A conserved quantity in thin body dynamics

Abstract

Thin, solid bodies with metric symmetries admit a restricted form of reparameterization invariance. Their dynamical equilibria include motions with both rigid and flowing aspects. On such configurations, a quantity is conserved along the intrinsic coordinate corresponding to the symmetry. As an example of its utility, this conserved quantity is combined with linear and angular momentum currents to construct solutions for the equilibria of a rotating, flowing string, for which it is akin to Bernoulli's constant.The mechanics of thin bodies are amenable to a description in which energies are effectively expressed with an integral over a surface or curve. On sufficiently long time scales, and when longitudinal waves are not important, the effective action is often taken to be metrically constrained, with a low-dimensional body stress tensor enforcing local distances between moving pieces of material. This approach, either at the level of the action or equations of motion, has been used to describe systems at many scales, from the overdamped motions of flagella [1,2] Thin bodies such as rods and solid membranes moving in a higher-dimensional space share characteristics with both rigid bodies and fluids. The metric constraints of inextensible solids preclude the broad reparameterization invariance inherent to fluids and fluid membranes, leaving only highly restricted reparameterization invariance associated with any metric symmetries the body may possess. Effectively one has global, rather than local, symmetries on the body. However, these intrinsic symmetries are not trivially identifiable with rigid transformations of embedding coordinates, because a lower-dimensional body can adopt extrinsic curvature in the surrounding space. Thus, unlike space-filling rigid continua, the body may display additional material flows in equilibrium in addition to, and distinct from, rigid motions. Simple examples include a string being unspooled [12], a cable being laid down on the ocean floor [14], or the flowing skirt of a rotating dancer [17]. Figure 1 is a schematic of such a combined rigid-flowing configuration.We will describe such a system using several types of coordinates. Aside from a time coordinate, there are three important sets of spatial coordinates to consider. First, there are time-independent coordinates attached to pieces of material. For an inextensible body, these can also be associated with geometric parameters, such as arc length. Second, there are coordinates for which material motions tangent to the body have been removed. These are effectively Eulerian from the intrinsic viewpoint of the body, but they are attached to a body moving through an embedding space, so have some Lagrangian character as well. This viewpoint is called the "orthogonal gauge" in [18]

Pseudomomentum: origins and consequences

“…One consequence is that when the flow can be expressed as a steady velocity field, the streamline component of the pseudomomentum equation expresses the same content as the energy equation. This is partly why one of the present authors misleadingly identified the conserved quantity associated with the material symmetry of a flowing string with Bernoulli's constant in [65].…”

“…Alternatively, one could similarly construct a quadrature for Z, as in the prior work [50]. A related work on rotating, flowing strings [65] used an alternate method in which the conserved Z-component of linear momentum was used to classify the equilibria. This quantity can be captured in the present example by inserting a spatial translational shift δx = Ẑ into (74).…”

Pseudomomentum: origins and consequences

“…One consequence is that when the flow can be expressed as a steady velocity field, the streamline component of the pseudomomentum equation expresses the same content as the energy equation. This is partly why one of the present authors misleadingly identified the conserved quantity associated with the material symmetry of a flowing string with Bernoulli's constant in [61].…”

“…The first term in the pseudomomentum (56) is the tangential component of the contact force. Thus, we may also obtain the tangential reaction forces at the rod ends and at the points of discontinuity in the rod stiffness and the channel curvature directly from this definition, the jump condition (61), and the prescription (62)(63):…”

“…Alternatively, one could similarly construct a quadrature for Z, as in the prior work [47]. A related work on rotating, flowing strings [61] used an alternate method in which the conserved Z-component of linear momentum was used to classify the equilibria. This quantity can be captured in the present example by inserting a spatial translational shift δx = Ẑ into (74).…”