❊st❛❜✐❧✐❞❛❞❡ ✈❡rt✐❝❛❧ ♥♦ ♣r♦❜❧❡♠❛
❝✐r❝✉❧❛r ❞❡ ❙✐t♥✐❦♦✈
▼❛r❝❡❧♦ ❋❛r✐❛s ❈❛❡t❛♥♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛
❛♦
■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡ ❊st❛tíst✐❝❛
❞❛
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦
♣❛r❛
♦❜t❡♥çã♦ ❞♦ tít✉❧♦
❞❡
▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛
Pr♦❣r❛♠❛✿ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ▼❛♥✉❡❧ ❱❛❧❡♥t✐♠ ❞❡ P❡r❛ ●❛r❝✐❛
❉✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦ ♦ ❛✉t♦r r❡❝❡❜❡✉ ❛✉①í❧✐♦ ✜♥❛♥❝❡✐r♦ ❞❛ ❈◆Pq
❊st❛❜✐❧✐❞❛❞❡ ✈❡rt✐❝❛❧ ♥♦ ♣r♦❜❧❡♠❛ ❝✐r❝✉❧❛r ❞❡ ❙✐t♥✐❦♦✈
❊st❡ ❡①❡♠♣❧❛r ❝♦rr❡s♣♦♥❞❡ à r❡❞❛çã♦ ✜♥❛❧ ❞❛ ❞✐ss❡rt❛çã♦ ❞❡✈✐❞❛♠❡♥t❡ ❝♦rr✐❣✐❞❛ ❡ ❞❡❢❡♥❞✐❞❛ ♣♦r ▼❛r❝❡❧♦ ❋❛r✐❛s ❈❛❡t❛♥♦ ❡ ❛♣r♦✈❛❞❛ ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✳ ❊st❛ ✈❡rsã♦ ❞❡✜♥✐t✐✈❛ ❞❛ ❞✐ss❡rt❛çã♦ ❝♦♥té♠ ❛s ❝♦rr❡çõ❡s ❡ ❛❧t❡r❛çõ❡s s✉❣❡r✐❞❛s ♣❡❧❛ ❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛ ❞✉r❛♥t❡ ❛ ❞❡❢❡s❛ r❡❛❧✐③❛❞❛ ♣♦r ▼❛r❝❡❧♦ ❋❛r✐❛s ❈❛❡t❛♥♦ ❡♠ ✶✾✴✶✷✴✷✵✶✶✳
❈♦♠✐ssã♦ ❏✉❧❣❛❞♦r❛✿
• Pr♦❢❛✳ ❉r✳ ▼❛♥✉❡❧ ❱❛❧❡♥t✐♠ ❞❡ P❡r❛ ●❛r❝✐❛ ✭♦r✐❡♥t❛❞♦r✮ ✲ ■▼❊✲❯❙P • Pr♦❢✳ ❉r✳ ❈❧♦❞♦❛❧❞♦ ●r♦tt❛ ❘❛❣❛③③♦ ✲ ■▼❊✲❯❙P
❆♦s ♠❡✉s ♣❛✐s
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛q✉✐✱
❆♦s ♠❡✉s ♣❛✐s✱ ❋á❜✐♦ ❡ ▼❛r✐❛ ❞❡ ❋át✐♠❛✱ ♣❡❧♦ ❛♣♦✐♦✱ s✉♣♦rt❡ ❡ ❛♠♦r ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❛ ♠✐♥❤❛ ✈✐❞❛✳
➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ às ♠✐♥❤❛s ✐r♠ãs✱ ❛✈ós ❡ t✐❛ ♣♦r ♠❡ ❛❣✉❡♥t❛r❡♠ ❡ ♣❡❧♦ ❛♣♦✐♦ ❞❛❞♦ ❞✉r❛♥t❡ ❡ss❡s ❛♥♦s✳
Pr✐♥❝✐♣❛❧♠❡♥t❡✱ ❛♦s ♣r♦❢❡ss♦r❡s ▼❛♥✉❡❧ ❱❛❧❡♥t✐♠ ❞❡ P❡r❛ ●❛r❝✐❛ ♣❡❧❛ ❞✐❢í❝✐❧ t❛r❡❢❛ ❞❡ ♠❡ ♦r✐❡♥t❛r ❡ ❡♥s✐♥❛♠❡♥t♦s ❞✉r❛♥t❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✱ ❡ ❛♦ ♣r♦❢❡ss♦r ❈❧♦❞♦❛❧❞♦ ●r♦tt❛ ❘❛❣❛③③♦ ♣❡❧❛ ❞✐s❝✉ssõ❡s ❝♦♥str✉t✐✈❛s s♦❜r❡ ♠❛t❡♠át✐❝❛✱ ❡♥s✐♥❛♠❡♥t♦ ❡ ❛♣♦✐♦ ❞✉r❛♥t❡ ❛ ❡❧❛❜♦r❛çã♦ ❞❡ss❡ tr❛❜❛❧❤♦✳ ❆♦s ♣r♦✲ ❢❡ss♦r❡s ➶♥❣❡❧♦ ❇❛r♦♥❡ ◆❡tt♦✱ ❙ô♥✐❛ ❘❡❣✐♥❛ ▲❡✐t❡ ●❛r❝✐❛✱ ❘✐❝❛r❞♦ ❋r❡✐r❡ ♣❡❧❛s ❞✐❝❛s ❡ ❝♦rr❡çõ❡s ❞❛❞❛s ❛♦ tr❛❜❛❧❤♦✱ ❡ ❞✐s❝✉ssõ❡s ❝❛❧♦r♦s❛s ❞✉r❛♥t❡ ♦s s❡♠✐♥ár✐♦s ❞❛❞♦s ♣♦r ♠✐♠ ♣❛r❛ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆♦s ❛♠✐❣♦s ❞♦ ■▼❊❯❙P ❆❧❡①❛♥❞r❡ ❆❜❞♦✱ ❉❛♥✐❧♦ ❚♦♥✐♥✐✱ ❊❞✉❛r❞♦ ❖❞❛✱ ❘✐❝❛r❞♦ ❘✐❜❡✐r♦✱ ❚❤✐❛❣♦ ❆r❛ú❥♦✱ ❡ t❛♥t♦s ♦✉tr♦s ❝♦♠ ♦s q✉❛✐s ❡✉ ♣✉❞❡ ❝♦♥t❛r ❝♦♠ ♦ ❛♣♦✐♦✳ ❆♦s ❛♠✐❣♦s ❞♦ ❢✉t❡❜♦❧ ❞♦ ■▼❊❯❙P✱ ♦ q✉❛❧ ♠❡ s❡r✈✐❛ ❝♦♠♦ ✉♠ ❡s❝❛♣❡ ❞♦ str❡ss ❞✉r❛♥t❡ ♦ tr❛❜❛❧❤♦✳
❆♦s ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛✱ ▼❛♥✉❡❧ ❱❛❧❡♥t✐♠ P❡r❛ ●❛r❝✐❛✱ ❈❧♦❞♦❛❧❞♦ ●r♦tt❛ ❘❛❣❛③③♦ ❡ ❙②❧✈✐♦ ❋❡rr❛③✲▼❡❧❧♦✱ ♣♦r t❡r❡♠ ❛❝❡✐t♦ ❛ r❡❛❧✐③❛r ❛ ❜❛♥❝❛ ❡♠ ✉♠ ♣❡rí♦❞♦ ♣♦✉❝♦ ❝♦♥✈❡♥❝✐♦♥❛❧✳
❆ ❉❡✉s✱ ♣♦r ❡st❛r s❡♠♣r❡ ♣r❡s❡♥t❡✳ ❆ ❈◆P◗✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳
❘❡s✉♠♦
❊st✉❞❛♠♦s ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞♦ ♣r♦❜❧❡♠❛ r❡str✐t♦ ❞♦s três ❝♦r♣♦s✱ ❝❤❛♠❛❞♦ ♣r♦❜❧❡♠❛ ❝✐r❝✉❧❛r ❞❡ ❙✐t♥✐❦♦✈✱ q✉❛♥❞♦ ❞♦✐s ❝♦r♣♦s ❞❡ ♠❛ss❛s ✐❣✉❛✐s ✭❝❤❛♠❛❞❛s ❞❡ ♣r✐♠ár✐❛s✮ ❡stã♦ ❡♠ ✉♠❛ ór❜✐t❛ ❝✐r❝✉❧❛r ✭❝♦♥✜❣✉r❛çã♦ ❝❡♥tr❛❧ ❞❡ ❞♦✐s ❝♦r♣♦s✮✱ ❡♥q✉❛♥t♦ q✉❡ ✉♠ t❡r❝❡✐r♦ ❝♦r♣♦ ❞❡ ♠❛ss❛ ♥❡❣❧✐❣❡♥❝✐❛❞❛ ✭❝❤❛♠❛❞❛ ✐♥✜♥✐t❡s✐♠❛❧✮ ♦s❝✐❧❛ s♦❜r❡ ✉♠❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ ❞❛s ♣r✐♠ár✐❛s ✭❝❤❛♠❛r❡♠♦s ❡ss❡ ♠♦✈✐♠❡♥t♦ ❞❡ ✈❡rt✐❝❛❧ ♣❡r✐ó❞✐❝♦✮✳ ❆q✉✐ ❛♥❛❧✐s❛♠♦s ❛ ❡st❛❜✐❧✐❞❛❞❡ ❞❡ss❡ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦✱ ❝♦♠ r❡❧❛çã♦ ❛ ♣❡q✉❡♥❛s ♣❡rt✉r❜❛çõ❡s ♥❛s ❞✐r❡çõ❡s ♦rt♦❣♦♥❛✐s ❛ r❡t❛ ♦♥❞❡ ♦❝♦rr❡ ♦ ♠♦✈✐♠❡♥t♦✳ ❈❤❛♠❛r❡♠♦s ❛ ❛t❡♥çã♦ ❛♦ ❢❡♥ô♠❡♥♦ ❞❡ ❛❧t❡r♥â♥❝✐❛ ❡♥tr❡ ❡st❛❜✐❧✐❞❛❞❡ ❡ ✐♥st❛❜✐❧✐❞❛❞❡ ♥❛ ❢❛♠í❧✐❛ ❞♦ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦ ✈❡rt✐❝❛❧✱ ❝♦♥❢♦r♠❡ ✈❛r✐❛♠♦s ❛ ❛♠♣❧✐t✉❞❡ ❞♦ ♠♦✈✐♠❡♥t♦✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ Pr♦❜❧❡♠❛ ❞❡ ❙✐t♥✐❦♦✈✱ ❊st❛❜✐❧✐❞❛❞❡✱ ▼♦✈✐♠❡♥t♦s P❡r✐ó❞✐❝♦✳
❆❜str❛❝t
❲❡ st✉❞✐❡❞ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ t❤❡ r❡str✐❝t❡❞ t❤r❡❡✲❜♦❞② ♣r♦❜❧❡♠✱ ♥❛♠❡❞ ❝✐r❝✉❧❛r ♣r♦❜❧❡♠ ♦❢ ❙✐t♥✐✲ ❦♦✈✱ ✇❤❡♥ t✇♦ ❜♦❞② ♦❢ ❡q✉❛❧ ♠❛ss ✭❝❛❧❧❡❞ ♣r✐♠❛r✐❡s✮ ♠♦✈✐♥❣ ❛r♦✉♥❞ ❡❛❝❤ ♦t❤❡r ♦♥ ❝✐r❝✉❧❛r ♠♦t✐♦♥ ✭❝❡♥tr❛❧ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t✇♦ ❜♦❞②✮✱ ✇❤✐❧❡ t❤❡ t❤✐r❞ ❜♦❞② ♦❢ ♥❡❣❧✐❣✐❜❧❡ ♠❛ss ✭❝❛❧❧❡❞ ✐♥✜♥✐t❡s✐♠❛❧✮ ♣❡r❢♦r♠s ❛❧♦♥❣ ❛ str❛✐❣❤t ❧✐♥❡ ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ ♣❧❛♥❡ ♦❢ t❤❡ ♣r✐♠❛r✐❡s ✭s♦ ❝❛❧❧❡❞ ♣❡r✐♦❞✐❝ ✈❡rt✐❝❛❧ ♠♦t✐♦♥s✮✳ ❲❡ ❛♥❛❧②③❡ t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ♣❡r✐♦❞✐❝ ✈❡rt✐❝❛❧ ♠♦t✐♦♥s ✇✐t❤ r❡s♣❡❝t t♦ s♠❛❧❧ ♣❡rt✉r❜❛t✐✲ ♦♥s ♦rt❤♦❣♦♥❛❧ t♦ t❤❡ str❛✐❣❤t ❧✐♥❡ ✇❤❡r❡ t❤❡ ♠♦t✐♦♥s ♦❝❝✉rs✳ ❲❡ ❝❛❧❧ ❛tt❡♥t✐♦♥ t♦ t❤❡ ♣❤❡♥♦♠❡♥♦♠ ♦❢ ❛❧t❡r♥❛t✐♦♥ ♦❢ st❛❜✐❧✐t② ❛♥❞ ✐♥st❛❜✐❧✐t② ✇✐t❤✐♥ t❤❡ ❢❛♠✐❧② ♦❢ ♣❡r✐♦❞✐❝ ✈❡rt✐❝❛❧ ♠♦t✐♦♥s✱ ✇❤❡♥❡✈❡r t❤❡✐r ❛♠♣❧✐t✉❞❡ ✐s ✈❛r✐❡❞ ✐♥ ❛ ❝♦♥t✐♥✉♦✉s ♠❛♥♥❡r✳
❑❡②✇♦r❞s✿ ❙✐t♥✐❦♦✈ Pr♦❜❧❡♠✱ ❙t❛❜✐❧✐t②✱ P❡r✐♦❞✐❝ ▼♦t✐♦♥s✳
❙✉♠ár✐♦
▲✐st❛ ❞❡ ❋✐❣✉r❛s ①✐
▲✐st❛ ❞❡ ❚❛❜❡❧❛s ①✐✐✐
✶ ■♥tr♦❞✉çã♦ ✶
✶✳✶ ❉❡s❝r✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷
✶✳✷ ❈♦♥s✐❞❡r❛çõ❡s Pr❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸
✷ ❙✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s ✺
✷✳✶ ❋✉♥❞❛♠❡♥t♦s ❡ ❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺
✷✳✷ ❙✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷✳✷✳✶ ❉❡✜♥✐çõ❡s ❜ás✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽
✷✳✷✳✷ ❙✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♣❡r✐ó❞✐❝♦s ❛r❜✐trár✐♦s ✶✵
✷✳✸ ❙✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s ❡ ❈❛♥ô♥✐❝♦s ❞❡ ❊q✉❛çõ❡s ❉✐❢❡r❡♥❝✐❛✐s ▲✐♥❡❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✸✳✶ ❉❡✜♥✐çõ❡s ❜ás✐❝❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷
✷✳✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❞❡ s♦❧✉çõ❡s ❞❛s ❡q✉❛çõ❡s ❝❛♥ô♥✐❝❛s ❡ ❍❛♠✐❧t♦♥✐❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹
✷✳✸✳✸ ❚❡♦r❡♠❛ ❞❡ ▲②❛♣✉♥♦✈✲P♦✐♥❝❛ré ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✷✳✸✳✹ ❊q✉❛çõ❡s ✈❛r✐❛❝✐♦♥❛✐s ❞❡ P♦✐♥❝❛ré ❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻
✸ ❉❡s❝r✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ r❡str✐t♦ ✶✾
✸✳✶ ■♥tr♦❞✉çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸✳✷ ❉❡s❝r✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ ♣❧❛♥❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✸✳✷✳✶ ❉❡✜♥✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❡ ❛s ❡q✉❛çõ❡s ❞♦ ♠♦✈✐♠❡♥t♦ ♥♦ s✐st❡♠❛ ✐♥❡r❝✐❛❧ ✳ ✳ ✳ ✳ ✶✾
✸✳✷✳✷ ❊q✉❛çõ❡s ❞♦ ♠♦✈✐♠❡♥t♦ ♥♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s s✐♥ó❞✐❝❛s ❡ ❛ ✐♥t❡❣r❛❧ ❏❛❝♦❜✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶
✸✳✷✳✸ ❊q✉❛çõ❡s ❞♦ ♠♦✈✐♠❡♥t♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ♥ã♦ ❢ís✐❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✸✳✷✳✹ ❘❡❧❛çã♦ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❣❡r❛❧ ❞♦s três ❝♦r♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✷✳✺ ❈❧❛ss✐✜❝❛çã♦ ❡ ♠♦❞✐✜❝❛çõ❡s ♥♦ ♣r♦❜❧❡♠❛ r❡str✐t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽
✸✳✸ ❉❡s❝r✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❡s♣❛❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✸✳✶ ❋♦r♠✉❧❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❡ ❡q✉❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✸✳✹ ❈♦♥✜❣✉r❛çõ❡s ❝❡♥tr❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✹✳✶ ❙♦❧✉çõ❡s ❞❡ ❡q✉✐❧í❜r✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✹✳✷ ❊q✉❛çõ❡s ♣❛r❛ ❛ ❝♦♥✜❣✉r❛çã♦ ❝❡♥tr❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✸✳✹✳✸ ❊q✉✐❧í❜r✐♦ r❡❧❛t✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
① ❙❯▼➪❘■❖
✸✳✺ ❉❡s❝r✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ r❡str✐t♦ ❞♦s (N + 1)❝♦r♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸
✹ ❆❧t❡r♥â♥❝✐❛ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❡ ✐♥st❛❜✐❧✐❞❛❞❡ ♥❛ ❢❛♠í❧✐❛ ❞♦ ♠♦✈✐♠❡♥t♦ ✈❡rt✐❝❛❧ ✸✺
✹✳✶ ●❡♥❡r❛❧✐❞❛❞❡s ❞♦ ♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✹✳✷ ❆♣r♦①✐♠❛çã♦ ❞❛ ♠❛tr✐③ ❞❡ ♠♦♥♦❞r♦♠✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹✳✷✳✶ Pr♦♣r✐❡❞❛❞❡s ❞♦ s✐st❡♠❛ ❧✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✹✳✷✳✷ ❊①♣r❡ssã♦ ❛♣r♦①✐♠❛❞❛ ❞❛ ♠❛tr✐③ ❞❡ ♠♦♥♦❞r♦♠✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✹✳✸ ❘❡s✉❧t❛❞♦s ♥✉♠ér✐❝♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
✹✳✸✳✶ ❘❡s✉❧t❛❞♦s ♣❛r❛ 5< a <13 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✹✳✸✳✷ ❘❡s✉❧t❛❞♦s ♣❛r❛ a >546 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽
✺ ❈♦♥❝❧✉sõ❡s ✺✶
✺✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
❆ ❋✉♥çõ❡s ♠❛tr✐❝✐❛✐s ❡ ❧♦❣❛r✐t♠♦ ❞❡ ♠❛tr✐③ ✺✸
❆✳✶ ❋✉♥çõ❡s ♠❛tr✐❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
❆✳✶✳✶ ❋✉♥çõ❡s ♠❛tr✐❝✐❛✐s ❤♦❧♦♠ór✜❝❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸
❆✳✶✳✷ ❋✉♥çõ❡s ♠❛tr✐❝✐❛✐s ❛♥❛❧ít✐❝❛s ♣♦r ♣❛rt❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹
❆✳✶✳✸ ▲❡♠❛ ❞❛ ❝♦♠♣♦s✐çã♦ ❞❡ ❢✉♥çõ❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
❆✳✶✳✹ ▲♦❣❛r✐t♠♦ ❞❡ ▼❛tr✐③ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻
❇ ❆❧❣♦r✐t♠♦s ✐♠♣❧❡♠❡♥t❛❞♦s ✺✼
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✸✳✶ ❈♦♥✜❣✉r❛çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ✐♥❡r❝✐❛✐s ❡ r♦t❛❝✐♦♥❛✐s ✭m1 > m2✮✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✸✳✷ ❈♦♥✜❣✉r❛çã♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ✐♥❡r❝✐❛❧ (ξ, η) ❡ r♦t❛❝✐♦♥❛❧(x, y) ♥❛s ✈❛r✐á✈❡✐s ♥ã♦ ❢ís✐✲
❝❛s ✭µ1 > µ2✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺
✸✳✸ ❖ ♣r♦❜❧❡♠❛ ❣❡r❛❧ ❞♦s três ❝♦r♣♦s❀ ❛s ♠❛ss❛s sã♦ mi ❡ ♦s ✈❡t♦r❡s ♣♦s✐çõ❡sri✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✹ ❖ ♣r♦❜❧❡♠❛ r❡str✐t♦ tr✐❞✐♠❡♥s✐♦♥❛❧✱ ❡ s❡✉ ♣♦♥t♦s ❞❡ ❡q✉✐❧í❜r✐♦ Li✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾
✹✳✶ ❘❡tr❛t♦ ❞❡ ❢❛s❡ ♥❛ ✈❛r✐❡❞❛❞❡ ν˜✳ ▲✐♥❤❛ ♠❛✐s ❡s❝✉r❛ é ❛ s❡♣❛r❛tr✐③ ❞♦ ❡s♣❛ç♦ ✭H˜ = 0✮✳ ✸✼
✹✳✷ ●rá✜❝♦ ❞❛ ❢✉♥çã♦ ✭✹✳✹✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✹✳✸ ❚r❛❥❡tór✐❛ r❡❛❧✐③❛❞❛ ♣❡❧❛ ♠❛ss❛ ✐♥✜♥✐t❡s✐♠❛❧ ♥♦ ❡s♣❛ç♦ ♣❛r❛ ❛ ❛♠♣❧✐t✉❞❡ ✐♥✐❝✐❛❧ ❞❡ ✺✳✵✺✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✹✳✹ ❚r❛❥❡tór✐❛ r❡❛❧✐③❛❞❛ ♣❡❧❛ ♠❛ss❛ ✐♥✜♥✐t❡s✐♠❛❧ ♥♦ ❡s♣❛ç♦ ♣❛r❛ ❛ ❛♠♣❧✐t✉❞❡ ✐♥✐❝✐❛❧ ❞❡ ✺✳✵✺✷✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✹✳✺ ❚r❛❥❡tór✐❛ r❡❛❧✐③❛❞❛ ♣❡❧❛ ♠❛ss❛ ✐♥✜♥✐t❡s✐♠❛❧ ♥♦ ❡s♣❛ç♦ ♣❛r❛ ❛ ❛♠♣❧✐t✉❞❡ ✐♥✐❝✐❛❧ ❞❡ ✺✳✹✻✵✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✻ ❚r❛❥❡tór✐❛ r❡❛❧✐③❛❞❛ ♣❡❧❛ ♠❛ss❛ ✐♥✜♥✐t❡s✐♠❛❧ ♥♦ ❡s♣❛ç♦ ♣❛r❛ ❛ ❛♠♣❧✐t✉❞❡ ✐♥✐❝✐❛❧ ❞❡ ✺✳✹✼✵✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✼ ❈♦♠♣♦rt❛♠❡♥t♦ ❞♦s í♥❞✐❝❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ♦♥❞❡ ♦❝♦rr❡ ❛s ♣r✐♠❡✐r❛s ✧s❡❧❛s ❝♦♠♣❧❡①❛s✧✭r❡❣✐ã♦ ❡♠ ❜r❛♥❝♦ ♥♦ ❣rá✜❝♦✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾
✹✳✽ ❈♦♠♣♦rt❛♠❡♥t♦ ❞♦s í♥❞✐❝❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♥❛ ✈✐③✐♥❤❛♥ç❛ ❞❡ ♦♥❞❡ ♦❝♦rr❡ ❛s ♣r✐♠❡✐r❛s ✧s❡❧❛s ❝♦♠♣❧❡①❛s✧✭r❡❣✐ã♦ ❡♠ ❜r❛♥❝♦ ♥♦ ❣rá✜❝♦✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ❝❛r❛❝t❡ríst✐❝❛s ❞❛s s♦❧✉çõ❡s ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❡①♣♦❡♥t❡ ♦✉ ♠✉❧t✐♣❧✐❝❛❞♦r❡s ❞♦ s✐st❡♠❛ ✭✷✳✶✺✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸
✹✳✶ ■♥t❡r✈❛❧♦s ❞❡ ❡st❛❜✐❧✐❞❛❞❡s ♣❛r❛ ♦ ♣❛râ♠❡tr♦ a ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❙✐t♥✐❦♦✈✱ ❬❙❇❉✵✼❪✳ ❖s ❧✐♠✐t❡s ✐♥❢❡r✐♦r ❡ s✉♣❡r✐♦r sã♦ ♦s ❜♦r❞♦s ❞♦ ✐♥t❡r✈❛❧♦ ❞❡ ❡st❛❜✐❧✐❞❛❞❡✳ ❋♦r❛ ❞❡ss❡ ✐♥t❡r✈❛❧♦ ♦ ♠♦✈✐♠❡♥t♦ é ✐♥stá✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
✹✳✷ ❱❛❧♦r❡s ❞❛s ❛♠♣❧✐t✉❞❡s ✭❛❞✐♠❡♥s✐♦♥❛❧✮ ✐♥✐❝✐❛❧ ❡ r❡s♣❡❝t✐✈❛ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦ ✈❡rt✐❝❛❧ ❡♠ q✉❡ ❡❧❛ s❡ ❡♥❝♦♥tr❛ ❡ ♦s ✈❛❧♦r❡s ❞♦s í♥❞✐❝❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡✳ ✳ ✳ ✺✵
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❖ ♣r♦❜❧❡♠❛ ❞♦s N ❝♦r♣♦s r❡❢❡r❡✲s❡ à ❞❡s❝r✐çã♦ ❞♦ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ N ♣♦♥t♦s ♠❛t❡r✐❛✐s ♦✉
♣❛rtí❝✉❧❛s s♦❜ ✐♥✢✉ê♥❝✐❛ ❞❛ ❧❡✐ ❞❡ ♠♦✈✐♠❡♥t♦ ❞❡ ◆❡✇t♦♥✱ ♦♥❞❡ ❛s ú♥✐❝❛s ❢♦rç❛s ❛t✉❛♥t❡s sã♦ ❛s ❢♦rç❛s ❞❡ ❛tr❛çã♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ❡♥tr❡ ❛s ♠❡s♠❛s✳ ❖ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s q✉❡ ❞❡s❝r❡✈❡ ♦ ♣r♦❜❧❡♠❛ ♣♦❞❡ s❡r r❡s♦❧✈✐❞♦ ♣♦r q✉❛❞r❛t✉r❛ ♣❛r❛ N = 2✱ ♣♦✐s ♣♦❞❡ s❡r r❡❞✉③✐❞♦ ❛♦
♣r♦❜❧❡♠❛ ❞❡ ❑❡♣❧❡r ✭✈❡r ❡①❡♠♣❧♦ 2❞❛ s❡çã♦ 2.1✮✳
❆♦ ❡st✉❞❛r ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ▼❛rt❡✱ ◆❡✇t♦♥ r❡s♦❧✈❡✉ ❝♦♠♣❧❡t❛♠❡♥t❡ ♦ ♣r♦❜❧❡♠❛ ❞♦s 2 ❝♦r♣♦s✳
❊♠ ♣r✐♠❡✐r❛ ❛♣r♦①✐♠❛çã♦✱ ❛ ór❜✐t❛ ❞❡ ▼❛rt❡ é s♦❧✉çã♦ ❞❛s ❡q✉❛çõ❡s ❞♦ ♣r♦❜❧❡♠❛✱ ♦♥❞❡ s♦♠❡♥t❡ ❛s ❢♦rç❛s ❣r❛✈✐t❛❝✐♦♥❛✐s ❞♦ ❙♦❧ ❡ ❞❡ ▼❛rt❡ sã♦ ❧❡✈❛❞❛s ❡♠ ❝♦♥t❛✱ ❡ ❡ss❡ ♣r♦❜❧❡♠❛ ♣♦❞❡ s❡r r❡❞✉③✐❞♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❡ ❑❡♣❧❡r✳ ❯s❛♥❞♦ ♠ét♦❞♦s ❞❛ t❡♦r✐❛ ❞❡ ♣❡rt✉❜❛rçõ❡s✱ ❡❧❡ ❢♦✐ ❝❛♣❛③ ❞❡ ❡st✐♠❛r ❛❧❣✉♥s ❞♦s ❡❢❡✐t♦s ❞❡ ♦r❞❡♠ s✉♣❡r✐♦r ❡ ❞❡s❝r❡✈❡r ❛❧❣✉♠❛s ❛♥♦♠❛❧✐❛s ❞❛ ór❜✐t❛ ❞❡ ▼❛rt❡✳
❉❡♣♦✐s ❞✐ss♦✱ ◆❡✇t♦♥ ❝♦♠❡ç♦✉ ❛ ♦❧❤❛r ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ór❜✐t❛ ❞❛ ❧✉❛✳ ❊ss❡ ♣r♦❜❧❡♠❛ ❡♠❜♦r❛ ♣❛r❡❝❡ss❡ s✐♠♣❧❡s✱ ♠♦str♦✉✲s❡ ♠✉✐t♦ ♠❛✐s ❞✐❢í❝✐❧ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ór❜✐t❛ ❞❡ ▼❛rt❡✱ ♣♦✐s s✉❛ ♣r✐♠❡✐r❛ ❛♣r♦①✐♠❛çã♦ é ♦ ♣r♦❜❧❡♠❛ ❞♦s 3 ❝♦r♣♦s ✭❚❡rr❛✱ ▲✉❛ ❡ ❙♦❧✮✳ ❍♦❥❡ ❡♠ ❞✐❛ ❛ ór❜✐t❛ ❞❛ ▲✉❛
é ❡st✐♠❛❞❛ ❛tr❛✈és ❞❡ ✐♥t❡❣r❛çã♦ ♥✉♠ér✐❝❛ ♦✉ ♣♦r ❡①♣❛♥sã♦ ❛ss✐♥tót✐❝❛✳
P♦r ❝❛✉s❛ ❞❛s ❞✐✜❝✉❧❞❛❞❡s ❡♥❝♦♥tr❛❞❛s ❥á ♥♦ ♣r♦❜❧❡♠❛ ❞♦s3❝♦r♣♦s✱ ❤♦❥❡ é ❛♣♦✐❛❞❛ ❛♠♣❧❛♠❡♥t❡
❛ ❝r❡♥ç❛ ❞❡ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❞♦sN ❝♦r♣♦s s❡❥❛ ♥ã♦ ✐♥t❡❣rá✈❡❧ ♣❛r❛N ≥3✳ ❊♥tr❡t❛♥t♦ ❤á ✉♠❛ ❣r❛♥❞❡
q✉❛♥t✐❞❛❞❡ ❞❡ ❛rt✐❣♦s s♦❜r❡ ♦ ❛ss✉♥t♦ q✉❡ ❢❛❧❛♠ s♦❜r❡ s♦❧✉çõ❡s ❡s♣❡❝✐❛✐s✱ ❡st✐♠❛t✐✈❛s ❛ss✐♥tót✐❝❛s✱ ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ❝♦❧✐sõ❡s✱ ❡①✐stê♥❝✐❛ ♦✉ ♥ã♦ ❞❡ ✐♥t❡❣r❛✐s✱ s♦❧✉çõ❡s ❡♠ sér✐❡s✱ s✐♥❣✉❧❛r✐❞❛❞❡s✱ ❡t❝✳✳✳
❆ ❡①✐stê♥❝✐❛✱ ❡st❛❜✐❧✐❞❛❞❡ ❡ ❜✐❢✉r❝❛çã♦ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s ❞♦ ♣r♦❜❧❡♠❛ ❞♦s N ❝♦r♣♦s tê♠
s✐❞♦ ❛ss✉♥t♦s ❞❡ ✈❛st❛ ❧✐t❡r❛t✉r❛✱ ♣❛rt✐❝✉❧❛♠❡♥t❡ ❛♣ós ♦s tr❛❜❛❧❤♦s ❞❡ P♦✐♥❝❛ré q✉❡ ❡st✉❞♦✉ ❡①✲ t❡♥s✐✈❛♠❡♥t❡ s♦❜r❡ ❛s s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s✱ ♠❡r❡❝❡♥❞♦ ❞❡st❛q✉❡ s❡✉ ♥♦tá✈❡❧ ▲❡s ♠ét❤♦❞❡s ♥♦✉✈❡❧❧❡s ❞❡ ❧❛ ♠é❝❛♥✐q✉❡ ❝é❧❡st❡✱ r❡s❡r✈❛❞♦ ❛ ❡st❡ tó♣✐❝♦✳ ❍á ✉♠❛ q✉❛♥t✐❞❛❞❡ ✈❛st❛ ❞❡ ❧✐t❡r❛t✉r❛ s♦❜r❡ ❛ ❡①✐stê♥❝✐❛ ❡ ♥❛t✉r❡③❛ ❞♦ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦s N ❝♦r♣♦s✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❡♠ ✉♠
❝❛s♦ ♣❛rt✐❝✉❧❛r✱ ❝❤❛♠❛❞♦ ❞❡ ♣r♦❜❧❡♠❛ r❡str✐t♦✳
❖ ♣r♦❜❧❡♠❛ r❡str✐t♦ ❞♦s N ❝♦r♣♦s✱ é ✉♠❛ ✧❛♣r♦①✐♠❛çã♦✧❞♦ ♣r♦❜❧❡♠❛ ❣❡r❛❧✳ ■st♦ é✱ s✉♣♦♠♦s
q✉❡ k < N ❝♦r♣♦s ♥ã♦ ✐♥✢✉❡♥❝✐❛♠ ♥♦ ♠♦✈✐♠❡♥t♦ ❞♦s ♦✉tr♦s N −k ❝♦r♣♦s ❡ ♦s ♠♦✈✐♠❡♥t♦s ✉♠
❞♦s ♦✉tr♦s✱ ♣❛r❛ ✐ss♦ ❞❡✜♥✐♠♦s ❛s ♠❛ss❛s ❞❡ss❡sk❝♦r♣♦s ❝♦♠♦ s❡♥❞♦ ✧♥✉❧♦s✧✱ ♥♦ s❡♥t✐❞♦ ❞❡ s❡r❡♠
♠✉✐t♦ ♠❡♥♦r❡s q✉❡ ❛s ♠❛ss❛s ❞♦s ♦✉tr♦ N −k ❝♦r♣♦s✱ ♠♦t✐✈♦ ❞♦ q✉❛❧ ❝❤❛♠❛♠♦s ❡ss❡ ♣r♦❜❧❡♠❛
❝♦♠♦ ✉♠❛ ❛♣r♦①✐♠❛çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❣❡r❛❧✳ ❆❧é♠ ❞✐ss♦ s✉♣♦♠♦s q✉❡ ♦sk❝♦r♣♦s ❡st❡❥❛♠ ❡♠ ❛❧❣✉♠❛
❝♦♥✜❣✉r❛çã♦ ❝❡♥tr❛❧✱ ❝♦♠ s♦❧✉çã♦ ❝♦♥❤❡❝✐❞❛ ✭s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞♦s k ❝♦r♣♦s✮✱ ♣❛r❛ ♠❛✐♦r❡s
❞❡t❛❧❤❡s ❝♦♥❢❡r✐r ♦ ❝❛♣ít✉❧♦ s♦❜r❡ ❞❡s❝r✐çã♦ ❞♦ ♣r♦❜❧❡♠❛ r❡str✐t♦✳
❆q✉✐ ❞❡t❛❧❤❛r❡♠♦s ✉♠ ❞♦s ♠ét♦❞♦s ✉s❛❞♦s ♥❛ ❛♥á❧✐s❡ ❞♦ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦✱ ❡♠ ♣❛rt✐❝✉❧❛r ♥♦ ❡st✉❞♦ ❞❡ s✉❛ ❡st❛❜✐❧✐❞❛❞❡ ♦r❜✐t❛❧✳ ❊st✉❞❛r❡♠♦s ♦ ❝❤❛♠❛❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❙✐t♥✐❦♦✈ ❝✐r❝✉❧❛r✱ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ♣r♦❜❧❡♠❛ ❞♦s 3 ❝♦r♣♦s r❡str✐t♦ ✭♣r♦❜❧❡♠❛ ❞♦s 3 ❝♦r♣♦s ❡♠ q✉❡ ❛ ❢♦rç❛ ❣r❛✈✐t❛❝✐♦♥❛❧
❞❡ ✉♠ ❞♦s ❝♦r♣♦s ♥ã♦ ✐♥✢✉ê♥❝✐❛ ♦s ♦✉tr♦s ❞♦✐s ❝♦r♣♦s✮✳
❖ t❡r♠♦ ✧♣r♦❜❧❡♠❛ ❞❡ ❙✐t♥✐❦♦✈✧r❡❢❡r❡✲s❡ à s❡❣✉✐♥t❡ s✐t✉❛çã♦✿ ❞♦✐s ❝♦r♣♦s ❞❡ ♠❛ss❛s ✐❣✉❛✐s✱ q✉❡ ❝❤❛♠❛r❡♠♦s ❞❡ ♣r✐♠ár✐❛s✱ ❝♦♠ ór❜✐t❛s ❡❧ít✐❝❛s ❝♦♣❧❛♥❛r❡s✱ ❝♦❢♦❝❛✐s✱ ❝♦♠ ♦ ❢♦❝♦ ❝♦♠✉♠ ❧♦❝❛❧✐③❛❞♦ ❡♠ s❡✉ ❜❛r✐❝❡♥tr♦ ❖✱ ❡♥q✉❛♥t♦ ✉♠ t❡r❝❡✐r♦ ❝♦r♣♦ ❞❡ ♠❛ss❛ ✐♥✜♥✐t❡s✐♠❛❧✱ ❝❤❛♠❛❞♦ ❞❡ ❝♦r♣ús❝✉❧♦ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ✐♥✜♥✐t❡s✐♠❛❧✱ ♠♦✈❡✲s❡ ❛♦ ❧♦♥❣♦ ❞❛ r❡t❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❛♦ ♣❧❛♥♦ ❞❛s ♣r✐♠ár✐❛s q✉❡
✷ ■◆❚❘❖❉❯➬➹❖ ✶✳✶
♣❛ss❛ ♣♦r ❖✱ ❡ ♣r❡t❡♥❞❡✲s❡ ❡st✉❞❛r ♦s ♠♦✈✐♠❡♥t♦s ❞❛ ✐♥✜♥✐t❡s✐♠❛❧✳
❙✐t♥✐❦♦✈✱ ❬❙✐t✻✵❪ ❡st✉❞♦✉ ♦ q✉❡ ❤♦❥❡ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✧♣r♦❜❧❡♠❛ ❡❧í♣t✐❝♦✧✱ ❡♠ q✉❡ ❛ ❝♦♥✜✲ ❣✉r❛çã♦ ❝❡♥tr❛❧ ❞❛s ♣r✐♠ár✐❛s ❢♦r♠❛♠ ❞✉❛s ❡❧✐♣s❡s ❞✐s❥✉♥t❛s ❝♦♥❢♦❝❛❧✱ ❛ q✉❛❧ ❛♣❛r❡❝❡ ♦ ❝❤❛♠❛❞♦ ♠♦✈✐♠❡♥t♦ ♦s❝✐❧❛tór✐♦✳ ❖ ❝❛s♦ ❡♠ q✉❡ ❛s ♣r✐♠ár✐❛s ♠♦✈❡♠✲s❡ ❡♠ ✉♠❛ ór❜✐t❛ ❝✐r❝✉❧❛r ❥á ❢♦r❛ ❡st✉✲ ❞❛❞♦ ❛♥t❡r✐♦r♠❡♥t❡ ♣♦r P❛✈❛♥✐♥✐ ✭1907✮ ❡ ♣♦st❡r✐♦r♠❡♥t❡ ♣♦r ▼❛❝▼✐❧❧❛♥ ✭1911✮✳ ◆❡st❛ s✐t✉❛çã♦ ♦
♣r♦❜❧❡♠❛ ✜❝❛ ♠✉✐t♦ ♠❛✐s s✐♠♣❧❡s✱ ♣♦✐s t♦r♥❛✲s❡ ✐♥t❡r❣rá✈❡❧ ❡ é ♣♦ssí✈❡❧ ♦❜t❡r s✉❛ s♦❧✉çã♦ ❡♠ t❡r✲ ♠♦s ❞❛s ❢✉♥çõ❡s ❡❧ít✐❝❛s ❞❡ ❏❛❝♦❜✐✳ ❊♠❜♦r❛ ♦ ♣r♦❜❧❡♠❛ ❝✐r❝✉❧❛r t❡♥❤❛ s✐❞♦ ✧❛♣r❡s❡♥t❛❞♦✧❛♥t❡s ❞♦s ❡st✉❞♦s ❞❡ ❙✐t♥✐❦♦✈✱ ♦ t❡r♠♦ ♣♦♣✉❧❛r ♣❛r❛ ❡❧❡ ✜❝♦✉ s❡♥❞♦ Pr♦❜❧❡♠❛ ❞❡ ❙✐t♥✐❦♦✈ ❝✐r❝✉❧❛r✱ ❡♠❜♦r❛ ❛❧❣✉♥s ❛✉t♦r❡s ♦ ❝❤❛♠❡♠ ❞❡ ✧♣r♦❜❧❡♠❛ ❞❡ P❛✈❛♥✐♥✐✧♦✉ ✧♣r♦❜❧❡♠❛ ❞❡ ▼❛❝▼✐❧❧❛♥✧✳
✶✳✶ ❉❡s❝r✐çã♦ ❞♦ ♣r♦❜❧❡♠❛
❆♣ós ❞❛r♠♦s ✉♠ ❜r❡✈❡ ❤✐stór✐❝♦ s♦❜r❡ ♦ ♣r♦❜❧❡♠❛ r❡str✐t♦ ❡ s✉❛ r❡❧❛çã♦ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❣❡r❛❧✱ ♥❡st❡ ♣❛rá❣r❛❢♦ r❡s✉♠✐r❡♠♦s ♦ ♣r♦❜❧❡♠❛ q✉❡ ✐r❡♠♦s ❡st✉❞❛r ❡♠ ❞❡t❛❧❤❡s ♥♦ ❝❛♣ít✉❧♦ ✭✹✮✳
❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ r❡str✐t♦ ❝✐r❝✉❧❛r ❝♦♠ ♣r✐♠ár✐❛s ❞❡ ♠❛ss❛s m1 =m2 =m✳ ❙❡❥❛ Ox1x2x3
❝♦♦r❞❡♥❛❞❛s s✐♥ó❞✐❝❛s ✭♠❛✐♦r❡s ❞❡t❛❧❤❡s ✈❡r ❝❛♣ít✉❧♦ ✭✸✮✮✱ ❝♦♠ ♦r✐❣❡♠✱O✱ ❧♦❝❛❧✐③❛❞❛ ♥♦ ❜❛r✐❝❡♥tr♦
❞❛s ♣r✐♠ár✐❛s❀ ❛s ♣r✐♠ár✐❛sP1 ❡P2 ❛rr❛♥❥❛❞❛s ❞❡ t❛❧ ❢♦r♠❛ q✉❡ ❛ r❡t❛ q✉❡ ❛s ✉♥❡ s❡rá ♦ ❡✐①♦Ox1✱
❡ ❡❧❛s r❡❛❧✐③❛♠ r♦t❛çã♦ ❡♠ t♦r♥♦ ❞♦ ❡✐①♦Ox3✳ ❆s ❝♦♦r❞❡♥❛❞❛s ❛❞✐♠❡♥s✐♦♥❛✐s ❞❛ ✐♥✜♥✐t❡s✐♠❛❧ s❡rã♦
q1 =x1, q2=x2, q3 =x3.
❆ ❤❛♠✐❧t♦♥✐❛♥❛ ♣❛r❛ ❛s ❡q✉❛çõ❡s ❞♦ ♠♦✈✐♠❡♥t♦ ❞❛ ✐♥✜♥✐t❡s✐♠❛❧ é
H = 1 2 p
2
1+p22+p23
+p1q2−p2q1− 1 2
1
r1 + 1
r2
,
❡♠ q✉❡ r1 ❡r2 ❞❡♥♦t❛♠ ❛s ❞✐stâ♥❝✐❛s ❡♥tr❡ ❛ ✐♥✜♥✐t❡s✐♠❛❧ ❡ ❛s r❡s♣❡❝t✐✈❛s ♣r✐♠ár✐❛s✱p1, p2, p3
sã♦ ♦s ♠♦♠❡♥t♦s ❝♦♥❥✉❣❛❞♦s ❞❡ q1, q2, q3✳
❖ ❡s♣❛ç♦ ❞❡ ❢❛s❡✱ ♦♥❞❡ ❡stá ❞❡✜♥✐❞♦ ♦ ♠♦✈✐♠❡♥t♦ ❞❛ ✐♥✜♥✐t❡s✐♠❛❧ ♣♦ss✉✐ ✉♠❛ s✉❜✈❛r✐❡❞❛❞❡
˜
Σ ={(p, q);p1 =p2 =q1=q2= 0}
✱ q✉❡ é ✐♥✈❛r✐❛♥t❡ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✢✉①♦✳
❈❤❛♠❛r❡♠♦s ❞❡ ♠♦✈✐♠❡♥t♦ ✈❡rt✐❝❛❧ tr❛❥❡tór✐❛s ❞♦ ❝♦r♣ús❝✉❧♦ ♣❡rt❡♥❝❡♥t❡s ❛ Σ˜✱ ♦✉ s❡❥❛✱ ♦ ♠♦✲
✈✐♠♥t♦ ❞❡ 3♥♦ ❡✐①♦ Ox3.
❆ ✉♠ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦ ♥❡ss❛ ✈❛r✐❡❞❛❞❡ ♣♦❞❡♠♦s ❛ss♦❝✐❛r ✉♠ ♣❛râ♠❡tr♦✱ a✱ ❝❤❛♠❛❞♦ ❞❡
❛♠♣❧✐t✉❞❡ ❞♦ ♠♦✈✐♠❡♥t♦✱ ❡ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❝♦♠♦ ✈❛❧♦r ❞❡ss❡ ♣❛râ♠❡tr♦ ❛ ❛♠♣❧✐t✉❞❡ ❞♦ ♠♦✈✐✲ ♠❡♥t♦ ♣❡r✐ó❞✐❝♦ ✭a =maxt∈R|q3(t)|✮✱ ♦✉ ♦ ✈❛❧♦r ❛❜s♦❧✉t♦ ❞❡ p3 q✉❛♥❞♦ ❛ ✐♥✜♥✐t❡s✐♠❛❧ ♣❛ss❛ ♣❡❧♦
❜❛r✐❝❡♥tr♦✱ ♦✉ ♦ ✈❛❧♦r ❞❛ ❡♥❡r❣✐❛ h ❞♦ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦✳
❉❡♣❡♥❞❡♥❞♦ ❞❛s ❝♦♥❞✐çõ❡s ✐♥✐❝✐❛✐s ❞❛❞❛s ✭(p0, q0)∈Σ˜✮✱ ❤á três t✐♣♦s ❞❡ ♠♦✈✐♠❡♥t♦✿ ❡s❝❛♣❡ ❤✐✲
♣❡r❜ó❧✐❝♦ ✭limt→∞|q3|=∞❡limt→∞|p3|>0✮✱ ❡s❝❛♣❡ ♣❛r❛❜ó❧✐❝♦ ✭limt→∞|q3|=∞❡limt→∞|p3|=
0✮ ❡ ♠♦✈✐♠❡♥t♦ ♣❡r✐ó❞✐❝♦ ✈❡rt✐❝❛❧ ✭maxt∈R|q3(t)|<∞✳✮
❉❛❞♦ ✉♠ ❝✉❜♦ R = [γ1, γ2]×[β1, β2]×[η1, η2] ⊂ R3✱ α = (α1, α2, α3) ∈ R, ❝❤❛♠❛r❡♠♦s ❞❡
✉♠❛ ♣❡rt✉r❜❛çã♦ ❞❡ ✉♠❛ tr❛❥❡tór✐❛ ♣❡r✐ó❞✐❝❛ X(t, t0, q0) = (q1(t, t0, q0), q2(t, t0, q0), q3(t, t0, q0)) ❛
tr❛❥❡tór✐❛ Xα(t, t0, q0) = (q1(t, t0, q0+α), q2(t, t0, q0+α), q3(t, t0, q0+α))✳
❈❤❛♠❛r❡♠♦s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❞♦ ♠♦✈✐♠❡♥t♦ ✈❡rt✐❝❛❧✱X(t, t0, q0)s❡∀α∈R❛ tr❛❥❡tór✐❛Xα(t, t0, q0)
é ❧✐♠✐t❛❞❛✳ ❆q✉✐ ❡st✉❞❛r❡♠♦s ✉♠ ❢❡♥ô♠❡♥♦ ✐♥t❡r❡ss❛♥t❡ q✉❡ ♦❝♦rr❡ ♥♦ ♠♦✈✐♠❡♥t♦ ✈❡rt✐❝❛❧ ❞♦ ♣r♦✲ ❜❧❡♠❛ r❡str✐t♦✱ q✉❡ é ❛ ❛❧t❡r♥â♥❝✐❛ ❡♥tr❡ ❡st❛❜✐❧✐❞❛❞❡ ❡ ✐♥st❛❜✐❧✐❞❛❞❡ ❞♦s ♠♦✈✐♠❡♥t♦s ♣❡r✐ó❞✐❝♦s ✈❡rt✐❝❛✐s ❝♦♠ r❡❧❛çã♦ ❛♦ ♣❛râ♠❡tr♦a✱ ✐st♦ é✱ ❞❛❞♦ ✉♠ r❡tâ♥❣✉❧♦Rs✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱∀α∈R
❛ tr❛❥❡tór✐❛ Xα(t, a) ❛❧t❡r♥❛ ❡♥tr❡ ❡stá✈❡❧ ❡ ✐♥stá✈❡❧✳ ◆♦ ❝❛s♦ ❡♠ q✉❡ a é s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡
♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ss❛ ❛❧t❡r♥â♥❝✐❛ t♦r♥❛✲s❡ ❛✐♥❞❛ ♠❛✐s ✐♥t❡r❡ss❛♥t❡✱ ♣♦✐s✱ ♣❛r❛ a >> 1✱ ❛♣❛r❡❝❡
✶✳✸ ❈❖◆❙■❉❊❘❆➬Õ❊❙ P❘❊▲■▼■◆❆❘❊❙ ✸
✉♥✐tár✐♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❝á❧❝✉❧♦s ♥✉♠ér✐❝♦s✱ ♦ ♣r✐♠❡✐r♦ ✈❛❧♦r ❞♦ ✐♥t❡r✈❛❧♦ ♦♥❞❡ ✐ss♦ ♦❝♦rr❡ é ♣❛r❛
a≈546✳ P♦❞❡✲s❡✱ ❡♥tã♦✱ ♠♦str❛r ♥✉♠❡r✐❝❛♠❡♥t❡ q✉❡ ♦ ❢❡♥ô♠❡♥♦ ❞❡ ❛❧t❡r♥â♥❝✐❛ ♥❡st❛ ❢❛♠í❧✐❛ ♦❝♦rr❡
✐♥✜♥✐t❛s ✈❡③❡s ❡ ❞❡♣❡♥❞❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❞❡a✳
❆ ♣r✐♠❡✐r❛ ✈❡③ ❡♠ q✉❡ ❡ss❛ ❛❧t❡r♥â♥❝✐❛ ❢♦✐ ✐♥✈❡st✐❣❛❞❛ ❢♦✐ ❡♠ ❬P▼✽✽❪✱ ♠❛s ♦❜t✐✈❡r❛♠ ✉♠❛ ❝♦♥❝❧✉sã♦ ❡rrô♥❡❛✱ ♣♦✐s ❛✜r♠❛r❛♠ q✉❡ ♦ ♠♦✈✐♠❡♥t♦ ✈❡rt✐❝❛❧ s❡r✐❛ s❡♠♣r❡ ✐♥stá✈❡❧✳ ❊ss❛ ❛✜r♠❛çã♦ ❢♦✐ ❝♦♥❢r♦♥t❛❞❛ ❡♠1994♣♦r ❇❡❧❜r✉♥♦ ❡t ❛❧✳ ❬❇▲❖✾✹❪✱ ♦♥❞❡ ❛ ❛❧t❡r♥â♥❝✐❛ ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ❡ ✐♥st❛❜✐❧✐❞❛❞❡
❞♦ ♠♦✈✐♠❡♥t♦ ✈❡rt✐❝❛❧ ❢♦✐ ❞❡t❡❝t❛❞❛ ♥✉♠❡r✐❝❛♠❡♥t❡ ♥♦ ❝❛s♦ ❞❛ ✈❛r✐❛çã♦ ♠♦♥ót♦♥❛ ❞❛ ❛♠♣❧✐t✉❞❡a✳
◆❛ ✈❡r❞❛❞❡✱ ❛ ✧❛❧t❡r♥â♥❝✐❛ ✐♥✜♥✐t❛✧❡♥tr❡ ❡st❛❜✐❧✐❞❛❞❡ ❡ ✐♥st❛❜✐❧✐❞❛❞❡ ❞❡ ✉♠❛ ❢❛♠í❧✐❛ ❛ ✉♠ ♣❛râ♠❡tr♦ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s é tí♣✐❝❛ ♣❛r❛ s✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s✱ ♣♦ré♠ ✐ss♦ ♦❝♦rr❡ ❣❡r❛❧♠❡♥t❡ ❡♠ s✐st❡♠❛s ❞❡ ❞♦✐s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✳
❊♥tr❡t❛♥t♦✱ ❤á ✉♠❛ ❞✐❢❡r❡♥ç❛ ✐♠♣♦rt❛♥t❡ ❡♥tr❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❙✐t♥✐❦♦✈ ❝✐r❝✉❧❛r ❡ ♦✉tr♦s t✐♣♦s ❞❡ s✐st❡♠❛s ♥♦s q✉❛✐s ♦❝♦rr❡ ❡ss❛ ❛❧t❡r♥â♥❝✐❛✳ ◆♦ ♣r♦❜❧❡♠❛ ❞❡ ❙✐t♥✐❦♦✈ ❛ ❢❛♠í❧✐❛ ❞❡ s♦❧✉çõ❡s ♣❡r✐ó❞✐❝❛s t❡♠ ❝♦♠♦ ❧✐♠✐t❡ ✉♠❛ ór❜✐t❛ ❛♣❡r✐ó❞✐❝❛ ✐❧✐♠✐t❛❞❛ ✭♠♦✈✐♠❡♥t♦ ❞❡ ❡s❝❛♣❡ ♣❛r❛❜ó❧✐❝♦✮✱ ❡♥q✉❛♥t♦ ♥♦s ♦✉tr♦s s✐st❡♠❛s ♦ ❧✐♠✐t❡ é ✉♠❛ ór❜✐t❛ ❛♣❡r✐ó❞✐❝❛ ❧✐♠✐t❛❞❛✳ ❉❡✈✐❞♦ ❛ ❡ss❛ ❞✐❢❡r❡♥ç❛ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❙✐t♥✐❦♦✈ ❝✐r❝✉❧❛r ♥ã♦ ♣♦❞❡ s❡r ❡st✉❞❛❞♦ ❞❛ ♠❛♥❡✐r❛ ✉s✉❛❧✱ ❝♦♠♦ ❡♠ ●r♦tt❛ ❘❛❣❛③③♦✱ ❈✳✱ ❬●❘✾✼❪✳
✶✳✷ ❈♦♥s✐❞❡r❛çõ❡s Pr❡❧✐♠✐♥❛r❡s
❙❡❣✉♥❞♦ ❑❡♥♥❡t❤ ❘✳ ▼❡②❡r✱ ❬▼❡②✾✾❪✱ ❤á ✉♠ ✈❡❧❤♦ ❞✐t❛❞♦ ❡♠ ♠❡❝â♥✐❝❛ ❝❡❧❡st❡ q✉❡ ❛✜r♠❛ q✉❡ ✧♥❡♥❤✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✈❛r✐á✈❡✐s é ❜♦♠ ♦ ❜❛st❛♥t❡✧✳ ❱ár✐♦s ❝♦♥❥✉♥t♦s ❞❡ ✈❛r✐á✈❡✐s ❢♦r❛♠ ✉s❛❞♦s ❛té ❤♦❥❡ ❡♠ ❛rt✐❣♦s s♦❜r❡ ❡ss❡ ❛ss✉♥t♦✳ ❊ ♥ã♦ ❤á ❛❧❢❛❜❡t♦s s✉✜❝✐❡♥t❡s q✉❡ ♣❡r♠✐t❛ ❛ss♦❝✐❛r ♣❛r❛ ❝❛❞❛ ✈❛r✐á✈❡❧ ✉♠ sí♠❜♦❧♦ ❞✐❢❡r❡♥t❡✳ ❆♦ ✐♥tr♦❞✉③✐r♠♦s ♦ ♣r♦❜❧❡♠❛ ❞♦s 3 ❝♦r♣♦s r❡str✐t♦ ✭s❡çã♦ ✸✮✱
✉s❛r❡♠♦s ❛❧❣✉♥s s✐st❡♠❛s ❞❡ ❝♦♦r❞❡♥❛❞❛s ❞✐❢❡r❡♥t❡s✳ P❛r❛ ❛ ❞✐s❝✉ssã♦ ❣❡r❛❧ ✉s❛r❡♠♦s✿
z=
q p
, J =J2n=
0 In
−In 0
, ∇zH =∇H=
∂H ∂z1 ✳✳✳ ∂H ∂z2n
✭✶✳✶✮
❙❡ A ❢♦r ✉♠❛ ♠❛tr✐③✱ ❡♥tã♦ AT s❡rá s✉❛ tr❛♥s♣♦st❛ ❡ A∗ = ¯AT ✭❝♦♥❥✉❣❛❞❛ ❞❛ tr❛♥s♣♦st❛ ♦✉
❛❞❥✉♥t❛✮ ❡ A−1 s✉❛ ✐♥✈❡rs❛✳ ❈❤❛♠❛r❡♠♦s ❞❡ T r A♦ tr❛ç♦ ❞❡ A ❡ ❞❡det As❡✉ ❞❡t❡r♠✐♥❛♥t❡✳
❉✐③❡♠♦s q✉❡ ✉♠❛ ❢✉♥çã♦ f : R → C é ❝♦♥tí♥✉❛ ♣♦r ♣❛rt❡s s❡ ❡❧❛ t❡♠✱ s♦♠❡♥t❡✱ ✉♠ ♥ú♠❡r♦
✜♥✐t♦ ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❡♠ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦ ✜♥✐t♦✳
❯♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ♣♦r ♣❛rt❡s ✐♥t❡❣rá✈❡❧ t❡♠ ❝♦♠♦ ♣r♦♣r✐❡❞❛❞❡✱ ❡♠ q✉❛❧q✉❡r ♣♦♥t♦ ❞❡ ❞✐s✲ ❝♦♥t✐♥✉✐❞❛❞❡ t∗∈(t
1, t2)✱ ♦s ❧✐♠✐t❡s
Rt∗
t1 |f(t)|dt=limǫ→0+
Rt∗−ǫ
t1 |f(t)|dt,
Rt2
t∗ |f(t)|dt=limǫ→0+
Rt2
t∗+ǫ|f(t)|dt
❡①✐st❡♠ ❡ s❡♠♣r❡ q✉❡ t1, t2 sã♦ ❡s❝♦❧❤✐❞♦s t❛✐s q✉❡ ♦s ✐♥t❡r✈❛❧♦s(t1, t∗),(t∗, t2) ♥ã♦ ❝♦♥t❡♥❤❛♠
♦✉tr♦s ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡✳
◆♦ ❝❛s♦ ❞❡ ♠❛tr✐③❡s✱ ✉♠❛ ❢✉♥çã♦ ♠❛tr✐❝✐❛❧ ✭♦✉ s✐♠♣❧❡s♠❡♥t❡ ✧♠❛tr✐③✧✮ é ❝♦♥tí♥✉❛✱ ❝♦♥tí♥✉❛ ♣♦r ♣❛rt❡s✱ ✐♥t❡❣rá✈❡❧✱ ❡t❝✳ s❡ ❛s ❝♦♥❞✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s ♣❛r❛ s❡✉s ❝♦❡✜❝✐❡♥t❡s✳ ❖s ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐✲ ♥✉✐❞❛❞❡ ❞❛ ♠❛tr✐③ sã♦ ♦s ♣♦♥t♦s ❞❡ ❞❡s❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ ❝❛❞❛ ❝♦❡✜❝✐❡♥t❡✳ ■♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞❛ ♠❛tr✐③ ✐♠♣❧✐❝❛ ✐♥t❡❣r❛❜✐❧✐❞❛❞❡ ❞❡ s✉❛ ♥♦r♠❛✳
✶✳✸ ❖r❣❛♥✐③❛çã♦ ❞♦ ❚r❛❜❛❧❤♦
✹ ■◆❚❘❖❉❯➬➹❖ ✶✳✸
◆♦ ❈❛♣ít✉❧♦ ✭✸✮✱ ✐r❡♠♦s ❞❡s❝r❡✈❡r ♦ ♣r♦❜❧❡♠❛ ❞♦s 3 ❝♦r♣♦s r❡str✐t♦✱ ♠♦str❛r ❛s ❡q✉❛çõ❡s ❞❡
♠♦✈✐♠❡♥t♦✱ ❞❡✜♥✐r ❛s ❝♦♦r❞❡♥❛❞❛s s✐♥ó❞✐❝❛s ❡ ✜①❛s ✉t✐❧✐③❛❞❛s ♣❛r❛ ❞❡s❝r❡✈❡r ❛❧❣✉♥s s✐st❡♠❛s✳ ❆♣r❡✲ s❡♥t❛r❡♠♦s t❛♠❜é♠ ❛ r❡❧❛çã♦ ❡♥tr❡ ❡ss❡ ♣r♦❜❧❡♠❛ ❡ ♦ ♣r♦❜❧❡♠❛ ❞♦s 3 ❝♦r♣♦s ❣❡r❛❧✱ ❡s❝❧❛r❡❝❡♥❞♦
❈❛♣ít✉❧♦ ✷
❙✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s
❖ ❢♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦ é ♦ ❡st✉❞♦ ❣❡♦♠étr✐❝♦ ❞♦s ❡s♣❛ç♦s ❞❡ ❢❛s❡ ❞❡ s✐st❡♠❛s ♠❡❝â♥✐✲ ❝♦s✱ ♣♦rt❛♥t♦ q✉❡r❡♠♦s ❡st✉❞❛r ♣r♦❜❧❡♠❛s ♠❡❝â♥✐❝♦s ✉t✐❧✐③❛♥❞♦ ❛ ❣❡♦♠❡tr✐❛ ❞❡ s✉❛s s♦❧✉çõ❡s✳ ❯♠ s✐st❡♠❛ ♠❡❝â♥✐❝♦ ❍❛♠✐❧t♦♥✐❛♥♦ é ❞❛❞♦ ♣♦r ✉♠❛ ✈❛r✐❡❞❛❞❡ ❞❡ ❞✐♠❡♥sã♦ ♣❛r ✭❡s♣❛ç♦ ❞❡ ❢❛s❡✮✱ ✉♠❛ ❡str✉t✉r❛ s✐♠♣❧ét✐❝❛ ♥❡ss❡ ❡s♣❛ç♦ ✭❛ ✐♥t❡❣r❛❧ ❞❡ P♦✐♥❝❛ré✮ ❡ ✉♠❛ ❢✉♥çã♦ ❞❡✜♥✐❞❛ ♥❡ss❡ ♠❡s♠♦ ❡s♣❛ç♦ ✭❛ ❢✉♥çã♦ ❍❛♠✐❧t♦♥✐❛♥❛✮✳ P♦rt❛♥t♦✱ ♦ ❢♦r♠❛❧✐s♠♦ ❍❛♠✐❧t♦♥✐❛♥♦ é ❛ ❡str✉t✉r❛ ♠❛t❡♠át✐❝❛ ♥❛t✉r❛❧✱ ♣❛r❛ s❡ ❞❡s❡♥✈♦❧✈❡r ❛ t❡♦r✐❛ ❞❡ s✐st❡♠❛s ♠❡❝â♥✐❝♦s ❝♦♥s❡r✈❛t✐✈♦s✱ ♣♦✐s ♣❛r❛ ❡ss❡ t✐♣♦ ❞❡ s✐st❡♠❛s ♠❡❝â♥✐❝♦s t❡♠♦s t♦❞❛s ❛s ❡str✉t✉r❛s r❡q✉❡r✐❞❛s✳
❖ tr❛t❛♠❡♥t♦ ❍❛♠✐❧t♦♥✐❛♥♦ s✐♠♣❧✐✜❝❛ ✉♠❛ sér✐❡ ❞❡ ♣r♦❜❧❡♠❛s ♠❡❝â♥✐❝♦s q✉❡ ✈✐st♦ ♣♦r ♦✉tr❛ t❡♦r✐❛ s❡ t♦r♥❛ ✉♠ ♣♦r❜❧❡♠❛ ✐♥tr❛tá✈❡❧ ✭♣♦r ❡①❡♠♣❧♦✱ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❛tr❛çã♦ ❞❡ ❞♦✐s ❝❡♥tr♦s ✜①♦s✮✳ ❖ ♣♦♥t♦ ❞❡ ✈✐st❛ ❍❛♠✐❧t♦♥✐❛♥♦ t❡♠ ❛✐♥❞❛ ♠❛✐s ✈❛❧♦r ♣❡❧♦ ❢❛t♦ ❞❡ ❛❧❣✉♠❛s t❡♦r✐❛s ✐♠♣♦rt❛♥t❡s t❡r❡♠ s✐❞♦ ❢♦r♠✉❧❛❞❛s ✉t✐❧✐③❛♥❞♦✲s❡ ❞❡ss❡ ❢♦r♠❛❧✐s♠♦✱ ❝♦♠♦ ❛ t❡♦r✐❛ ❞❡ ♣❡rt✉r❜❛çõ❡s✱ ❛ t❡♦r✐❛ ❡r❣ó❞✐❝❛✱ ❛ ♠❡❝â♥✐❝❛ ❡st❛tíst✐❝❛✱ ❛ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✱ ❡t❝✳
◆❡st❡ ❝❛♣ít✉❧♦ ❛❜♦r❞❛r❡♠♦s ❜r❡✈❡♠❡♥t❡ s♦❜r❡ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❜ás✐❝❛s ❞❡ s✐st❡♠❛s ❍❛♠✐❧✲ t♦♥✐❛♥♦s ❝♦♠ ❡①❡♠♣❧♦s✱ ❬▼❍❖✵✾❪ ❡ s✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♣❡r✐ó❞✐❝♦s✱ ❬❨❙✼✺❪✳
✷✳✶ ❋✉♥❞❛♠❡♥t♦s ❡ ❡①❡♠♣❧♦s
❯♠ s✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ é ✉♠ s✐st❡♠❛ ❞❡ 2n❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♦r❞✐♥ár✐❛s ❞❛ ❢♦r♠❛
˙
q=Hp, p˙=−Hq, ✭✷✳✶✮
♦✉✱ ❡♠ ❝♦♠♣♦♥❡♥t❡s✱
˙
qi=
∂H ∂pi
(t, q, p), p˙i =−
∂H ∂qi
(t, q, p), i= 1, . . . , n, ✭✷✳✷✮
❡♠ q✉❡ H = H(t, q, p), q✉❡ s❡rá ❝❤❛♠❛❞❛ ❞❡ ❍❛♠✐❧t♦♥✐❛♥❛✱ é ✉♠❛ ❢✉♥çã♦ H : O → R ♦♥❞❡ O ⊂R×Rn×Rn ❡O é ❛❜❡rt♦✳ ❖s ✈❡t♦r❡sq= (q1, . . . , qn)❡ p= (p1, . . . , pn) sã♦ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡
❝❤❛♠❛❞♦s ❞❡ ✈❡t♦r❡s ♣♦s✐çã♦ ❡ ♠♦♠❡♥t♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❡ t é ♦ t❡♠♣♦✳ ❆s ✈❛r✐á✈❡✐s q ❡ p sã♦
❞✐t❛s ✈❛r✐á✈❡✐s ❝♦♥❥✉❣❛❞❛s✳
◆❛ ♥♦t❛çã♦ ✭✶✳✶✮✱ ✭✷✳✶✮ ♣♦❞❡ s❡r r❡❡s❝r✐t♦ ❝♦♠♦
˙
z=J∇H(t, z). ✭✷✳✸✮
❯♠ r❡s✉❧t❛❞♦ ❜ás✐❝♦ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s é ♦ t❡♦r❡♠❛ ❞❡ ❡①✐stê♥❝✐❛ ❡ ✉♥✐❝✐❞❛❞❡✳ ❖ t❡♦r❡♠❛ ❞✐③ q✉❡ s❡H é ❞❡ ❝❧❛ss❡ C1✱ ♣❛r❛ ❝❛❞❛ (t0, z0) ∈ O✱ ❤á ✉♠❛ ú♥✐❝❛ s♦❧✉çã♦ z=φ(t, t0, z0) ❞❡ ✭✷✳✸✮✱
❞❡✜♥✐❞❛ ♣❛r❛t♥✉♠ ✐♥t❡r✈❛❧♦ ❛❜❡rt♦ q✉❡ ❝♦♥té♠t0✱ ❡ q✉❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✐♥✐❝✐❛❧φ(t0, t0, z0) =z0.
◆♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❡♠ q✉❡H ✐♥❞❡♣❡♥❞❡ ❞❡t✱ ♦✉ s❡❥❛✱ H:O →R,♦♥❞❡ O é ✉♠ ❛❜❡rt♦ ❞❡R2n,❛
❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ✭✷✳✸✮ é ❛✉tô♥♦♠❛ ❡ ♦ s✐st❡♠❛ ❍❛♠✐❧t♦♥✐❛♥♦ é ❞✐t♦ ❝♦♥s❡r✈❛t✐✈♦✳ ◆❡st❡ ❝❛s♦ ✈❛❧❡
✻ ❙■❙❚❊▼❆❙ ❍❆▼■▲❚❖◆■❆◆❖❙ ✷✳✶
❛ ❝❤❛♠❛❞❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ tr❛♥s❧❛çã♦ ♣❛r❛ ♦ ✢✉①♦✿φ(t−t0,0, z0) =φ(t, t0, z0)✳ ◆❡st❡ ❝❛s♦✱ t❛♠❜é♠✱
♣♦❞❡✲s❡ ❡❧✐♠✐♥❛r ❛ ❞❡♣❡♥❞ê♥❝✐❛ ❞❡t0♥❛ s♦❧✉çã♦✱ ♣♦❞❡♥❞♦✲s❡ ❡s❝r❡✈❡r ❛ s♦❧✉çã♦ ❝♦♠♦φ(t, z0)✱ ❛ss✐♠
♣♦❞❡♠♦s ❞✐③❡r q✉❡ ❛s s♦❧✉çõ❡s sã♦ ❝✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s ❡♠ O ⊂R2n, ❡ O é ❝❤❛♠❛❞♦ ❞❡ ❡s♣❛ç♦
❞❡ ❢❛s❡✳
❈♦♥s✐❞❡r❡ ❛❣♦r❛ ♦ ❝❛s♦ ❛✉tô♥♦♠♦ ❞❡ ✭✷✳✸✮✱
˙
z=J∇H(z). ✭✷✳✹✮
❯♠❛ ✐♥t❡❣r❛❧ ♣r✐♠❡✐r❛✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡ ✉♠❛ ✐♥t❡❣r❛❧ ♣❛r❛ ✭✷✳✹✮ é ✉♠❛ ❢✉♥çã♦ s✉❛✈❡F :O →R,
q✉❡ é ❝♦♥st❛♥t❡ ❛♦ ❧♦♥❣♦ ❞❛s s♦❧✉çõ❡s ❞❡ ✭✷✳✹✮✱ ✐st♦ é✱ F(φ(t, z0)) = F(z0) =constante✳ ❊①❡♠♣❧♦s
❞❡ ✐♥t❡❣r❛✐s sã♦ ❛ ❡♥❡r❣✐❛✱ ♦ ♠♦♠❡♥t♦✳ ❆s s✉♣❡r❢í❝✐❡s ❞❡ ♥í✈❡✐s F−1(c) ⊂ R2n✱ c ✉♠❛ ❝♦♥st❛♥t❡✱
sã♦ ❡①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s ✐♥✈❛r✐❛♥t❡s✱ ✐st♦ é✱ sã♦ ❝♦♥❥✉♥t♦s t❛✐s q✉❡✱ s❡ ❛s s♦❧✉çõ❡s ❝♦♠❡ç❛♠ ♥❡❧❡s✱ ♣❡r♠❛♥❡❝❡♠ ♥❡ss❡ ❝♦♥❥✉♥t♦✳ ❊♠ ❣❡r❛❧✱ ♦s ❝♦♥❥✉♥t♦s ❞❡ ♥í✈❡✐s sã♦ ✈❛r✐❡❞❛❞❡s ❞❡ ❞✐♠❡♥sã♦ 2n−1✳
❆❣♦r❛ ✐r❡♠♦s ❞❛r ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ s✐st❡♠❛s ❍❛♠✐❧t♦♥✐❛♥♦s ❧✐❣❛❞♦s ❛ ♣r♦❜❧❡♠❛s ❡♠ ♠❡❝â♥✐❝❛ ❝❡❧❡st❡✳
✶✳ ❈♦♥s✐❞❡r❡ N ♣♦♥t♦s ♠❛t❡r✐❛✐s ♠♦✈❡♥❞♦✲s❡ ♥✉♠ s✐st❡♠❛ ❞❡ r❡❢❡rê♥❝✐❛ ◆❡✇t♦♥✐❛♥♦✱ R3✱ ❡♠
q✉❡ ❛ ú♥✐❝❛ ❢♦rç❛ ❛t✉❛♥t❡✱ s❡❥❛ ❛ ❢♦rç❛ ❞❡ ❛tr❛çã♦ ❡♥tr❡ ♦s ♠❡s♠♦s✳ ❙❡❥❛qi ♦ ✈❡t♦r ♣♦s✐çã♦ ❞❛
i−és✐♠❛ ♣❛rtí❝✉❧❛ ❝♦♠ ♠❛ss❛ mi❀ ❡♥tã♦ ♣❡❧❛ s❡❣✉♥❞❛ ❧❡✐ ❞❡ ◆❡✇t♦♥ ❡ ♣❡❧❛ ❧❡✐ ❞❛ ❣r❛✈✐t❛çã♦
✉♥✐✈❡rs❛❧ t❡♠♦s ❛ s❡❣✉✐♥t❡ ❡q✉❛çã♦ ❞♦ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ❛i−és✐♠❛ ♠❛ss❛
miq¨i = N
X
j=1
Gmimj(qi−qj)
||qj−qi||3
= ∂U
∂qi ✭✷✳✺✮
♦♥❞❡
U =− X 1≤i<j≤N
Gmimj
||qi−qj|| ✭✷✳✻✮
Gé ❛ ❝♦♥st❛♥t❡ ❣r❛✈✐t❛❝✐♦♥❛❧ ✉♥✐✈❡rs❛❧ ❡U é ♦ ♣♦t❡♥❝✐❛❧✳ ❖ s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s
✭✷✳✺✮ ❞❡✜♥❡ ♦ ♣r♦❜❧❡♠❛ ❞♦s N ❝♦r♣♦s ✭❢♦r♠✉❧❛çã♦ ◆❡✇t♦♥✐❛♥❛✮✳
❙❡❥❛ q = (q1, . . . , qN) ∈ R3N✳ ❖ s✐st❡♠❛ ✭✷✳✺✮ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❡♠ s✉❛ ❢♦r♠❛ ✈❡t♦r✐❛❧ ❞❛
s❡❣✉✐♥t❡ ♠❛♥❡✐r❛
Mq¨− ∇U(q) = 0,
♦♥❞❡ M = diag(m1, m1, m1, . . . , mN, mN, mN)❀ ❛ ❢♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛ ❞♦ ♣r♦❜❧❡♠❛ é
♦❜t✐❞❛ ✐♥tr♦❞✉③✐♥❞♦ ♦s ✈❡t♦r❡s ❞♦s ♠♦♠❡♥t♦s ❧✐♥❡❛r❡s✳ ❉❡✜♥❛ p = (p1, . . . , pN) ∈ R3N ♣♦r
p=Mq˙✱ ♣♦rt❛♥t♦ pi =miq˙i é ♦ ♠♦♠❡♥t♦ ❞❛ i−és✐♠❛ ♣❛rtí❝✉❧❛✳ ❆s ❡q✉❛çõ❡s ❞♦ ♠♦✈✐♠❡♥t♦
t♦r♥❛♠✲s❡ ❡♥tã♦
˙
q=Hp =M−1p, p˙=−Hq=−Uq ✭✷✳✼✮
♦✉ ❡♠ ❝♦♠♣♦♥❡♥t❡s✱
˙
qi=
∂H ∂pi
=pi/mi, p˙i =−
∂H ∂qi
=−∂U
∂qi
=
N
X
i=1
Gmimj(qj−qi)
||qi−qj||3 ✭✷✳✽✮
♦♥❞❡ ♦ ❍❛♠✐❧t♦♥✐❛♥♦ é
H = 1 2p
TM−1p+U =
N
X
i=1 ||pi||2
2mi
✷✳✶ ❋❯◆❉❆▼❊◆❚❖❙ ❊ ❊❳❊▼P▲❖❙ ✼
❍ é ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛ ❞❡ ♣❛rtí❝✉❧❛s✳ ❊❧❛ é ✉♠❛ ✐♥t❡❣r❛❧ ❞♦ s✐st❡♠❛✱ ❥á q✉❡
dH
dt = ∂H∂qq˙+∂H∂pp˙= 0.
❖s ✈❡t♦r❡s q ❡ psã♦ ❞✐t♦s ❞❡ ✈❛r✐á✈❡✐s ❝♦♥❥✉❣❛❞❛s✳
❖ ♣r♦❜❧❡♠❛ ❞♦s N ❝♦r♣♦s é ✉♠ s✐st❡♠❛ ❞❡ 3N ✭6N✮ ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ✭♣r✐♠❡✐r❛
♦r❞❡♠ ♥❛ ❢♦r♠✉❧❛çã♦ ❍❛♠✐❧t♦♥✐❛♥❛✮✱ ♣♦rt❛♥t♦ ✉♠❛ s♦❧✉çã♦ ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ r❡q✉❡r 6N −1
✐♥t❡❣r❛✐s ✐♥❞❡♣❡♥❞❡♥t❡s ❞♦ t❡♠♣♦✳ P❛r❛N >2t❡r t❛♥t❛s ✐♥t❡❣r❛✐s ❣❧♦❜❛✐s é q✉❛s❡ ✐♠♣♦ssí✈❡❧✳
❊♥tr❡t❛♥t♦✱ ♣❛r❛N >2❡①✐st❡♠ 10✐♥t❡❣r❛✐s ♣❛r❛ s✐st❡♠❛✳
❙❡❥❛
L=p1+. . .+pN
❖ ♠♦♠❡♥t♦ ❧✐♥❡❛r t♦t❛❧✳ ❉❡ ✭✷✳✽✮ s❡❣✉❡ q✉❡L˙ = 0.■ss♦ ♥♦s ❞❛r C¨ = 0✱ ❡♠ q✉❡
C=m1q1+. . .+mNqN
é ♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ❞♦ s✐st❡♠❛✱ ♣♦✐s C˙ = L✳ P♦rt❛♥t♦ ♦ ♠♦♠❡♥t♦ ❧✐♥❡❛r t♦t❛❧ é ❝♦♥st❛♥t❡✱
❡ ♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ ❞♦ s✐st❡♠❛ ♠♦✈❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ✉♥✐❢♦r♠❡ r❡t✐❧í♥❡♦✳ ❙❡ t♦♠❛r♠♦s ✉♠❛ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♥♦ q✉❛❧ ♦ ❝❡♥tr♦ ❞❡ ♠❛ss❛ é ✜①❛❞♦ ♥❛ ♦r✐❣❡♠✱ ❛s6✐♥t❡❣r❛✐s s❡ r❡❞✉③❡♠
❛✿
X
miqi(1)= 0,
X
miq(2)i = 0,
X
miqi(3)= 0 ✭✷✳✶✵✮
X
miq˙i(1)= 0,
X
miq˙i(2)= 0,
X
miq˙(3)i = 0. ✭✷✳✶✶✮
❍á✱ t❛♠❜é♠✱ 3 ✐♥t❡❣r❛✐s ❛s q✉❛✐s ♠♦str❛♠ q✉❡ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r é ❝♦♥st❛♥t❡ ❡♠ t♦r♥♦
❞❡ q✉❛❧q✉❡r ❡✐①♦ ✜①❛❞♦ ♥♦ ❡s♣❛ç♦✳ ❙❡ t♦♠❛r♠♦s ♦s ❡✐①♦s ❝♦♠♦ ♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✱ ❡ss❛ ✐♥t❡❣r❛✐s sã♦✿
X
mi(qi(2)q˙i(3)−qi(3)q˙(2)i ) = 0,
X
mi(qi(3)q˙(1)i −q(1)i q˙(3)i ) = 0,
X
mi(q(1)i q˙(2)i −qi(2)q˙i(1)) = 0.
✭✷✳✶✷✮ ❊ ❛ ♦✉tr❛ ✐♥t❡❣r❛❧ q✉❡ s♦❜r❛ é ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞♦ s✐st❡♠❛✳
✷✳ Pr♦❜❧❡♠❛ ❞❡ ❑❡♣❧❡r
❯♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❡2 ❝♦r♣♦s é ♦❜t✐❞♦ q✉❛♥❞♦ ✉♠ ❝♦r♣♦ ❞❡ ♠❛ss❛M é s✉♣♦st♦
✜①❛❞♦ ♥❛ ♦r✐❣❡♠ ✭♣♦r ❡①❡♠♣❧♦✱ ♦ ❙♦❧✮✳ ◆❡st❡ ❝❛s♦ ❛ ❡q✉❛çã♦ ◆❡✇t♦♥✐❛♥❛ ❞♦ ♠♦✈✐♠❡♥t♦ ❞♦ ♦✉tr♦ ❝♦r♣♦ ❞❡ ♠❛ss❛m é ❞❛ ❢♦r♠❛
mq¨= GM mq
||q||3 , ou q¨=
µq
||q||3 =∇U(q), ✭✷✳✶✸✮
❡♠ q✉❡ q∈R3 é ♦ ✈❡t♦r ♣♦s✐çã♦ ❞♦ ❝♦r♣♦ ❞❡ ♠❛ss❛ m♥✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ✜①♦✱µ é
❛ ❝♦♥st❛♥t❡GM✱ ❡ U ♦ ♣♦t❡♥❝✐❛❧
U =−||µq||.
❙❡ ❞❡✜♥✐r♠♦s ♦ ♠♦♠❡♥t♦ p = ˙q ∈R3✱ ❡♥tã♦ ❛ ❡q✉❛çã♦ ◆❡✇t♦♥✐❛♥❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❡♠ s✉❛
✽ ❙■❙❚❊▼❆❙ ❍❆▼■▲❚❖◆■❆◆❖❙ ✷✳✷
˙
q =Hp=p,p˙=−Hq = ||µqq||3, ♦♥❞❡
H= ||p2||2 − ||µq||.
❍ é ❝❤❛♠❛❞❛ ❛ ❍❛♠✐❧t♦♥✐❛♥❛ ❞♦ ♣r♦❜❧❡♠❛ ❞❡ ❑❡♣❧❡r✳ ❆ ❢♦r♠✉❧❛çã♦ ◆❡✇t♦♥✐❛♥❛ é ✉♠ s✐st❡♠❛ ❞❡ três ❡q✉❛çõ❡s ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠✱ ❡♥q✉❛♥t♦ ❛s ❡q✉❛çõ❡s ❍❛♠✐❧t♦♥✐❛♥❛s sã♦ s❡✐s ❡q✉❛çõ❡s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳
❍ é ✉♠❛ ✐♥t❡❣r❛❧ ❞♦ ♠♦✈✐♠❡♥t♦✱ ✐st♦ é✱ é ❝♦♥st❛♥t❡ ❛♦ ❧♦♥❣♦ ❞❛s s♦❧✉çõ❡s✱ ♣♦✐s
dH dt = 0.
❉❡✜♥✐♥❞♦ A=q×p,❝♦♠♦ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r✱ t❡♠♦s
˙
A= ˙q×p+q×p˙=p×p+q×(−||µqq||3) = 0,
♣♦rt❛♥t♦ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r A é ❝♦♥st❛♥t❡ ❛♦ ❧♦♥❣♦ ❞❛s s♦❧✉çõ❡s✱ ❧♦❣♦ ❛s três ❝♦♠♣♦♥❡♥t❡s
❞❡ Asã♦ ✐♥t❡❣r❛✐s✳ ❙❡A= 0✱ ❡♥tã♦
d dt
q
||q||
= (q||×qq||˙)×3 q = A||q×||q3 = 0.
P♦rt❛♥t♦ s❡ ♦ ♠♦♠❡♥t♦ ❛♥❣✉❧❛r é ♥✉❧♦✱ ♦ ♠♦✈✐♠❡♥t♦ é r❡t✐❧í♥❡♦✳ ❖ ✉s♦ ❞❡ss❛ r❡t❛ ❞♦ ♠♦✈✐♠❡♥t♦ ❝♦♠♦ ✉♠ ❞♦s ❡✐①♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ tr❛♥s❢♦r♠❛✲s❡ ♦ ♣r♦❜❧❡♠❛ ❡♠ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ✉♠ ❣r❛✉ ❞❡ ❧✐❜❡r❞❛❞❡✱ ❡ ♣♦rt❛♥t♦ ✐♥t❡❣rá✈❡❧✳ ◆❡st❡ ❝❛s♦ ❛s ✐♥t❡❣r❛✐s sã♦ ❡❧❡♠❡♥t❛r❡s ❡ ♣♦❞❡✲s❡ ♦❜t❡r ❢ór♠✉❧❛s s✐♠♣❧❡s ♣❛r❛ ❡❧❛s✳
❙❡ A 6= 0✱ ❡♥tã♦ q ❡ p = ˙q sã♦ ♦rt♦❣♦♥❛✐s ❛ A✱ ♣♦rt❛♥t♦ ♦ ♠♦✈✐♠❡♥t♦ ♦❝♦rr❡ ♥✉♠ ♣❧❛♥♦
♦rt♦❣♦♥❛❧ ❛ A✱ ♣❧❛♥♦ ✐♥✈❛r✐❛♥t❡✳ ◆❡st❡ ❝❛s♦ ♦ ♣r♦❜❧❡♠❛ é r❡❞✉③✐❞♦ ❛ ✉♠ ♣r♦❜❧❡♠❛ ❝♦♠ ❞♦✐s
❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❝♦♠ ✉♠❛ ✐♥t❡❣r❛❧✱ ❡ t❛❧ ♣r♦❜❧❡♠❛ é ✐♥t❡❣rá✈❡❧✳
✷✳✷ ❙✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s
✷✳✷✳✶ ❉❡✜♥✐çõ❡s ❜ás✐❝❛s◆❡st❛ s❡çã♦ ❡st❛r❡♠♦s ♣r❡♦❝✉♣❛❞♦s ❝♦♠ s✐st❡♠❛s ❞♦ t✐♣♦✿
dxi
dt =ai1(t)x1+. . .+ain(t)xn, i= 1, . . . , n, ✭✷✳✶✹✮
❡♠ q✉❡ ❛ ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡ t é r❡❛❧ ❡ ♦s ❝♦❡✜❝✐❡♥t❡s aij(t) ✭j, i = 1, . . . , n✮ sã♦ ❣❡r❛❧♠❡♥t❡
❢✉♥çõ❡s ❛ ✈❛❧♦r❡s ❝♦♠♣❧❡①♦s✱ ❝♦♥tí♥✉❛s ♣♦r ♣❛rt❡ ❡ ✐♥t❡❣rá✈❡✐s ❡♠ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦ ✜♥✐t♦(t1, t2).
❊♠ ♥♦t❛çã♦ ♠❛tr✐❝✐❛❧✱ ♦ s✐st❡♠❛ ✭✷✳✶✹✮✱ t♦r♥❛✲s❡
dx
dt =A(t)x, ✭✷✳✶✺✮
❡♠ q✉❡x= (x1, . . . , xn)T ❡ A(t) = [aij(t)]é ❛ ♠❛tr✐③ n×n❝♦♠ ❝♦❡✜❝✐❡♥t❡s aij(t).
❙❡❥❛♠ xi(t) ✭i = 1, . . . , k✮ k s♦❧✉çõ❡s ❛r❜✐trár✐❛s ❞❛ ❡q✉❛çã♦ ✭✷✳✶✺✮ ✭xi(t) é ✉♠ ✈❡t♦r ❝♦❧✉♥❛✮
✷✳✷ ❙■❙❚❊▼❆❙ ❉❊ ❊◗❯❆➬Õ❊❙ ❉■❋❊❘❊◆❈■❆■❙ ▲■◆❊❆❘❊❙ ✾
s❛t✐s❢❛③ ♦ s❡❣✉✐♥t❡ s✐st❡♠❛ ♠❛tr✐❝✐❛❧
dX
dt =A(t)X ✭✷✳✶✻✮
❡
X(0) = [α1. . . αk].
❙✉♣♦♥❤❛ k =n ❡ det X(0)6= 0. ◗✉❛❧q✉❡r ♦✉tr❛ s♦❧✉çã♦ X1(t)✱ n×n✱ ❞❛ ❡q✉❛çã♦ ✭✷✳✶✻✮ ♣♦❞❡
s❡r ❡①♣r❡ss❛ ❝♦♠♦
X1(t) =X(t)C, ✭✷✳✶✼✮
♦♥❞❡ C=X−1(0)X 1(0).
◆❡st❡ ❝❛s♦ t❡♠♦s ❛ ❢ór♠✉❧❛ ❞❡ ▲✐♦✉✈✐❧❧❡✲❏❛❝♦❜✐ ♦✉ ❚❡♦r❡♠❛ ❞❡ ▲✐♦✉✈✐❧❧❡
det X1(t) =det X1(0)exp
Z t
0
T r A(t)dt. ✭✷✳✶✽✮
❉❡✜♥✐çã♦✳ ❯♠ ❝♦♥❥✉♥t♦ ❢✉♥❞❛♠❡♥t❛❧ ❞❡ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ✭✷✳✶✺✮ é q✉❛❧q✉❡r ❝♦♥❥✉♥t♦ ❞❡ns♦❧✉✲
çõ❡sx1(t), . . . , xn(t)q✉❡ sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❆ ♠❛tr✐③X(t)❝♦♠ ❝♦❧✉♥❛sx1(t), . . . , xn(t),
♣♦rt❛♥t♦✱ s❛t✐s❢❛③ ✭✷✳✶✻✮✱ ❡ é ❝❤❛♠❛❞❛ ❞❡ ♠❛tr✐③ ❢✉♥❞❛♠❡♥t❛❧✳ ◆♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❡♠ q✉❡ X(0) =In
❝❤❛♠❛♠♦s ❡ss❛ ♠❛tr✐③ ❞❡ ♠❛tr✐③ ♣r✐♥❝✐♣❛❧✳
❱❛♠♦s ✐♥t❡r♣r❡t❛r ❛❣♦r❛ ✭✷✳✶✻✮ ❝♦♠♦ ✉♠ ♦♣❡r❛❞♦r q✉❡ ❞❛❞❛ ✉♠❛ ♠❛tr✐③A(t) ❞❡✈♦❧✈❡ ❛ ♠❛tr✐③
♣r✐♥❝✐♣❛❧ X(t)✱ ✐st♦ é ✉♠❛ ♠❛tr✐③ X(t) q✉❡ s❛t✐s❢❛ç❛ ✭✷✳✶✻✮ ❡X(0) =I✳ ❖✉ s❡❥❛✱ ✈❛♠♦s ❞❡✜♥✐r ✉♠
❝♦♥❥✉♥t♦ Ω ❞❡ t♦❞❛s ❛s ♠❛tr✐③❡sA(t)✱n×n✱ ❝♦♥tí♥✉❛s ♣♦r ♣❛rt❡s ❡ ✐♥t❡❣rá✈❡✐s ❡♠[0, T], T >0 ❡
♦ ❝♦♥❥✉♥t♦ ∆ ❞❛s ♠❛tr✐③❡s n×n ♥ã♦✲s✐♥❣✉❧❛r❡s ❡ ❝♦♥tí♥✉❛s✱ ❞❡✜♥✐❞❛ ♥✉♠ ✐♥t❡r✈❛❧♦ [0, T]✱T >0✱
❝♦♠ ❞❡r✐✈❛❞❛ ❝♦♥tí♥✉❛ ♣♦r ♣❛rt❡s✱ t❛❧ q✉❡X(0) =I ❡X(t)s❛t✐s❢❛③❡♥❞♦ ✭✷✳✶✻✮✳ P♦❞❡♠♦s ❝♦♥s✐❞❡r❛r
♥❛t✉r❛❧♠❡♥t❡
Ω ={A: [0, T]→ Mn×n(R);A continua por partes e integravel},
∆ ={X : [0, T]→GL(n);X(0) =I, X continua, com derivada continua por partes eX˙ =AX para algum A∈Ω},
❉❡✜♥❛ ✉♠❛ ♥♦r♠❛ ❡♠Ω ❡ ✉♠❛ ♠étr✐❝❛ ❡♠∆♣♦r
||A||=
Z T
0 |
A(t)|dt, ρ(X1, X2) =sup0≤t≤T|X1(t)−X2(t)|+
Z T
0 | ˙
X1(t)−X˙2(t)|dt, ✭✷✳✶✾✮
♦♥❞❡ |Z|❞❡♥♦t❛ ❛ ♥♦r♠❛ ❞❛ ♠❛tr✐③Z =|νij|:
|Z|=qPi,j|νij|2.
▼♦str❛r❡♠♦s q✉❡ ♠❛tr✐③❡s A1(t)❡ A2(t) ✧♣ró①✐♠❛s✧♥❛ ♥♦r♠❛||.||❞❡t❡r♠✐♥❛♠ ♠❛tr✐③❡s ♣r✐♥❝✐✲
♣❛✐s Z1(t) ❡ Z2(t) ✧♣ró①✐♠❛s✧♥❛ ♠étr✐❝❛ ρ ❡ ✈✐❝❡✲✈❡rs❛✱ ♦✉ s❡❥❛✱ ✈❛❧❡✿
Pr♦♣♦s✐çã♦ ✭❬❨❙✼✺❪✮ ✶✳ Ω❡ ∆sã♦ ❤♦♠❡♦♠♦r❢♦s✱ ♦✉ s❡❥❛✱ ❡①✐st❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❥❡t♦r❛ ❝♦♥tí♥✉❛
❝♦♠ ✐♥✈❡rs❛ ❝♦♥tí♥✉❛ q✉❡ ❧❡✈❛ ♠❛tr✐③❡s ❞❡ Ω❡♠ ♠❛tr✐③❡s ❞❡ ∆✳
❉❡♠♦♥str❛çã♦✳ ❜✐❥❡t✐✈✐❞❛❞❡✿
❆ ❜✐❥❡t✐✈✐❞❛❞❡ ❞♦ ♦♣❡r❛❞♦r s❛✐ ❞✐r❡t♦ ❞❛ ❞❡✜♥✐çã♦ ❞❡ Ω ❡ ∆✳ ❆ ✐♥❥❡t✐✈✐❞❛❞❡ s❛✐ ❞♦ t❡♦r❡♠❛ ❞❛
✉♥✐❝✐❞❛❞❡ ❞❡ s♦❧✉çõ❡s✱ ❞❛❞♦A(t)♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ❞❛s s♦❧✉çõ❡s ❡①✐st❡ ✉♠ ú♥✐❝♦X(t)∈∆s❛t✐s❢❛③❡♥❞♦
✭✷✳✶✻✮✳ ❆ s♦❜r❡❥❡t✐✈✐❞❛❞❡ s❛✐ ❞♦ ❢❛t♦ q✉❡ ❞❛❞♦ X(t)∈∆✱X(t) é ♥ã♦ s✐♥❣✉❧❛r ❡ s❛t✐s❢❛③ ✭✷✳✶✻✮ ♣❛r❛
✶✵ ❙■❙❚❊▼❆❙ ❍❆▼■▲❚❖◆■❆◆❖❙ ✷✳✷
A(t) = dXdt X−1(t).
❈♦♥t✐♥✉✐❞❛❞❡✿
❉❛❞♦ X(t) ∈ ∆, ∃δ > 0 t❛❧ q✉❡ ♣❛r❛ t♦❞♦ X1(t) ∈ ∆ ❝♦♠ ρ(X1, X) < δ, t❡♠♦s q✉❡ ♦
sup0≤t≤T|X1(t)−X(t)| < δ ❝♦♠♦ X(t) ❡ X1(t) sã♦ ♥ã♦ s✐♥❣✉❧❛r❡s✱ ♦✉ s❡❥❛✱ det X(t) ≥ M > 0
✭0≤t≤T✮∀X ∈∆s❡❣✉❡ q✉❡∃δ1s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦ t❛❧ q✉❡sup0≤t≤T|X1−1(t)−X−1(t)|< δ1 ||A1−A||=
RT
0 |A1(t)−A(t)|dt=
RT
0 |X˙(X−1−X −1
1 ) + ( ˙X−X˙1)X1−1|dt≤ ≤sup0≤t≤T|X−1−X1−1|
RT
0 |X˙|dt+sup0≤t≤T|X1−1|
RT
0 |X˙ −X˙1|dt.
❊ ♦ ❧❛❞♦ ❞✐r❡✐t♦ ❞❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦❞❡ ✜❝❛r tã♦ ♣❡q✉❡♥♦ q✉❛♥t♦ s❡ q✉❡✐r❛ ♣❛r❛ ρ(X1, X) s✉✜❝✐✲
❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❝♦♠♣❧❡t❛♥❞♦ ❛ ❞❡♠♦♥str❛çã♦✳
❙❡❣✉❡ ❞♦ t❡♦r❡♠❛ q✉❡✱ s❡ ||A1 −A|| é s✉✜❝✐❡♥t❡♠❡♥t❡ ♣❡q✉❡♥♦✱ ❡♥tã♦ ρ(X1, X) t❛♠❜é♠ s❡rá✱
❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ♦sup|X1−X|,♦✉ s❡❥❛✱ ♣❛r❛ ♣❡q✉❡♥❛s ✈❛r✐❛çõ❡s ❞❛ ♠❛tr✐③ ❞❡ ❝♦❡✜❝✐❡♥t❡s ❤á
✉♠❛ ♣❡q✉❡♥❛ ✈❛r✐❛çã♦ ❞❛ ♠❛tr✐③ s♦❧✉çã♦✳
❱❛♠♦s ❛❣♦r❛ ❝♦♥s✐❞❡r❛r ♦ s✐st❡♠❛ ❛❞❥✉♥t♦ ❞❡ ✭✷✳✶✺✮✱ ❡♠ ♥♦t❛çã♦ ✈❡t♦r✐❛❧ t❡♠♦s✿
dz dt =−A
∗(t)z ✭✷✳✷✵✮
❡♠ q✉❡ z = (z1, . . . , zn)T ❡ A∗(t) =|¯aji(t)| é ❛ ♠❛tr✐③ n×n❝♦♠ ❝♦❡✜❝✐❡♥t❡s ¯aji(t). ❙❡ A(t) é
✉♠❛ ♠❛tr✐③ r❡❛❧ ❡♥tã♦ t❡♠♦s A∗(t) = AT(t). ◆❡st❡ ❝❛s♦ ♦ s✐st❡♠❛ ✭✷✳✷✵✮ é ❞✐t♦ ♦ ❛ss♦❝✐❛❞♦ ❝♦♠
✭✷✳✶✺✮✳ ❙❡❥❛Z(t) ❛ ♠❛tr✐③ ♣r✐♥❝✐♣❛❧ ❞♦ s✐st❡♠❛ ✭✷✳✷✵✮✳ ❙❡❣✉❡ ❞❡ ✭✷✳✶✻✮ q✉❡
dX∗
dt =X∗A∗(t),
t❡♠♦s ❡♥tã♦ q✉❡ X∗(t)Z(t) =In.❉❡ ❢❛t♦✱X∗(0)Z(0) =In,❡ ❞❡r✐✈❛♥❞♦ t❡♠♦s d
dt(X∗Z) = dX ∗
dt Z+X∗dZdt =X∗A(t)Z−X∗A(t)Z = 0.
P♦rt❛♥t♦ ❛s ♠❛tr✐③❡s ♣r✐♥❝✐♣❛✐s ❞❡ ✭✷✳✶✻✮ ❡ ✭✷✳✷✵✮ tê♠ ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s
Z(t) = [X∗(t)]−1. ✭✷✳✷✶✮
P❛r❛ q✉❛✐sq✉❡r ❞✉❛s s♦❧✉çõ❡sx(t) ❡ z(t) ❞❛s ❡q✉❛çõ❡s ✭✷✳✶✺✮ ❡ ✭✷✳✷✵✮✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❡♠♦s
< x(t), z(t)>=constante. ✭✷✳✷✷✮
❉❡ ❢❛t♦✱
d
dt < x, z >=< dxdt, z >+< x,dtdz >=< Ax, z >−< x, A∗z >= 0.
➱ ❡✈✐❞❡♥t❡ ❞❡ ✭✷✳✷✷✮ q✉❡ ♣❛r❛ q✉❛❧q✉❡r s♦❧✉çã♦z(t)❞❛ ❡q✉❛çã♦ ✭✷✳✷✵✮ ❛ ❢✉♥çã♦ψ(t, x) = (x, z(t))
é ✉♠❛ ✐♥t❡❣r❛❧ ❞❛ ❡q✉❛çã♦ ✭✷✳✶✺✮✳ ❆ ✈♦❧t❛ t❛♠❜é♠ ✈❛❧❡✳
✷✳✷✳✷ ❙✐st❡♠❛s ❞❡ ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ❧✐♥❡❛r❡s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♣❡r✐ó❞✐❝♦s ❛r✲ ❜✐trár✐♦s
◆❡st❛ s❡çã♦ ❛❞♠✐t✐r❡♠♦s q✉❡ ♣❛r❛ ❛❧❣✉♠ T >0❡ t♦❞♦ tt❡♠♦s
A(t+T) =A(t). ✭✷✳✷✸✮
❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ♦s ❝♦❡✜❝✐❡♥t❡s ❞♦ s✐st❡♠❛ ✭✷✳✶✺✮ sã♦ ❢✉♥çõ❡s T−♣❡r✐ó❞✐❝❛s ❞❡ t, ❡♠ ❝♦♥✲
s❡q✉ê♥❝✐❛ s✉❛ ♠❛tr✐③ ♣r✐♥❝✐♣❛❧ s❛t✐s❢❛③ ❛ ✐❞❡♥t✐❞❛❞❡
