cap makes that it is a certain distance from two of the on frictionless curved manifolds simply by way the same contribution to the zˆ moment of inertia vertices, then all of the constraints can be main- and the zˆ component of the angular momentum, tained even as the curvature changes. Such a Massachusetts Institute of Technology, Cambridge, so the additional factors of two in the numerator rigid body with irredundant constraints will be MA 02139, USA.E-mail: [email protected] and denominator cancel. Note that the counterro- called a quasi-rigid body. A quasi-rigid body has www.sciencemag.org SCIENCE VOL 299 21 MARCH 2003 1865 R ESEARCH A RTICLE a well-defined configuration even though the metric, q is a tuple of manifold coordinates, and For the sphere, choose spherical coordi- distances between all the constituents are not v is a tuple of the associated generalized veloc- nates, with the equator bisecting the struts (Fig. fixed. One particular class of quasi-rigid bodies ities (7). A Lagrangian for a quasi-rigid body 2). The longitude of the vertex at time t is ␺(t). has a tree-like topology. From each vertex ex- can be obtained by defining generalized coor- The length of the geodesic struts, which are tend an arbitrary number of branches or struts. dinates for the system that incorporate the time- spherical arcs, is specified by the angular extent At the end of any strut, more struts may be dependent constraints among the constituent e(t), measured from the center of the sphere. attached. Masses may be attached to the verti- particles, deriving the associated transformation The separation angle between the struts and the ces. The configuration of the system is deter- of the generalized velocities, and rewriting the line of symmetry is ␣(t). Conservation of mo- mined if the lengths of all the struts and the free Lagrangians for the constituent masses in mentum, with zero initial momentum, is angles they make with the connecting struts are terms of these system coordinates (8). enough to determine the motion of the system specified. The first example shows that more Consider the dynamics of such a quasi- as it deforms general quasi-rigid bodies are allowed; they rigid body on two illustrative manifolds: the D␺͑t͒ ϭ need not be tree-like and the struts need not be plane and the sphere (9). Because of the ͑ ␣͑ ͒ ͑ ͒ ͑ ͒ ␣͑ ͒ Ϫ ␣͑ ͒ ͑ ͒͒ geodesics. Struts of quasi-rigid bodies can be symmetry of the body and the manifold, these 2m1 D t sine t cose t sin t cos t De t ϩ ͑ ␣͑ ͒͒2 ϩ ͑ ␣͑ ͒͒2͑ ͑ ͒͒2 constructed from a local lattice or truss of nearly three-mass quasi-rigid bodies will move only m0 2m1 cos t 2m1 sin t cose t rigid rods.
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