Swim-like motion of bodies immersed in an ideal fluid

Swim-like motion of bodies immersed in an ideal fluid

Abstract

The connection between swimming and control theory is attracting increasing attention in the recent literature. Starting from an idea of Alberto Bressan [8] we study the system of a planar body whose position and shape are described by a finite number of parameters, and is immersed in a 2-dimensional ideal and incompressible fluid in terms of gauge field on the space of shapes. We focus on a class of deformations measure preserving which are diffeomeorphisms whose existence is ensured by the Riemann Mapping Theorem. We face a crucial problem: the presence of possible non vanishing initial impulse. If the body starts with zero initial impulse we recover the results present in literature (Marsden, Munnier and oths). If instead the body starts with an initial impulse different from zero, the swimmer can self-propel in almost any direction if it can undergo shape changes without any bound on their velocity. This interesting observation, together with the analysis of the controllability of this system, seems innovative.

Swimming, Ideal fluid, Control, Gauge theory.
\subjclass

74F10, 74L15, 76B99, 76Z10

Introduction

In this work we are interested in studying the self-propulsion of a deformable body in a fluid. This kind of systems is attracting an increasing interest in recent literature. Many authors focus on two different type of fluids. Some of them consider swimming at micro scale in a Stokes fluid [2, 4, 5, 6, 27, 35, 40], because in this regime the inertial terms can be neglected and the hydrodynamic equations are linear. Others are interested in bodies immersed in an ideal incompressible fluid [8, 18, 23, 30, 33] and also in this case the hydrodynamic equations turn out to be linear.
We deal with the last case, in particular we study a deformable body -typically a swimmer or a fish- immersed in an ideal and irrotational fluid. This special case has an interesting geometric nature and there is an attractive mathematical framework for it. We exploit this intrinsically geometrical structure of the problem inspired by [39, 40] and [32], in which they interpret the system in terms of gauge field on the space of shapes. The choice of taking into account the inertia can apparently lead to a more complex system, but neglecting the viscosity the hydrodynamic equations are still linear, and this fact makes the system more manageable. The same fluid regime and existence of solutions of these hydrodynamic equations has been studied in [18] regarding the motion of rigid bodies.
We start from an early idea of Alberto Bressan [8] and some unpublished developments, according to which the shape changes can be described by a finite number of parameters. These kind of systems, where the controls are precisely given by further degrees of freedom of the systems, have been first studied deeply by Aldo Bressan, see e.g. [9, 10, 11]. In this framework we show that the composed system “fluid-swimmer” is Lagrangian geodesic. Next, coupling this fact with some techniques developed in [33], we are able to show that the kinetic energy of the system (i.e. the Lagrangian) is bundle-like, a concept by Bruce Reinhart [38] and introduced in control theory by Franco Rampazzo in [37]. This leads us to express the equations of motion as linear control equations, where any quadratic term is vanishing, radically simplifying our final analysis on the system. The geometric construction of the control dynamic equations follows substantially the line of thought of [13, 29].
At a first glance, the deformations of the swimmer are naturally given by diffeomorphisms, that are infinite dimensional objects. By considering a planar setting and making use of complex analysis, as suggested in [14, 33] the Riemann Mapping Theorem plays a crucial role in describing the shape changes of the swimmer. It turns out that the diffeomorphisms can be parametrized by appropriate complex converging series. In the literature other authors exploit the same way of describing the shape changes by conformal maps, for example in [14] in the environment of the Stokes approximate regime or in [33] in the case of an ideal and irrotational fluid, in which they take into account only a finite number of terms to represent the diffeomorphisms. We follow substantially an analogous approach to merge this idea with the setting of Alberto Bressan. The choice of using a finite number of parameters means that the kind of deformations that we consider is more restricted but still enough to describe a wide range of swimmers.
In order to have a more manageable system that the one in [33], we establish a connection between the use of complex and real shape parameters. We show that, if we consider small shape changes, a well precise choice of the real and imaginary part of the shape parameters leads to obtain exactly the same deformation proposed in [32], which use a rather different parametrization governed by suitable small deformation. Therefore we get a description of our system with a finite number of parameters/coordinates, which is useful to apply the idea of controlling the shape coordinates to steer the swimmer between two different configurations. In this environment we recover the well known Scallop Theorem [3] in the case in which we suppose to have only one real shape parameter. Thanks to the idea of using a finite number of parameters we can reduce our dynamic equations to a control system. The controllability issue has been recently linked to the problem of swimming [2, 3, 22, 27] since it helps in solving effectively motion planning or optimal control problems.

We point out that in the Stokes regime there are interesting results, for example in [27] the authors study the controllability of a swimmer which performs small deformations around the sphere, or in [23] in which he considers the swimming mechanism as a ‘broken-line’-like structure, formed by an ordered sequence of finitely many sets. Even in [2, 16] they study the controllability of a slender swimmer composed by links immersed in a viscous fluid at low Reynolds number.
In the present work we deal with the ideal and irrotational case, neglecting viscosity. In particular differently from what was done in other works, we focus our attention on a crucial problem: the presence of an initial impulse. The case of zero initial linear momentum is studied in literature concerning systems of different nature: both in the multi-particle or many-bodies field, [19, 21, 20], and shape changing bodies, [30, 32, 33, 34], as the equation of motion are a driftless affine control system whose controllability can be studied using classical techniques. Instead, the case of a non vanishing initial impulse leads us to a more complex system since the equations of motion involve also a non zero drift term and their controllability is more tricky to study. Therefore we have two contributions to the motion of the system: the first one that is purely geometrical and determined by the structure of the problem, and the second one, strictly linked to the presence of a non vanishing initial impulse.
The controllability of this kind of systems is studied in detail, and among other facts it is worth noting that we need at least three real shape parameters to make the system controllable.

We have three state parameters, three conjugate variables and at least three controls. Despite the evident complexity of the computations linked to this number of variables, we managed to obtain interesting results.

The plan of the paper is the following. In Section 1 we present in some detail the geometric aspects useful to formulate our problem. The proper geometrical setting of the swimmer in a 2-dimensional fluid is faced in Section 2. Section 3 contains an exhaustive study, in a complex setting, of the deformation of the body, together with the construction of the equation of motion. We deal with all the controllability issues in Section 4.

1 Preliminaries

This section covers some auxiliary mathematical topics, in particular from Lie groups, fiber bundles and connections that we shall need later. This summary will be helpful to set the notation, fill in some gaps, and to provide a guide to the literature for needed background.

1.1 Lie Groups

Let us start from some geometric and algebraic notions on Lie groups, that arise in discussing conservation laws for mechanical and control systems and in the analysis of systems with some underlying symmetry.

Definition 1.1

A Lie group is a smooth manifold that is a group with identity element , and for which the group operations of multiplication, for , and inversion, , are smooth.

Before giving a brief description of some of the theory of Lie groups we mention an important example: the group of linear isomorphisms of to itself. This is a Lie group of dimension called the general linear group and denoted by . The conditions for a Lie group are easily checked. This is a manifold, since it is an open subset of the linear space of all linear maps of to itself; the group operations are smooth, since they are algebraic operations on the matrix entries.

Definition 1.2

A matrix Lie group is a set of invertible matrices that is closed under matrix multiplication and that is a submanifold of .

Lie groups are frequently studied in conjunction with Lie algebras, which are associated with the tangent spaces of Lie groups as we now describe.

Definition 1.3

For any pair of matrices , we define the matrix Lie bracket .

Proposition 1.1

The matrix Lie bracket operation has the following two properties:

  • For any matrices and , (skew-symmetry).

  • For any matrices , , and ,
    . (This is known as the Jacobi identity.)

As is known, properties (i) and (ii) above are often thought as the definition of more general Lie brackets (than ) on vector spaces called Lie algebras.

Definition 1.4

A (matrix) Lie algebra is a set of matrices that is a vector space with respect to the usual operations of matrix addition and multiplication by real numbers (scalars) and that is closed under the matrix Lie bracket operation .

Proposition 1.2

For any matrix Lie group , the tangent space at the identity is a Lie algebra.
As usual, for matrix Lie groups one denotes

We now describe an example that plays an important role in mechanics and control.

The plane Euclidean Group
Consider the Lie group of all matrices of the form

(1.1)

where and . This group is usually denoted by and is called the special Euclidean group. The corresponding Lie algebra, , is three-dimensional and is spanned by

(1.2)

The special Euclidean group is of central interest in mechanics since it describes the set of rigid motions and coordinate transformations on the plane. Let be a matrix Lie group and let be the corresponding Lie algebra. The dimensions of the differentiable manifold and the vector space are of course the same, and there must be a one-to-one local correspondence between a neighborhood of in and a neighborhood of the identity element in . An explicit local correspondence is provided by the exponential mapping , which we now describe. For any (the space of matrices). is defined by

(1.3)

This map for can be defined by the exponential of the elements of the Lie algebra . More precisely

(1.4)
(1.5)

Since for all , we have that

that is clearly elements of .
We now define the action of a Lie group on a manifold . Roughly speaking, a group action is a group of transformations of indexed by elements of the group and whose composition in is compatible with group multiplication in .

Definition 1.5

Let be a manifold and let be a Lie group. A left action of a Lie group on is a smooth mapping such that

  • for all ,

  • for all and ,

  • is a diffeomorphism for each .

A Lie group acts on its tangent bundle by the tangent map. We can consider the left or the right action of on by: or , where and denote left and right translations, respectively; so if is a curve in , then there exists a time dependent such that

(1.6)

and similarly for the right action.
Given left action of a Lie group on , , and an element of the Lie algebra then is a flow on , the corresponding vector field on is called infinitesimal generator of corresponding to , is denoted by

(1.7)

1.2 Fiber Bundles and Connections

Fiber bundles provide a basic geometric structure for the understanding of many mechanical and control problems.

A fiber bundle essentially consists of a given space (the base) together with another space (the fiber) attached at each point, plus some compatibility conditions. More formally, we have the following:

Definition 1.6

Let be a differentiable base manifold and a Lie group. A differentiable manifold is called principal fiber bundle if the following conditions are satisfied:

  • acts on to the left, freely and differentiably:

    (1.8)

    writing

  • and the canonical projection is differentiable

  • is locally trivial, namely every point has a neighborhood such that is isomorphic to , in the sense that is a diffeomorphism such that satisfies

An important additional structure on a bundle is a connection. Suppose we have a bundle and consider (locally) a section of this bundle, i.e., a choice of a point in the fiber over each point in the base. We call such a choice a “field”. The idea is to single out fields that are “constant”. For vector fields on a linear space, for example, it is clear what we want such fields to be; for vector fields on a manifold or an arbitrary bundle, we have to specify this notion. Such fields are called “horizontal”. A connection is used to single out horizontal fields, more precisely fields which live in a subspace of the the tangent space, and is chosen to have other desirable properties, such as linearity.

Definition 1.7

Let be a principal fiber bundle. the kernel of denoted by , is the subspace of tangent to the fiber through and is called vertical subspace. A connection on the principal fiber bundle is a choice of a tangent subspace at each point called horizontal subspace such that:

  • For all and ,

  • depends differentiably on

Hence, for any , we have that determines an isomorphism : for all and we have that . In other words the choice of an horizontal subspace can be seen also as the choice of a vector valued “connection one form” which vanishes on the horizontal vectors.

It follows the definition

Definition 1.8

An Ehresmann connection A is a vector valued one form such that

  • A is vertical valued: is a linear map for each point

  • A is a projection: for all .

In the special case in which is a principal fiber bundle the previous conditions on read:

  • for all and , where is the infinitesimal generator of the left action of on (1.7).

  • is equivariant:

    for all and where denotes the given action of on and where denotes the adjoint action of on defined as

Therefore it is evident that the horizontal subspace is the kernel of .
n the case in which there is a metric in our manifold , we have a special way to define the horizontal subspace: it is the orthogonal with respect to the metric to the vertical subspace.

(1.9)

In this special case our connection is called mechanical connection (see [31] and therein references). We now would like to express the connection in coordinates, in order to do this we first introduce the following definition

Definition 1.9

Let us consider the following diagram

where

The functions like are sections and we call the set of all sections from in .

Alternatively often a connection is introduced as a derivation as follows. Let be a map

Let now be a local basis of sections of the principal fiber bundle, in this basis the connection one-form can be expressed as

If we change basis in , say , the connection changes, i.e.

therefore and satisfy the following relation

(1.10)

Let be a smooth curve in passing through the point . Let be any point in the fiber of over . We would like to find a smooth curve in such that , , and (i.e., the tangent vectors to the curve are horizontal).

From the usual theory of differential equations it follows that such a curve exists and is unique, at least locally at any point (i.e., for small values of ). The curve is called a horizontal lift of . Regarding the tangent vectors, for any and any vector there exists a unique vector such that . The vector is called the horizontal lift of .

Given an Ehresmann connection we can define the horizontal lift of curves in , hence we can also define a notion of parallel transport that allows us to identify different fibers of .

Figure 1: Horizontal lift of the curve

Note that, in general, the parallel transport will be path-dependent. If we start with two different curves and , such that and , then the horizontal lifts and at a point will not meet, in general, at a point in the fiber , i.e., we will have . This gap on the fiber is called holonomy and depends on the choice of the connection and on the topology of the base manifold. In particular if it is connected the holonomy depends on the basepoint only up to conjugation [24].

2 Geometrical setting

In this section we present the geometrical framework underlying dynamical control systems. We derive the equations of motion and discuss how to use the geometrical tools introduced before to gain informations on our system.

2.1 Geometry of control equations

In this subsection we derive the local dynamic equations for the control system where is a smooth -dimensional foliation on , and is the Riemaniann metric on the manifold , as done in [29]. As is well known, on a set adapted for the foliation, coincides with the model foliation of by -dimensional hyperplanes. Let , be a local chart of in , distinguished for , so that maps into the trivial fibration . Set ; given a path , we suppose that for every , the reaction forces that implement the (ideal) constraint are workless with respect to the set of the virtual displacements compatible with the constraint .

Let be a foliated Riemanian manifold, let be an open set adapted for an let . If is the kinetic energy of the unconstrained system , then the kinetic energy of the system subject to the time dependent constraint is . The related dynamic equations are, in Lagrangian formalism

(2.1)

These can be put in Hamiltonian form by performing a partial legendre transformation on the - variables. When we identify with and with , the above Lagrange equations are equivalent to

(2.2)

We call these equations control equations. Let

(2.3)

be the local block representation of the metric in , where are symmetric and invertible respectively and matrices.

To every denote with the subspace orthogonal to with respect to . Referring to the local expression of in , it is easy to see that is the space orthogonal to the vectors with respect to the metric .

Therefore can be equivalently assigned through the -valued connection one form defined in 1.8

(2.4)

whose kernel and range are respectively and . Now we consider the orthogonal splitting of a vector into its horizontal ad vertical components

Using the above decomposition, we get the induced splitting of the kinetic energy metric tensor into its vertical and horizontal part:

(2.5)

where .

Definition 2.1

The Riemannian metric is bundle-like for the foliation iff on a neighborhood with adapted coordinates the above orthogonal splitting of holds with .

The importance of this notion will be clear in the following subsection (2.1.1). Using this notation we want to rewrite the control equations.

From

we obtain

(2.6)

Substituting (2.6) in (2.3) and recalling that we have

(2.7)

Therefore the control equations are

(2.8)

We now introduce, following [29], the global version of the above dynamic equations when is the total space of a surjective submersion . Let be the vertical subbundle and the dual of . Denote with the cotangent projection and set , . If are local fibered coordinates on , are local fibered coordinates on . Moreover, denote with the local coordinates on the -fiber over . Now, to every , is a fiber canonically simplettomophic to , representing the phase space of the constrained system restricted to the -fiber over .

Suppose that a control vector field is given on and that the path is an integral curve of . Then the dynamic equations (2.6) and (2.7) are the local expression of a vector field over that projects on by . Moreover the field is tangent to the fiber of only if the control is trivial: vanishing. Let us suppose that the control is given by a curve in that is the integral curve of the vector field . Thus the movement of the system is described by a differentiable curve such that . Note that is the natural increase of the curve in the fiber tangent to . Composing with the Legendre transform and with the projection we obtain the parametric curve which represent the evolution of the system taking into account the control.

Let denote the horizontal lift of the Ehresmann connection, introduced in the previous section, and the cotangent projection, using the above definitions we introduce the function

Theorem 2.1

To every control vector field on , the corresponding dynamic vector field can be expressed as the sum of three terms:

(2.9)

with

(2.10)
(2.11)
(2.12)

where is the Hamiltonian vector field corresponding to the case of locked control, is the Hamiltonian vector field on associated to and hor is the horizontal lift of an Ehresmann connection on entirely determined by and the metric. These equations are exactly the control equations (2.8).

Proof: [13]

The importance of initial impulse

In what follows let us suppose that the metric is bundle like.

Proposition 2.2

The control system (2.8) is of two different types depending on the value of the initial value of the conjugate variables .

  1. Case
    The system (2.8) is an affine non linear driftless control system;

  2. Case
    The system (2.8) is an affine non linear control system with drift.

Proof: Since we have supposed to have a bundle like metric we have that

Therefore the control equation (2.8) becomes

(2.13)

Case .
The function is the unique solution of (2.13) according to the Cauchy theorem. Thus (2.13) becomes a driftless control system.

It is clear that this last equation is entirely determined by the connection (see (2.4)). Therefore in the case of null initial impulse case only the geometry of the system determines its motion.
Case .
In this case the equation (2.13) has no trivial solution that is . Thus (2.13) is a non linear control system with drift determined exactly by the presence of a non zero

The presence of the drift is crucial because in this case the motion of the system is determined both by the connection (given by the geometry) and by the impulse, that is non zero. This proves the importance of the initial value of .

In this work we analyze both the cases. The one with zero initial impulse is well studied in literature for many systems [33, 32, 19, 21, 20]. The one with is becoming of increasing interest since the presence of the impulse influences deeply the motion, as we have seen. We deal with this problem that is more complex and tricky to study because of the presence of the drift.

2.2 Geometric and dynamic phase

As we have seen, in the general theory, connections are associated with bundle mappings, which project larger spaces onto smaller ones. The larger space is the bundle, and the smaller space is the base. Directions in the larger space that project to a point are vertical directions. The connection is a specification of a set of directions, the horizontal directions, at each point, which complements the space of vertical directions. In general, we can expect that for a horizontal motion in the bundle corresponding to a cyclic motion in the base, the vertical motion will undergo a shift, called a phase shift, between the beginning and the end of its path. The magnitude of the shift will depend on the curvature of the connection and the area that is enclosed by the path in the base space: it is exactly the holonomy. This shift in the vertical element is often given by an element of a group, such as a rotation or translation group, and is called also the geometric phase. Referring to what said in the previous subsection, the motion is determined only by the geometrical properties of the system if it starts with zero initial impulse. In many examples, the base space is the control space in the sense that the path in the base space can be chosen by suitable control inputs to the system, i.e. changes in internal shape. In the locomotion setting, the base space describes the internal shape of the object, and cyclic paths in the shape space correspond to the movements that lead to translational and rotational motion of the body.
Nevertheless the shape changes are not the only ones to determine a net motion of the body. More generally, this motion can always be decomposed into two components: the geometric phase, determined by the shape of the path and the area enclosed by it, and the dynamic phase, driven by the internal kinetic energy of the system characterized by the impulse. It is important to stress the difference between the two phases. The geometric phase is due entirely to the geometric structure of the system. Instead the dynamic phase is present if and only if the system has non zero initial impulse or if the impulse is not a conserved quantity, in our context we refer to what is explained in subsection 2.1.1. More precisely if the curvature of the connection is null, not necessarily the system does not move after a cyclic motion in the base: a net motion can result if the system starts with non zero initial impulse, and this motion is entirely due to the dynamic phase.

Figure 2: Geometric phase and dynamic phase.

Figure (2) shows a schematic representation of this decomposition for general rigid body motion. In this figure the sphere represents the base space, with a loop in the shape space shown as a circular path on the sphere. The closed circle above the sphere represents the fiber of this bundle attached to the indicated point. Given any path in the base (shape) space, there is an associated path, called the horizontal lift, that is independent of the time parametrization of the path and of the initial vertical position of the system. Following the lifted path along a loop in the shape space leads to a net change in vertical position along the fiber. This net change is just the geometric phase. On top of that, but decoupled from it, there is the motion of the system driven by the impulse, (if it is not zero) which leads to the dynamic phase. Combining these two provides the actual trajectory of the system.

2.3 Gauge potential

Let us consider a planar body immersed in a 2 dimensional fluid, which moves changing its shape. For the moment we do not specify the kind of fluid in which it is immersed that can be either ideal and incompressible or a viscous one with low Reynolds number. Our aim is to show that the motion of this deformable body through the fluid is completely determined by the geometry of the sequence of shapes that the idealized swimmer assumes, and to determine it. This idea was introduced by Shapere and Wilczek in [40] [39] and developed in [14], where they apply geometrical tools to describe the motion of a deformable body in a fluid, focusing their attention on the Stokes regime.
The configuration space of a deformable body is the space of all possible shapes. We should distinguish between the space of shapes located somewhere in the plane and the more abstract space of unlocated shapes. The latter space can be obtained from the space cum locations by making the quotient with the group of rigid motions in the plane, i.e declaring two shapes with different centers of mass and orientation to be equivalent. The first problem we wish to solve can be stated as follows: what is the net rotation and translation which results when a deformable body undergoes a given sequence of unoriented shapes? The problem is intuitively well posed: when a body changes its shape in some way a net rotation and translation is induced. For example, if the system is composed simply by the body, its net rigid motion can be computed by making use of the law of conservation of momentum, if instead the body is immersed in an ideal incompressible fluid this motion can be found by solving the Euler equations for the fluid flow with boundary condition on the surface of the body with the shape corresponding to the given deformation.
These remarks may seem straightforward, but we encounter a crucial ambiguity trying to formulate the problem more specifically. Namely how can we specify the net motion of an object which is continuously changing shape? To quantify this motion it is necessary to attach a reference frame to each unlocated shape. This is equivalent to choosing a standard location for each shape; more precisely to each unlocated shape there now corresponds a unique located shape. Once a choice of standard locations for shapes has been made, then we shall say that the rigid motion required to move between two different configurations is the displacement and rotation necessary to align their centers and axes. In what follows we shall develop a formalism, already used in [40] [39], which ensures us that the choice of axes for the unlocated shapes is completely arbitrary and that the rigid motion on the physical space is independent from this choice. This will be clear soon below.
For a given sequence of unlocated shapes , the corresponding sequence of located shapes are related by

(2.14)

where is a rigid motion. This relation expresses how to recover the located shapes given the unlocated ones, i.e. . It is clear that we are dealing with a fiber bundle: the located shapes live on the big manifold and the unlocated ones, , live on the base manifold obtained by the quotient of the manifold by the plane euclidean group , i.e .

To make (2.14) more explicit we introduce a matrix representation for the group of Euclidean motions, of which is a member. A two dimensional rigid motion consisting of a rotation followed by a translation may be represented as a matrix

(2.15)

where is an ordinary rotation matrix, is a 2 component column vector. This is the matrix representation of the plane euclidean group action on the manifold where the located shapes live on.
Now in considering the problem of self propulsion we shall assume that our swimmer has control over its form but cannot exert net forces and torques on itself. A swimming stroke is therefore specified by a time-dependent sequence of forms, or equivalently unlocated shapes . The located shape will then be expressed exactly by formula (2.14).

Our problem of determining the net rigid motion of the swimmer thus resolves itself into the computation of given . In computing this displacement it is most convenient to begin with infinitesimal motions and to build up finite motions by integrating. So let us define the infinitesimal motion by

(2.16)

In this formula we can recognize the differential equation corresponding to formula (1.6), from which we understand that take values in the Lie algebra of the plane euclidean group: . For any given infinitesimal change of shape , formula (2.16), describes the net overall translation and rotation which results. We can integrate it to obtain

(2.17)

where denotes a reverse path ordering, known in literature as chronological series [1]:

The assignment of center and axes can be arbitrary, so we should expect that physical results are independent of this assignment. How does this show up in our formalism? A change in the choice of centers and axes can equally well be thought of as a change (rigid motion) of the standard shapes, let us write

(2.18)

The located shapes being unchanged, (2.14) requires us to define [40] [39]

From this, the transformation law of follow

(2.19)

from which we can recognize the transformation laws (1.10) of an Ehresmann connection called also Gauge potential. Our freedom in choosing the assignment of axes shows up as a freedom of gauge choice on the space of standard shapes. Accordingly the final relationship between physical shapes is manifestly independent of such choices.
Our aim will be to compute this gauge potential in function of the unlocated shapes that our swimmer is able to control.

3 Swimming in an ideal fluid

We focus on a swimmer immersed in an ideal and incompressible fluid. The dynamical problem of its self propulsion has been reduced to the calculation of the gauge potential . In our model we assume that the allowed motions, involving the same sequence of forms will include additional time-dependent rigid displacements. In other words the actual motion will be the composition of the given motion sequence and rigid displacements.

3.1 System of coordinates

Let be a reference Galilean frame by which we identify the physical space to . At any time the swimmer occupies an open smooth connected domain and we denote by the open connected domain of the surrounding fluid. The coordinates in are denoted with and are usually called spatial coordinates. Let us call .
Attached to the swimmer, we define also a moving frame . Its choice is made such that its origin coincides at any time with the center of mass of the body. This frame represents the choice of the axes in the space of unlocated shapes. As we have shown before, the computation of the net rigid motion of the swimmer due to shape changes is independent from this choice that accordingly is arbitrary. The fact that this frame has always its origin in the center of mass is a matter of convenience: indeed this choice, and others (see Remark 3.1), tell us that the body frame is the one in which the kinetic energy of the body is minimal [26].
We denote by the related so called body coordinates. In this frame and at any time the swimmer occupies a region and the fluid the domain .
We define also the computational space, that is the Argand-Gauss plane which we will need only to perform explicit calculations, endowed with the frame and in which the coordinates are denoted . In this space is the unit disk and .

Remark 3.1 (Minimal Kinetic Energy)

In order to simplify the calculations, since, as said in the previous section, the choice of the body frame is arbitrary, we use the one of the minimal kinetic energy. It is the one in which the velocity of the center of mass is null. This condition is clearly satisfied if its origin coincides with the center of mass at any time. Moreover in this frame the angular momentum with respect to the body frame has to be null.

Remark 3.2

The orientation of the body frame remains arbitrary and does not effect the fact that it is the frame of minimal kinetic energy. One of the most used conventions to define a possible orientation of such a system is to choose as axes the eigenvectors of the moment of inertia of our body. Obviously as we have said before this choice does not effect the located shape, since it is independent on the choice of the frame.

3.2 Shape changes

Banach spaces of sequences. Inspired by [33], we denote any complex sequence by where for any , , . Most of the complex sequences we will consider live in the Banach space

endowed with its natural norm . This space is continuously embedded in

whose norm is , where is the unit disk of the computational space.

Definition 3.1

x We call the intersection of the unit ball of with the space .

This space will play an important role in the description of the shape changes that will follow.
Finally we introduce also the Hilbert space

whose norm is . According to Parseval’s identity we have

Therefore we have the following space inclusions

We have introduced these spaces because they will be crucial in the description of the shape changes of the idealized swimmer.

Description of the shape changes

Following the line of thoughts of [33] and [14] the shape changes of the swimmer are described with respect to the moving frame by a diffeomorphism , depending on a shape variable which maps the closed unit disk of the computational space onto the domain in the body frame. The diffeomorphisms allows us to associate to each sequence a shape of the swimmer in the body frame. We can write, according to our notation, that for any (see definition 3.1),

(3.1)

and .
We now explain how to build the map for any given sequence , see Fig 3.

Figure 3: The physical space and the body frame.
Theorem 3.1 (Riemann Mapping Theorem)

Let be a simply connected open bounded subset of with . Then there exists an holomorphic isomorphism with . Any other isomorphisms with are of the form with a rotation. All functions can be extended to an homeomorphism of onto if and only if is a Jordan curve.

Defining , if , from the isomorphism we have also an isomorphism from to the exterior ; we apply to the inversion obtaining the open simply connected , we find another Riemann- isomorphism with . Then we consider . The function is injective around zero, therefore , it follows that has a pole of the first order in zero and therefore has a Laurent expansion

(3.2)

We now have the area theorem [36]: if a function like is injective on the punctured disk then we have

(3.3)

If we want an isomorphism of on we take

(3.4)

We now suppose that the boundary of is a Jordan curve, i.e. simple closed curve in the plane, therefore the function can be extended to homeomorphism on the boundary. Now can be extended continuously to all setting in the interior of

(3.5)

Since on we deduce that the following map is continuous in for all :

(3.6)
Proposition 3.2

For all , and are both well defined and invertible. Further, is a diffeomorphism, is a conformal mapping and is an homeomorphism form onto .

Proof: [33]

Remark 3.3

Despite the generality of the Riemann Mapping Theorem, the way in which we decided to represent our diffeomorphism, lead us to some restrictions. Indeed in order to be sure that also is well defined -from proposition 3.2- we need to impose the restrictive condition , see (3.3), meaning that the shape variables have to be finitely bounded for both the norms of and . To summarize we can say that to use the shape variable allows us to describe all of the bounded non-empty connected shapes of the body that are not too far from the unit ball.

3.3 Rigid motions

The overall motion of our body in the fluid is, as said before, the composition of its shape changes with a rigid motion. The shape changes have been described in the previous subsection and, as we will see, the Gauge potential