Table Of ContentLATEXBodies of zero total cross section and bodies invisible in one direction
A. Aleksenko1,∗, A. Plakhov1,2
1 Department of Mathematics, Aveiro University, Aveiro 3810, Portugal
2 Aberystwyth University, Aberystwyth SY23 3BZ, UK, on leave from Department of Mathematics,
University of Aveiro, Aveiro 3810-193, Portugal
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0 ∗Email: [email protected], [email protected]
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2
Abstract Ω×[0, h]. Then R (B)6= 0.
v0
n
We consider a body in a parallel flow of non- Proof. Usingthattheparticle trajectory does not
a
J interacting particles. The interaction of particles intersectthesectionΩ×{c}andΩisconvex, onecon-
6 with the body is perfectly elastic. We introduce the cludesthattheparticleinitiallymovesinthecylinder
1 notions of a body of zero resistance and an invisible above this section, then intersects the lateral surface
body and prove that all such bodies do exist. of the cylinder and moves freely afterwards. This
]
s implies that v+(x,v ) 6= v , hence R (B)6= 0.
c B 0 0 v0
i Introduction
t
p Suppose that there is parallel flow of non- 1 Zero resistance bodies and invisible bodies
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. interacting particles falling on B ⊂ R3 which is Theorem. There exist (a) a body that has zero
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c a bounded connected set with piecewise smooth resistance in the direction v0; (b) a body invisible in
si boundary. Initially, the velocity of a particle equals the direction v0.
y v0 ∈ S2; then it makes several reflections from B, Proof. (a) Consider two identical coplanar equi-
ph and finally moves freely with the velocity vB+(x,v0), lateral triangles ABC and A′B′C′, with C being the
[ where x ∈ {v0}⊥ indicates the initial position of midpoint of the segment A′B′, and C′, the midpoint
1 the particle. One can imagine that the flow is of AB. The vertical line CC′ is parallel to v0. Let
v highly rarefied or consists of light rays. The force A′′ (B′′) be the point of intersection of segments AC
6 of pressure of the flow on the body, or resistance andA′C′ (BCandB′C′,respectively); seeFig.1. The
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of the body in the direction v , is proportional to
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1.2 tRiov0(eBqu)a:l=s tRh{ev0d}⊥en(vs0ity−ovfB+t(hxe,vfl0)o)wd/xm, ewdhiuemre athned rdax- v0 dx C
0 means the Lebesgue measure in {v }⊥. The problem A′ B′
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of minimal resistance is concerned with minimizing
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E
: the resistance in a prescribed class of bodies. There
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i is a large literature on this problem, starting from
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the famous Newton’s aerodynamic problem [1].
r
a Wesaythatthebodyisinvisible inthe direction v0
if the trajectory of each particle outside a prescribed A′′ B′′
bounded set coincides with a straight line. We prove
that there exist bodies of zero resistance and bodies F
invisible in one direction.
Consider the class of bodies B that are contained
in the cylinder Ω×[0, h] and contain a cross section
Ω×{c}, c ∈ [0, h]. For the sake of brevity, we shall
B
call them bodies inscribed in the cylinder. Multiple A C′ dx
reflections areallowed. IfΩistheunitcirclethenthe
Figure 1: The basic construction.
infimum of resistance equals zero, inf |R (B)| = 0
B v0
(see [2]). The infimum is not attained, that is, zero
resistance bodies do not exist. This follows from the body B generated by rotation of the triangle AA′A′′
following simple proposition. (or BB′B′′) around the axis CC′ is shown on Fig.2a.
Lemma. Let Ω be a convex set with nonempty It has zero resistance in the direction v .
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interior and let B be a body inscribed in the cylinder Let the particle first hit the segment A′A′′ at a
equalities are sharp: there exist zero resistance bod-
ies and invisible bodies with exactly 4 reflections.
Proof. (a) If m = 1 then the final velocity of
each particle does not coincide with the initial one,
v+(x,v )6= v , therefore R (B) 6= 0. That is, a zero
B 0 0 v0
resistance body requires at least two reflections.
(b) Note that a thin parallel beam of particles
changes the orientation under each reflection. To
be more precise, let x(t) = x + v t, v(t) = v
0 0
be the initial motion of a particle, and let x(t) =
Figure 2: Bodies of zero resistance. x(i)(x)+v(i)(x)t, v(t) = v(i)(x)beitsmotionbetween
the ith and (i+1)th reflections, i = 0, 1,...,m. Let
the body be invisible in the direction v ; then one
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point E. (If the particle first hits B′B′′, the argument has v(0) = v(m) = v , x(0) = x, and x(m) −x ⊥ v .
0 0
isthesame.) Afterthereflection,thedirectionofmo-
At each reflection and for any fixed x, the orien-
tion forms the angle π/3 with the vertical. Next, the tation of the triple (cid:0)∂x(i), ∂x(i), v(i)(cid:1) changes. The
particle hits the segment B′′B at the point F such ∂x1 ∂x2
that |A′E| = |B′′F|, and after the second reflection initial and final orientations, (cid:0)∂∂xx(0), ∂∂xx(0), v(0)(cid:1) and
1 2
moves vertically downward. That is, the final veloc- (cid:0)∂x(m), ∂x(m), v(m)(cid:1), coincide, therefore m is even.
∂x ∂x
1 2
ity equals v . One easily sees that with m = 2, the initial and
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(b) A body invisible in the direction v can be ob- finalparts of a trajectory cannot belong to a straight
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tained by doubling a zero resistance body; see Fig.3. line; this proves that m = 4.
Acknowledgements
v0 This work was supported by Centre for Re-
search on Optimization and Control (CEOC) from
the ”Fundac¸a˜o para a Ciˆencia e a Tecnologia”
(FCT), cofinanced by the European Community
FundFEDER/POCTI,andbyFCT:researchproject
PTDC/MAT/72840/2006.
References
[1] I. Newton. Philosophiae Naturalis Principia
Mathematica, 1686.
[2] A.Yu. Plakhov. Newton’s problem of the body
of minimal resistance with a bounded number of
collisions.Russ.Math.Surv.58,191-192(2003).
[3] A. Aleksenko, A. Plakhov, Bodies of zero resis-
tance and bodies invisible in one direction, 2008,
arXiv:0809.0108v3 [math.DS]
Figure 3: A body invisible in the direction v . It
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is obtained by taking 4 truncated cones out of the
cylinder.
Denote by m = m(B,v ) the maximal number of
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reflections of an individual particle from the body.
Theorem (a) If the body B has zero resistance in
the direction v then m(B,v )≥ 2. (b) If B is invisi-
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ble in the direction v then m(B,v ) ≥ 4. These in-
0 0
