Curvature Impact Ordering Problem (CIO Problem)

Cu a u e Impac O de ing P oblem (CIO
P oblem)
Abs ac
The Cu a u e Impac O de ing (CIO) P oblem asks whe he he e exis s a
gene al geome ic–dynamical law ha de e mines how cu a u e o a bounda y
su ace b eaks he empo al o de ing be ween “ i s con ac ” and “ i s de achmen ”
o colliding pa icles. On la bounda ies, poin s ha collide ea lie ypically de ach
ea lie in a well-o de ed way. On cu ed bounda ies, howe e , local a ia ions o he
su ace no mal can cause he o de ing o impac and de achmen imes o become
non- i ial o e en in e ed. The CIO P oblem seeks a uni ied desc ip ion o his
o de ing b eakdown in e ms o cu a u e, inciden condi ions, and local dynamics.
1 Physical Se up
Conside a igid body mo ing in R3and a smoo h bounda y su ace S⊂R3. Fo sim-
plici y, we desc ibe he body a he le el o ep esen a i e ma e ial poin s (o pa icles)
ha ollow classical ajec o ies un il con ac .
Le
•Sbe a wice-di e en iable su ace wi h posi ion ec o x∈Sand uni no mal n(x),
•κ(x) deno e local cu a u e in o ma ion a x(e.g. p incipal cu a u es o mean
cu a u e),
•a se o ma e ial poin s {Xi}mo e owa d Swi h inciden eloci ies in
i.
Fo each poin i, de ine
in
i: i s con ac ime wi h S, (1)
ou
i: ime a which he poin loses con ac and de aches,(2)
ou
i: pos -impac eloci y a de achmen .(3)
On an ideal la plane S la wi h cons an no mal n, he local con ac dynamics a e
symme ic in angen ial di ec ions, and “ i s con ac – i s de achmen ” o de ing is ypi-
cally p ese ed o neighbo ing impac poin s:
in
i< in
j⇒ ou
i≤ ou
j(locally, in he absence o s ong angen ial asymme ies).
(4)
On a cu ed su ace, howe e , he no mal di ec ion a ies wi h posi ion:
n=n(x), κ =κ(x),(5)
and his a ia ion can induce non- i ial di e ences in impac du a ion, no mal impulse,
and ebound di ec ion.
1
2 Co e Ques ion o he CIO P oblem
2.1 O de ing on a la su ace
Fo a la su ace S la and a small clus e o impac poin s, we can de ine a local o de ing
o impac s and de achmen s. Le (i, j) be wo neighbo ing ma e ial poin s ha make
con ac wi h S la a posi ions xi,xjand imes in
i, in
j. In many s anda d con ac models
( igid body wi h ic ion, sho con ac ime, no complica ed s icking), one expec s ha
in
i< in
j⇒ ou
i≲ ou
j,(6)
so ha he empo al o de o “ i s -in, i s -ou ” is locally main ained.
2.2 O de ing b eakdown on a cu ed su ace
On a cu ed su ace Swi h non-ze o cu a u e κ(x), he si ua ion may change quali a-
i ely. Two nea by poin s iand jcan sa is y
in
i< in
j,(7)
ye , due o cu a u e-induced di e ences in no mal di ec ion and local geome y, he
con ac du a ions and ebound di ec ions may sa is y
ou
i> ou
j,(8)
so ha he empo al o de ing e e ses.
The Cu a u e Impac O de ing P oblem asks:
Does he e exis a gene al geome ic–dynamical law ha ela es he cu a-
u e ield κ(x), he inciden condi ions ( in
i), and he esul ing de achmen
imes ( ou
i) in such a way ha he o de ing o impac and de achmen can be
p edic ed o classi ied?
Mo e p ecisely, he p oblem is o de e mine whe he one can w i e a ela ion o he
o m
O de ing{ in
i},{ ou
i}=Fκ(x),{ in
i},con ac model pa ame e s,(9)
whe e Fcha ac e izes when
in
i< in
j⇒ ou
i≶ ou
j(10)
holds, and when he implica ion can ail o in e .
3 Cu a u e s. Impac Dynamics
A cu ed su ace in oduces spa ial a ia ions o he no mal di ec ion:
n(x+δx)≈n(x) + (∇n)δx,(11)
whe e he g adien o he no mal is di ec ly ela ed o he cu a u e enso . As a esul :
2
•Di e en poin s along he su ace expe ience di e en no mal componen s o impac
eloci y.
•Local con ac ime, comp ession dep h, and eac ion o ce can di e , e en i he
global body mo ion is simila .
•Poin s ha con ac ea lie can, in some geome ies, de ach la e han poin s ha
con ac la e , o ice e sa.
This sugges s ha he mapping
( in
i,xi, in
i)−→ ou
i(12)
may become highly sensi i e o cu a u e and local geome y, and ha cu a u e can
b eak he nai e “ i s -in, i s -ou ” in ui ion inhe i ed om la su aces.
4 Fo mal S a emen o he CIO P oblem
We now s a e he CIO P oblem in a mo e o mal way.
CIO P oblem. Le Sbe a smoo h su ace wi h cu a u e enso K(x), and le a igid
body (o a dis ibu ion o ma e ial poin s) impac Swi h p esc ibed ini ial condi ions.
De ine o each con ac poin i he impac and de achmen imes ( in
i, ou
i) unde a gi en
con ac dynamics model.
(1) Exis ence o a gene al law. Does he e exis a gene al law o unc ional
G:K(x), in
i,ma e ial & con ac pa ame e s7→ o de ing pa e n o { in
i, ou
i}
(13)
ha p edic s when he empo al o de ing be ween impac and de achmen is p e-
se ed, dis o ed, o in e ed?
(2) Cu a u e h esholds o o de ing b eakdown. Can one iden i y c i ical con-
di ions (e.g. in e ms o cu a u e magni ude, cu a u e g adien s, inciden speed,
angen ial componen s) unde which
in
i< in
j⇒ ou
i≤ ou
j(14)
necessa ily holds, and condi ions unde which his implica ion can ail?
(3) Classi ica ion o o de ing egimes. Is i possible o classi y egimes such as:
•O de ing p ese ed: “ i s -in, i s -ou ” holds o all neighbo ing con ac poin s.
•O de ing weakly dis o ed: small de ia ions bu no global in e sion.
•O de ing in e ed o mixed: local egions whe e la e impac s de ach ea lie
han ea lie impac s.
in e ms o geome ic and dynamical pa ame e s?
A comple e solu ion o he CIO P oblem would p o ide a uni ying amewo k o un-
de s anding how cu a u e, impac geome y, and local dynamics coope a e o de e mine
he empo al s uc u e o con ac and ebound.
3
5 Rema ks and Possible Di ec ions
•The CIO P oblem si s a he in e sec ion o di e en ial geome y, con ac mechanics,
and dynamical sys ems.
•E en simpli ied models ( igid body, ic ionless con ac , small de o ma ions) can
exhibi non- i ial o de ing e ec s when he su ace cu a u e is non-ze o.
•Nume ical expe imen s wi h con olled cu a u e p o iles (e.g. sphe ical, pa abolic,
and mo e complex shapes) may help iden i y empi ical laws o conjec u es o o -
de ing ansi ions.
The CIO P oblem is hus p oposed as an open ques ion: o ind, o p o e he non-
exis ence o , a gene al law ha links cu a u e and impac dynamics o he o de ing o
con ac and de achmen e en s.
4