❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❈❛♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦
❇r❛q✉✐stó❝r♦♥❛
❆♥❛ ▲✉ís❛ ❙❛❞❡r ❚❛❣❧✐♦❧❛tt♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ✕ P❘❖❋▼❆❚ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡
❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦
Tagliolatto, Ana Luísa Sader
Braquistócrona / Ana Luísa Sader Tagliolatto. - Rio Claro, 2015
54 f. : il., figs.
Dissertação (mestrado) - Universidade Estadual Paulista, Instituto de Geociências e Ciências Exatas
Orientador: Suzinei Aparecida Siqueira Marconato
1. Cálculo das variações. 2. Cicloide. 3. Proposta didática. 4. Cálculo variacional. I. Título.
517.4 T128b
❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖
❆♥❛ ▲✉ís❛ ❙❛❞❡r ❚❛❣❧✐♦❧❛tt♦
❇r❛q✉✐stó❝r♦♥❛
❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❯♥✐✈❡rs✐tár✐❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛✲ ❞♦r❛✿
Pr♦❢❛✳ ❉r❛✳ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦ ❖r✐❡♥t❛❞♦r❛
Pr♦❢❛✳ ❉r❛✳ ▼❛r✐❛ ❆♣❛r❡❝✐❞❛ ❇❡♥á
❉❈▼ ✲ ❋❋❈▲❘P✴❯❙P ❘✐❜❡✐rã♦ Pr❡t♦✴❙P
Pr♦❢✳ ❉r✳ ❏❛✐r ❙✐❧✈ér✐♦ ❞♦s ❙❛♥t♦s
❉❈▼ ✲ ❋❋❈▲❘P✴❯❙P ❘✐❜❡✐rã♦ Pr❡t♦✴❙P
❆❣r❛❞❡❝✐♠❡♥t♦s
❆♦ ♠❡✉ ❡s♣♦s♦ q✉❡ s❡♠♣r❡ ❛♣♦✐♦✉✱ ✐♥❝❡♥t✐✈♦✉ ❡ ❛❝♦♠♣❛♥❤♦✉ ♥♦s ❡st✉❞♦s✳
➚ ♣r♦❢❡ss♦r❛ ❙✉③✐♥❡✐ ❆♣❛r❡❝✐❞❛ ❙✐q✉❡✐r❛ ▼❛r❝♦♥❛t♦ ♣❡❧❛ ❝♦♠♣❡tê♥❝✐❛ ❡ ♣❛❝✐ê♥❝✐❛ ❞✉r❛♥t❡ ❛ ♦r✐❡♥t❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳
❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞❛ ❯◆❊❙P ❞❡ ❘✐♦ ❈❧❛r♦ ♣❡❧❛ ❝♦♥tr✐❜✉✐çã♦ ♥❡ss❡ ♣r♦❝❡ss♦ ❞❡ ❢♦r♠❛çã♦ ❝♦♥t✐♥✉❛❞❛ ❞❡ ♣r♦✜ss✐♦♥❛✐s q✉❡ ❛t✉❛♠ ♥❛ ❡❞✉❝❛çã♦ ❜ás✐❝❛✳
❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞❛ s❡❝r❡t❛r✐❛ ❞❡ ♣ós✲❣r❛❞✉❛çã♦ ❞❛ ❯◆❊❙P ❞❡ ❘✐♦ ❈❧❛r♦ ❡ ❞❛ ❜✐✲ ❜❧✐♦t❡❝❛ ❞♦ ■▼❊❈❈✴❯◆■❈❆▼P ♣❡❧❛ ❛t❡♥çã♦ ❡ ❡♠♣❡♥❤♦ ❡♠ ❛❥✉❞❛r ♣r♦♥t❛♠❡♥t❡✳
❆♦ ❝♦❧❡❣❛ ❆♥❞ré ▲✉✐s ◆♦✈❛❡s ♣❡❧♦ ❛✉①í❧✐♦ ❝♦♠ ♦ ▲❆❚❊❳✳
◗✉❡ ❛q✉❡❧❡ q✉❡ ❝♦♥s✐❣❛ s♦❧✉❝✐♦♥❛r ❡st❡ ♣r♦❜❧❡♠❛ ❝♦♥q✉✐st❡ ♦ ♣rê♠✐♦ q✉❡ ♣r♦♠❡t❡♠♦s✳ ❊st❡ ♣rê♠✐♦ ♥ã♦ é ♦✉r♦ ♥❡♠ ♣r❛t❛ ❬✳✳✳❪ ❛s ❤♦♥r❛s✱ ♦s ❡❧♦❣✐♦s ❡ ♦s ❛♣❧❛✉s♦s❀ ❬✳✳✳❪ ❡①❛❧t❛r❡♠♦s✱ ♣ú❜❧✐❝❛ ❡ ♣r✐✈❛❞❛♠❡♥t❡✱ ♣♦r ♣❛❧❛✈r❛ ❡ ♣♦r ❝❛rt❛✱ ❛ ♣❡rs♣✐❝á❝✐❛ ❞♦ ♥♦ss♦ ❣r❛♥❞❡ ❆♣♦❧❧♦✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦ ❢❛♠♦s♦ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛ ❡ ❞✐❢❡r❡♥t❡s s♦❧✉çõ❡s ❛tr❛✈és ❞❛ t❡♦r✐❛ ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ❡ ❛tr❛✈és ❞❡ ❝♦♥❝❡✐t♦s ❞❛ ❣❡♦♠❡tr✐❛ ❡ ❢ís✐❝❛✱ ❡♥✈♦❧✈❡♥❞♦ s✐t✉❛çõ❡s ❝♦♠ ❝♦♥❞✐çõ❡s ❛♥á❧♦❣❛s às ❞❛ ❜r❛q✉✐stó❝r♦♥❛✳ ❯♠❛ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ ❛❞❡q✉❛❞❛ ❛ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ q✉❡ é ❛❞❛♣tá✈❡❧ ❛ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧ ❢♦✐ t❛♠❜é♠ ❛♣r❡s❡♥t❛❞❛✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦ ✐t ✇❛s ♣r❡s❡♥t❡❞ t❤❡ ❢❛♠♦✉s ❜r❛❝❤✐st♦❝❤r♦♥❡ ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ❞✐✛❡✲ r❡♥t s♦❧✉t✐♦♥s t❤r♦✉❣❤ t❤❡ t❤❡♦r② ♦❢ ✈❛r✐❛t✐♦♥❛❧ ❝❛❧❝✉❧✉s ❛♥❞ t❤r♦✉❣❤ t❤❡ ❝♦♥❝❡♣ts ♦❢ ❣❡♦♠❡tr② ❛♥❞ ♣❤②s✐❝s✱ ✐♥✈♦❧✈✐♥❣ s✐t✉❛t✐♦♥s ✇✐t❤ s✐♠✐❧❛r ❝♦♥❞✐t✐♦♥s t♦ t❤♦s❡ ♦❢ ❜r❛❝❤✐s✲ t♦❝❤r♦♥❡✳ ❆❞❡q✉❛t❡ ❞✐❞❛❝t✐❝ ♣r♦♣♦s❛❧ t♦ ❤✐❣❤ s❝❤♦♦❧ st✉❞❡♥ts ✇❤✐❝❤ ✐s ❛❧s♦ s✉✐t❛❜❧❡ ❢♦r ♠✐❞❞❧❡ s❝❤♦♦❧ st✉❞❡♥ts ✇❛s ♣r❡s❡♥t❡❞✳
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ ❈✉r✈❛ ✐s♦❝rô♥✐❝❛ ❬✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✷ ➪r❡❛ ❞❡❧✐♠✐t❛❞❛ ♣♦r ✉♠ ❛r❝♦ ❞❡ ❝✐❝❧♦✐❞❡ ❬✶❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸ ❈✉r✈❛ ❣❡r❛❞❛ ♣♦r ♣ê♥❞✉❧♦ ❝♦♠ ❛r❝♦s ❞❡ ❝✐❝❧♦✐❞❡ ❝♦♠♦ ❜❛t❡♥t❡s ❬✶❪✳ ✳ ✳ ✳ ✷✷ ✷✳✹ P❛r❛♠❡tr✐③❛çã♦ ❞❛ ❝✐❝❧♦✐❞❡ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✺ t= 2π
3 ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✻ t=π ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷✳✼ t= 3π
2 ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✽ t= 2π ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹
✷✳✾ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❝✐❝❧♦✐❞❡ ❬✷❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✸✳✶ ❊s❝♦❧❤❛ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❇r❛q✉✐stó❝r♦♥❛✳ ✷✺ ✸✳✷ P❛râ♠❡tr♦s ♥♦ ♣r♦❜❧❡♠❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❋❡✐①❡ ❞❡ ❝✐❝❧♦✐❞❡s ❬✸❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✹ ❊sq✉❡♠❛ ♣❛r❛ ♦ ❢❡♥ô♠❡♥♦ ❞❛ r❡❢r❛çã♦ ❞❡ ✉♠ r❛✐♦ ❞❡ ❧✉③✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✳✺ ▼❡✐♦ ó♣t✐❝♦ ❡ ❛ tr❛❥❡tór✐❛ ❞❡s❝r✐t❛ ♣♦r ✉♠ r❛✐♦ ❞❡ ❧✉③ ♣❛rt✐♥❞♦ ❞❡ ❆ ❡
❝❤❡❣❛♥❞♦ ❡♠ ❇ ❬✹❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✻ ❋♦t♦❣r❛✜❛ q✉❡ ♠♦str❛ ❛ r❡✢❡①ã♦ ❡ ❛ r❡❢r❛çã♦ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❧✉③ ✐♥❝✐❞❡♥t❡
❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ á❣✉❛ ❤♦r✐③♦♥t❛❧ ❬✺❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✼ ❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✸✳✻ ❬✺❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✸✳✽ ➶♥❣✉❧♦ q✉❡ ♦ ❝❛♠✐♥❤♦ ❞❡s❝r✐t♦ ♣❡❧♦ r❛✐♦ ❞❡ ❧✉③ ❡ ❛ ✈❡rt✐❝❛❧ ❬✹❪✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✾ P♦ssí✈❡✐s tr❛❥❡tór✐❛s ❞❡ ✉♠ s❛❧✈❛✲✈✐❞❛s ♣❛r❛ s♦❝♦rr❡r ✉♠❛ ✈ít✐♠❛✳ ✳ ✳ ✳ ✸✼ ✹✳✶ ❊s❜♦ç♦ ❞❡ ✉♠❛ P✐st❛ ❍❛❧❢ P✐♣❡ ❬✻❪✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✹✳✷ ❆s ❝✉r✈❛s ♥♦s ✐♥t❡r✈❛❧♦s[0; 0,8π]❡[0,8π+4; 1,6π+4]r❡♣r❡s❡♥t❛♠ ❛r❝♦s
❙✉♠ár✐♦
✶ ■♥tr♦❞✉çã♦ ✶✼
✷ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛ ✶✾
✷✳✶ ❯♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✷ ❆ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❝✐❝❧♦✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷
✸ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦ ✷✺
✸✳✶ ❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸✳✷ ❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ r❡❢r❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✸ ❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ s❛❧✈❛♠❡♥t♦ ♥❛ ♣r❛✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼
✹ Pr♦♣♦st❛ ❞✐❞át✐❝❛ ✸✾
✹✳✶ ❆❜♦r❞❛❣❡♠ ❛tr❛✈és ❞❡ ❛♣❧✐❝❛çã♦✿ r❛♠♣❛ ❞❡ s❦❛t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✷ ❆❜♦r❞❛❣❡♠ ❛tr❛✈és ❞♦ ✉s♦ ❞❡ ❢❡rr❛♠❡♥t❛ t❡❝♥♦❧ó❣✐❝❛✿ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ✹✷ ✹✳✸ ❆❜♦r❞❛❣❡♠ ❡①♣❡r✐♠❡♥t❛❧✿ ❝♦♥str✉çã♦ ❞❡ r❛♠♣❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
✺ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✹✾
❘❡❢❡rê♥❝✐❛s ✺✶
✶ ■♥tr♦❞✉çã♦
❙❡ ♣❡r❣✉♥t❛r♠♦s ❛ ❛❧❣✉é♠ q✉❛❧ é ♦ ❝❛♠✐♥❤♦ ♠❛✐s rá♣✐❞♦ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞❡s♥✐✲ ✈❡❧❛❞♦s✱ ♣♦ss✐✈❡❧♠❡♥t❡ r❡s♣♦♥❞❡rá q✉❡ é ❛ r❡t❛ ❛♦ ✐♠❛❣✐♥❛r q✉❡ ♦ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦ é s❡♠♣r❡✱ t❛♠❜é♠✱ ♦ ♠❛✐s rá♣✐❞♦✳ ❖❜s❡r✈❛r q✉❡ ❡①✐st❡ ✉♠ ❝❛♠✐♥❤♦ ♠❛✐♦r q✉❡✱ ♣♦ré♠✱ t♦r♥❛ ♦ t❡♠♣♦ ❞❡ ♣❡r❝✉rs♦ ♠❡♥♦r ♣♦❞❡ ❝❛✉s❛r ❡str❛♥❤❡③❛✳
❆♣❡s❛r ❞❡ s❡ tr❛t❛r ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ❛♥t✐❣♦✱ ❞♦ ✜♥❛❧ ❞♦ sé❝✉❧♦ ❳❱■■✱ ❡ ❜❡♠ ❝♦♥❤❡✲ ❝✐❞♦ ♥♦ ♠❡✐♦ ❛❝❛❞ê♠✐❝♦✱ ❛ ❝♦♥st❛t❛çã♦ ❡①♣❡r✐♠❡♥t❛❧ ❛✐♥❞❛ s✉r♣r❡❡♥❞❡ ♣❡ss♦❛s q✉❡ ❛ ✈❡❡♠ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③✳
❖ ♣r♦❜❧❡♠❛ ❞❛ ❇r❛q✉✐stó❝r♦♥❛ é ✉♠❛ q✉❡stã♦ ♠❡❝â♥✐❝♦✲❣❡♦♠étr✐❝❛ s♦❜r❡ ❛ ❝✉r✈❛ ❞❡ ❞❡s❝✐❞❛ ♠❛✐s rá♣✐❞❛✳ ❆ ♣❛❧❛✈r❛ ❜r❛q✉✐stó❝r♦♥❛ ❞❡r✐✈❛ ❞❛s ♣❛❧❛✈r❛s ❣r❡❣❛s ❇r❛❝❤✐s✲ t♦s✱ q✉❡ s✐❣♥✐✜❝❛ ♠❡♥♦r✱ ❡ ❈❤r♦♥♦s✱ q✉❡ s✐❣♥✐✜❝❛ t❡♠♣♦✳ ❈♦♥s✐st❡ ❡♠ ❞❡t❡r♠✐♥❛r ❛ ❝✉r✈❛ q✉❡ ✉♥❡ ❞♦✐s ♣♦♥t♦s ❞❛❞♦s ❛ ❞✐❢❡r❡♥t❡s ❞✐stâ♥❝✐❛s ♥❛ ❤♦r✐③♦♥t❛❧✱ ❡ ♥ã♦ ♥❛ ♠❡s♠❛ ❧✐♥❤❛ ✈❡rt✐❝❛❧✱ ♣❡❧❛ q✉❛❧ ✉♠❛ ♣❛rtí❝✉❧❛ ♠ó✈❡❧✱ s♦❜ ♦ s❡✉ ♣ró♣r✐♦ ♣❡s♦ ❡ ❝♦♠❡ç❛♥❞♦ ♦ s❡✉ ♠♦✈✐♠❡♥t♦ ♥♦ ♣♦♥t♦ s✉♣❡r✐♦r✱ ❞❡s❝❡ ♦ ♠❛✐s r❛♣✐❞❛♠❡♥t❡ ♣♦ssí✈❡❧ ❛té ❛♦ ♣♦♥t♦ ✐♥❢❡r✐♦r✳
❊st❡ tr❛❜❛❧❤♦ ❡stá ♦r❣❛♥✐③❛❞♦ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ♥♦ ❝❛♣ít✉❧♦ ✷ é ❛♣r❡s❡♥t❛❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛✱ ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞❡ s✉❛ ♣r♦♣♦s✐çã♦✱ ❝✉r✐♦s✐❞❛❞❡s s♦❜r❡ ❛ ❝✉r✈❛ q✉❡ é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❡ s✉❛ ♣❛r❛♠❡tr✐③❛çã♦❀ ♦ ❝❛♣ít✉❧♦ ✸ ✐♥❝❧✉✐ ❞✐❢❡✲ r❡♥t❡s s♦❧✉çõ❡s✱ ❛ s❛❜❡r✱ ❛tr❛✈és ❞❛ t❡♦r✐❛ ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ❡ ❛tr❛✈és ❞❡ ❝♦♥❝❡✐t♦s ❞❛ ❣❡♦♠❡tr✐❛ ❡ ❢ís✐❝❛✱ ❡♥✈♦❧✈❡♥❞♦ s✐t✉❛çõ❡s ❝♦♠ ❝♦♥❞✐çõ❡s ❛♥á❧♦❣❛s às ❞❛ ❜r❛q✉✐stó✲ ❝r♦♥❛❀ ♣♦r ✜♠✱ ♦ ❝❛♣ít✉❧♦ ✹ tr❛③ ✉♠❛ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ ❡♥✈♦❧✈❡♥❞♦ ❡st❡ ♣r♦❜❧❡♠❛ ❡ é ❛❞❡q✉❛❞❛ ❛ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
✷ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛
✷✳✶ ❯♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛
❖ ❞❡s❛✜♦ ❞❡ ❡♥❝♦♥tr❛r ❛ ❜r❛q✉✐stó❝r♦♥❛ ❢♦✐ ♣r♦♣♦st♦ ❡♠ ❥✉♥❤♦ ❞❡ ✶✻✾✻ ♣♦r ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ✭✶✻✻✼ ✕ ✶✼✹✽✮ ♥❛ r❡✈✐st❛ ❆❝t❛ ❊r✉❞✐t♦r✉♠ ❞❡ ▲❡✐♣③✐❣ ❡ ❛♣r❡s❡♥t❛✲s❡ tr❛❞✉③✐❞♦ ❞♦ ❧❛t✐♠ ❡♠ ❬✶✵❪✿
❉❛❞♦s ❞♦✐s ♣♦♥t♦s ❆ ❡ ❇ ❡♠ ✉♠ ♣❧❛♥♦ ✈❡rt✐❝❛❧✱ ❢❛③❡r ❝♦rr❡s♣♦♥❞❡r ❛ ✉♠❛ ♣❛rtí❝✉❧❛ ♠ó✈❡❧ ▼ ❛ tr❛❥❡tór✐❛ ❆▼❇ ♣❡❧❛ q✉❛❧ ❛ ♣❛rtí❝✉❧❛✱ ❞❡s❝❡♥❞♦ s♦❜r❡ ♦ s❡✉ ♣ró♣r✐♦ ♣❡s♦✱ ♣❛ss❛ ❞♦ ♣♦♥t♦ ❆ ♣❛r❛ ♦ ♣♦♥t♦ ❇ ♥♦ ❡s♣❛ç♦ ❞❡ t❡♠♣♦ ♠❛✐s ❝✉rt♦✳ ❬✶✵❪
❈♦♥✈✐❞♦✉ ♦s ♠❛t❡♠át✐❝♦s ❞❛ é♣♦❝❛ ❛ r❡s♦❧✈❡r❡♠ ❡ ❛✐♥❞❛ ❛✜r♠♦✉ q✉❡ ❡♠❜♦r❛ ♦ s❡❣♠❡♥t♦ ❆❇ ❢♦ss❡✱ ❞❡ ❢❛t♦✱ ♦ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦ ❡♥tr❡ ♦s ♣♦♥t♦s ❆ ❡ ❇✱ ♥♦ ❡♥t❛♥t♦✱ ♥ã♦ s❡r✐❛ ❡ss❡ ♦ ❝❛♠✐♥❤♦ ♣❡r❝♦rr✐❞♦ ♥♦ ♠❡♥♦r t❡♠♣♦✳ ❆✜r♠❛ ❛✐♥❞❛ q✉❡ t❛❧ ❝✉r✈❛ é ❜❡♠ ❝♦♥❤❡❝✐❞❛ ❞♦s ❣❡ô♠❡tr❛s ❡ ❞❡st❛ ❢♦r♠❛ ❡①♣õ❡ q✉❡ ❥á t✐♥❤❛ ❡♥❝♦♥tr❛❞♦ ❛ s♦❧✉çã♦✳
P♦st❡r✐♦r♠❡♥t❡✱ ❡♠ ❥❛♥❡✐r♦ ❞❡ ✶✻✾✼✱ ❏♦❤❛♥♥ ❢❛③ ✉♠❛ ♥♦✈❛ ♣✉❜❧✐❝❛çã♦ ✭●r♦❡♥✐♥❣❡♥✮ r❡❡s❝r❡✈❡♥❞♦ ♦ ♣r♦❜❧❡♠❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
❉❡t❡r♠✐♥❛r ❛ ❝✉r✈❛ q✉❡ ❥✉♥t❛ ❞♦✐s ♣♦♥t♦s ❞❛❞♦s✱ ❛ ❞✐❢❡r❡♥t❡s ❞✐stâ♥❝✐❛s ♥❛ ❤♦r✐③♦♥t❛❧ ❡ ♥ã♦ ♥❛ ♠❡s♠❛ ❧✐♥❤❛ ✈❡rt✐❝❛❧✱ ♣❡❧❛ q✉❛❧ ✉♠❛ ♣❛rtí❝✉❧❛ ♠ó✈❡❧✱ s♦❜ ♦ s❡✉ ♣ró♣r✐♦ ♣❡s♦✱ ❡ ❝♦♠❡ç❛♥❞♦ ♦ s❡✉ ♠♦✈✐♠❡♥t♦ ♥♦ ♣♦♥t♦ s✉♣❡r✐♦r✱ ❞❡s❝❡ ♠❛✐s r❛♣✐❞❛♠❡♥t❡ ❛té ❛♦ ♣♦♥t♦ ✐♥❢❡r✐♦r✳ ❬✶✵❪
❆❧é♠ ❞✐ss♦✱ ♣r♦❧♦♥❣❛ ♦ ♣r❛③♦ ♣❛r❛ q✉❡ ❛s s♦❧✉çõ❡s ❢♦ss❡♠ ❛♣r❡s❡♥t❛❞❛s✱ ❛t❡♥❞❡♥❞♦ ❛ ✉♠ ♣❡❞✐❞♦ ❞❡ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐③ ✭✶✻✹✻ ✕ ✶✼✶✻✮✱ ú♥✐❝♦ ❛ ❡s❝r❡✈❡r✲❧❤❡ ❛✜r✲ ♠❛♥❞♦ t❡r r❡s♦❧✈✐❞♦ ♦ ♣r♦❜❧❡♠❛✳ ❉❡st❛ ❢♦r♠❛✱ ❛ q✉❡stã♦ ♣♦❞❡r✐❛ s❡r t♦r♥❛❞❛ ♣ú❜❧✐❝❛ ♥❛ ❋r❛♥ç❛ ❡ ■tá❧✐❛ ❡✱ ❛✐♥❞❛✱ ♣❛r❛ ❛q✉❡❧❡s q✉❡ ♥ã♦ t✐✈❡r❛♠ ❛❝❡ss♦ à ❆❝t❛✳
❈♦♠ r❡❧❛çã♦ ❛♦ ♠ér✐t♦ ❡♠ r❡s♦❧✈❡r t❛❧ q✉❡stã♦✱ ❏♦❤❛♥♥ ❛✜r♠❛ q✉❡✿
❉✐✜❝✐❧♠❡♥t❡ ❤á ❛❧❣♦ q✉❡ ♠❛✐s ❣r❛♥❞✐♦s❛♠❡♥t❡ ❡st✐♠✉❧❡ ❡s♣ír✐t♦s ♥♦❜r❡s ❡ ❡♥❣❡♥❤♦s♦s ♣❛r❛ tr❛❜❛❧❤♦s q✉❡ ❝♦♥❞✉③❛♠ ❛♦ ❛✉♠❡♥t♦ ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦
✷✵ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛
❞♦ q✉❡ ♣r♦♣♦r ♣r♦❜❧❡♠❛s s✐♠✉❧t❛♥❡❛♠❡♥t❡ ❞✐❢í❝❡✐s ❡ út❡✐s✱ ❡ q✉❡ ❛tr❛✈és ❞❛ s♦❧✉çã♦ ❞♦s ♠❡s♠♦s✱ ❡ ♣♦r ♥❡♥❤✉♠ ♦✉tr♦ ♠♦❞♦✱ ❧❤❡s ♣❡r♠✐t❛♠ ❛t✐♥❣✐r ❛ ❢❛♠❛ ❡ ❝♦♥str✉✐r ♣❛r❛ s✐ ♣ró♣r✐♦s ♠♦♥✉♠❡♥t♦s ❡t❡r♥♦s ♣❛r❛ ❛ ♣♦st❡r✐❞❛❞❡❀ ❬✳✳✳❪ ♦❢❡r❡❝❡♠♦s àq✉❡❧❡ ❤♦♠❡♠ ❞❡ ♥♦❜r❡ s❛♥❣✉❡✱ ✉♠ ♣rê♠✐♦✱ ❝♦♠♣♦st♦ ♣♦r ❤♦♥r❛s✱ ❡❧♦❣✐♦s ❡ ❛♣❧❛✉s♦s❀ ❛ss✐♠ ❝♦r♦❛r❡♠♦s✱ ❤♦♥r❛r❡♠♦s ❡ ❡①❛❧t❛r❡♠♦s✱ ♣ú❜❧✐❝❛ ❡ ♣r✐✈❛❞❛♠❡♥t❡✱ ♣♦r ❝❛rt❛ ❡ ♣♦r ♣❛❧❛✈r❛✱ ❛ ♣❡rs♣✐❝á❝✐❛ ❞♦ ♥♦ss♦ ❣r❛♥❞❡ ❆♣♦❧❧♦✳ ❬✶✵❪
❖ t❡①t♦ ❞❛ s❡❣✉♥❞❛ ♣✉❜❧✐❝❛çã♦ ❞❡ ❇❡r♥♦✉❧❧✐ ❝✉r✐♦s❛♠❡♥t❡ ❝✐t❛ ❇❧❛✐s❡ P❛s❝❛❧ ✭✶✻✷✸ ✕ ✶✻✻✷✮✱ q✉❡ ❢♦✐ ✉♠ ❣r❛♥❞❡ ❡st✉❞✐♦s♦ ❞❛ ❝✐❝❧♦✐❞❡✱ ❡ P✐❡rr❡ ❞❡ ❋❡r♠❛t ✭✶✻✵✶ ✕ ✶✻✻✺✮✱ q✉❡ ❞á ♥♦♠❡ ❛♦ ♣r✐♥❝í♣✐♦ ❞♦ t❡♠♣♦ ♠í♥✐♠♦✶✳ ❆✐♥❞❛ ❞❡✐①❛ ❡①♣r❡ss♦ q✉❡ ✉t✐❧✐③❛ ❛ ❤✐♣ót❡s❡ ❞❡ ●❛❧✐❧❡✉✷ ❡♠ s✉❛ s♦❧✉çã♦ ❡ q✉❡ ❞❡s❝♦♥s✐❞❡r❛ ❛ ❢r✐❝çã♦✱ ❧♦❣♦ ✧✈❡❧♦❝✐❞❛❞❡s ❛❞q✉✐r✐❞❛s ♣♦r ✉♠ ❝♦r♣♦ ♣❡s❛❞♦ ❡♠ q✉❡❞❛ sã♦ ♣r♦♣♦r❝✐♦♥❛✐s à r❛✐③ q✉❛❞r❛❞❛ ❞❛ ❛❧t✉r❛ ♣❡r❝♦rr✐❞❛ ❡♠ q✉❡❞❛✧❬✶✵❪✳
❊♠ ♠❛✐♦ ❞❡ ✶✻✾✼✱ ❛ ❆❝t❛ ❊r✉❞✐t♦r✉♠ ♣✉❜❧✐❝♦✉ q✉❛tr♦ s♦❧✉çõ❡s ❝✉❥♦s ❛✉t♦r❡s ❡r❛♠ ▲❡✐❜♥✐③✱ ♦ ♣ró♣r✐♦ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✱ s❡✉ ✐r♠ã♦ ♠❛✐s ✈❡❧❤♦ ❏❛❝♦❜ ❇❡r♥♦✉❧❧✐ ✭✶✻✺✹ ✕ ✶✼✵✺✮ ❡ ✉♠❛ r❡s♦❧✉çã♦ ❛♥ô♥✐♠❛ ❝✉❥❛ ❛✉t♦r✐❛ ❢♦✐ r❡❝♦♥❤❡❝✐❞❛ ❝♦♠♦ s❡♥❞♦ ❞❡ ■s❛❛❝ ◆❡✇✲ t♦♥ ✭✶✻✹✸ ✕ ✶✼✷✼✮✳ ✧❖ ▲❡ã♦ s❡ r❡❝♦♥❤❡❝❡ ♣❡❧❛s ♠❛r❝❛s ❞❡ s✉❛s ❣❛rr❛s✦✧é ✉♠ ❝♦♠❡♥tár✐♦ ❛tr✐❜✉í❞♦ ❛ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ r❡❢❡r✐♥❞♦✲s❡ ❛ ◆❡✇t♦♥✱ ❛ ♣r♦♣ós✐t♦ ❞❛ s♦❧✉çã♦ ❛♥ô♥✐♠❛ ❛♣r❡s❡♥t❛❞❛✳ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ é ❝♦♥s✐❞❡r❛❞♦ ♦ ♣r✐♠❡✐r♦ ❛ r❡s♦❧✈❡r ❛ q✉❡stã♦✿ ♠♦str♦✉ q✉❡ ❛ s♦❧✉çã♦ é ✉♠❛ ❝✐❝❧♦✐❞❡✳
❆ ❝✐❝❧♦✐❞❡ ❤❛✈✐❛ s✐❞♦ ❛♠♣❧❛♠❡♥t❡ ❡st✉❞❛❞❛ ❛♥t❡r✐♦r♠❡♥t❡✱ ✐♥❝❧✉s✐✈❡ ♣♦r ●❛❧✐❧❡✉ ●❛❧✐❧❡✐ ✭✶✺✻✹ ✕ ✶✻✹✸✮ ❡ ❈❤r✐st✐❛❛♥ ❍✉②❣❡♥s ✭✶✻✷✾ ✕ ✶✻✾✺✮✳ ❊st❡ ú❧t✐♠♦ ❡♥❝♦♥tr♦✉ ❛♣❧✐✲ ❝❛çã♦ ♥❛ ❝♦♥str✉çã♦ ❞❡ r❡❧ó❣✐♦s ✉t✐❧✐③❛♥❞♦ ♦ ❢❛t♦ ❞❛ ❝✉r✈❛ s❡r ✐só❝r♦♥❛ ✭t❛✉tó❝r♦♥❛✮✱ ♦✉ s❡❥❛✱ ❢❛③❡r ❝♦♠ q✉❡ ✉♠ ❝♦r♣♦ ❡♠ ❝♦♥❞✐çõ❡s ✐❞❡❛✐s✱ s✉❥❡✐t♦ ❛♣❡♥❛s à ❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ❡ r❡str✐t♦ ❛♦ ♣❡r❝✉rs♦ ❞❛ ❝✉r✈❛✱ ❛t✐♥❥❛ ♦ ♣♦♥t♦ ❜❛✐①♦ ❛♣ós ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ q✉❡ ✐♥❞❡♣❡♥❞❛ ❞❛ ❛❧t✉r❛ ❞❛ q✉❛❧ ❢♦✐ s♦❧t♦✱ ❝♦♥❢♦r♠❡ ❛♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ✷✳✶✳
✶❖ ♣r✐♥❝í♣✐♦ ❞❡ ❋❡r♠❛t✱ ♦✉ ♣r✐♥❝í♣✐♦ ❞♦ t❡♠♣♦ ♠í♥✐♠♦✱ ❡♥✉♥❝✐❛❞♦ ❡♠ ✶✻✺✼✱ ❛✜r♠❛ q✉❡ ❛ ❧✉③✱ ❛♦
♣r♦♣❛❣❛r✲s❡ ❞❡ ✉♠ ♣♦♥t♦ ♣❛r❛ ♦✉tr♦✱ ❡s❝♦❧❤❡ ♦ ❝❛♠✐♥❤♦ ♣❛r❛ ♦ q✉❛❧ ♦ t❡♠♣♦ ❞❡ ♣❡r❝✉rs♦ é ♠í♥✐♠♦ ♠❡s♠♦ q✉❡✱ ♣❛r❛ t❛❧✱ s❡ t❡♥❤❛ ❞❡ ❞❡s✈✐❛r r❡❧❛t✐✈❛♠❡♥t❡ ❛♦ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦✳
❯♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ✷✶
❋✐❣✉r❛ ✷✳✶✿ ❈✉r✈❛ ✐s♦❝rô♥✐❝❛ ❬✶❪✳
❆ ❝✉r✈❛ q✉❡ r❡s♣♦♥❞❡ ♦ ♣r♦❜❧❡♠❛ ❝♦❧♦❝❛❞♦ é r✐❝❛ ❡♠ ♣r♦♣r✐❡❞❛❞❡s ❝✉r✐♦s❛s ❡ ♣♦r ❣❡r❛r t❛♥t❛s ❝♦♥tr♦✈érs✐❛s ❢♦✐ ❝❤❛♠❛❞❛ ✧❛ ❍❡❧❡♥❛ ❞❛ ❣❡♦♠❡tr✐❛✧♦✉ ✧♦ ♣♦♠♦ ❞❛ ❞✐s✲ ❝ór❞✐❛✧✳ ❆❧❣✉♠❛s ❞❡ss❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ sã♦ ❡♥❝♦♥tr❛❞❛s ❡ ❞❡♠♦♥str❛❞❛s ❡♠ ❬✶❪✱ ♣á❣✐♥❛s ✶✻✶ ✕ ✶✽✷✱ ❡stã♦ ❧✐st❛❞❛s ❛ s❡❣✉✐r✿
• ❛ ár❡❛ ❞❡❧✐♠✐t❛❞❛ ♣♦r ✉♠ ❛r❝♦ ❞❡ ❝✐❝❧♦✐❞❡ ❡ ♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s é ✐❣✉❛❧ ❛ três
✈❡③❡s ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ q✉❡ ❧❤❡ ❞á ♦r✐❣❡♠ ✭✜❣✉r❛ ✷✳✷✮❀
❋✐❣✉r❛ ✷✳✷✿ ➪r❡❛ ❞❡❧✐♠✐t❛❞❛ ♣♦r ✉♠ ❛r❝♦ ❞❡ ❝✐❝❧♦✐❞❡ ❬✶❪✳
• ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠ ❛r❝♦ ❞❡ ❝✐❝❧♦✐❞❡ é q✉❛tr♦ ✈❡③❡s ♦ ❞✐â♠❡tr♦ ❞♦ ❝ír❝✉❧♦ r♦❧❛♥t❡
q✉❡ ❛ ❣❡r♦✉❀
• s❡ ♣❡♥❞✉r❛r ✉♠ ♣ê♥❞✉❧♦ ❡ ❝♦❧♦❝❛r ❞♦✐s ❛r❝♦s ❞❡ ✉♠❛ ❝✐❝❧♦✐❞❡ ❝♦♠♦ ❜❛t❡♥t❡s✱ ❡st❡
❞❡s❝r❡✈❡rá ✉♠❛ ❝✐❝❧♦✐❞❡ ✐❣✉❛❧ à q✉❡ ❣❡r♦✉ ♦s ❛r❝♦s ✭✜❣✉r❛ ✷✳✸✮❀
• q✉❛♥❞♦ ♦ ♣❡s♦ ❞❡ ✉♠ ♣ê♥❞✉❧♦ ♠♦✈❡✲s❡ ❛♦ ❧♦♥❣♦ ❞❡ ✉♠❛ ❝✐❝❧♦✐❞❡✱ ❛✐♥❞❛ q✉❡
✷✷ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛
❋✐❣✉r❛ ✷✳✸✿ ❈✉r✈❛ ❣❡r❛❞❛ ♣♦r ♣ê♥❞✉❧♦ ❝♦♠ ❛r❝♦s ❞❡ ❝✐❝❧♦✐❞❡ ❝♦♠♦ ❜❛t❡♥t❡s ❬✶❪✳
✷✳✷ ❆ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❝✐❝❧♦✐❞❡
❈♦♥❢♦r♠❡ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✷❪✱ s❡❥❛♠ C ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦ r✱ s ✉♠❛ r❡t❛ ❡ P ✉♠
♣♦♥t♦ ❞❡ C✳ ❉❡♥♦♠✐♥❛♠♦s ❝✐❝❧♦✐❞❡ ❛ ❝✉r✈❛ ❞❡s❝r✐t❛ ♣❡❧♦ ♣♦♥t♦ P q✉❛♥❞♦ C r♦❧❛ s♦❜r❡
❛ r❡t❛ s✱ s❡♠ ❞❡s❧✐③❛r✳
❉❡✜♥✐çã♦ ✷✳✶✳ ❉❡♥♦♠✐♥❛♠♦s ❝✐❝❧♦✐❞❡ ❛ ❝✉r✈❛ ❞❡✜♥✐❞❛ ♣♦r ✉♠ ♣♦♥t♦ ❞❡ ✉♠❛ ❝✐r❝✉♥✲ ❢❡rê♥❝✐❛ q✉❡ r♦❧❛ s❡♠ ❞❡s❧✐③❛r s♦❜r❡ ✉♠❛ r❡t❛✳ ❯♠❛ ❝✐❝❧♦✐❞❡ ✐♥✐❝✐❛❞❛ ♥❛ ♦r✐❣❡♠ ❞❡ ✉♠ s✐st❡♠❛ ❞❡ ❡✐①♦s✱ ❝r✐❛❞♦ ♣♦r ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ r✱ ❝♦♥s✐st❡ ♥♦s ♣♦♥t♦s ✭①✱②✮
t❛✐s q✉❡✿
(
x=r(t−sen(t))
y=r(1−cos(t)) ❡♠ q✉❡ t é ✉♠ ♣❛râ♠❡tr♦ r❡❛❧✳
❱❛♠♦s ❛❞♠✐t✐r q✉❡ ❛ r❡t❛ s é ♦ ❡✐①♦ OX✱ ♦ ❝ír❝✉❧♦ C ✐♥✐❝✐❛ ♦ ♠♦✈✐♠❡♥t♦ ❡st❛♥❞♦
s❡✉ ❝❡♥tr♦ ♥♦ ♣♦♥t♦ (0, r)❡ q✉❡ ♦ ♣♦♥t♦ P ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ♦r✐❣❡♠ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r✲
❞❡♥❛❞❛s ♥♦ ✐♥í❝✐♦ ❞♦ ♠♦✈✐♠❡♥t♦✳
❚r❛❝❡♠♦s ❞♦✐s ❝ír❝✉❧♦s✿ C1✱ r❡♣r❡s❡♥t❛♥❞♦ C ❡♠ s✉❛ ♣♦s✐çã♦ ✐♥✐❝✐❛❧✱ ❡ C2✱ r❡♣r❡s❡♥✲
t❛♥❞♦ C ❛♣ós t❡r r♦❧❛❞♦ ❛❧❣✉♥s ✐♥st❛♥t❡s✳
❙❡❥❛♠O1 ❡O2 ♦s ❝❡♥tr♦s ❞❡C1 ❡C2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀P = (x, y)♦ ♣♦♥t♦ ❞❛ ❝✐❝❧♦✐❞❡
❡♠ C2❀ ❆ ♦ ♣♦♥t♦ ❡♠ q✉❡ C2 t♦❝❛ ♦ ❡✐①♦ OX❀ Q = (x,0) ❡ T = (0, y) ❛s ♣r♦❥❡çõ❡s
♦rt♦❣♦♥❛✐s ❞❡ P s♦❜r❡ OX ❡ OY ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀M ❡N ❛s ♣r♦❥❡çõ❡s ♦rt♦❣♦♥❛✐s ❞❡ P s♦❜r❡ O2O1 ❡ O2A✱ r❡s♣❡❝t✐✈❛♠❡♥t❡❀ t ❛ ♠❡❞✐❞❛ ❞♦ â♥❣✉❧♦ q✉❡ O2P ❢❛③ ❝♦♠ O2A✱
❆ ♣❛r❛♠❡tr✐③❛çã♦ ❞❛ ❝✐❝❧♦✐❞❡ ✷✸
❋✐❣✉r❛ ✷✳✹✿ P❛r❛♠❡tr✐③❛çã♦ ❞❛ ❝✐❝❧♦✐❞❡ ❬✷❪✳
◆♦t❡ q✉❡ ♦ s❡❣♠❡♥t♦ OA t❡♠ ♦ ♠❡s♠♦ ❝♦♠♣r✐♠❡♥t♦ q✉❡ ♦ ❛r❝♦ ❞❡ A ❛ P s♦❜r❡
♦ ❝ír❝✉❧♦ C2✱ q✉❡ ❝♦♥s✐st❡ ❞♦s ♣♦♥t♦s ❞❡ C q✉❡ ❥á ✜③❡r❛♠ ❝♦♥t❛t♦ ❝♦♠ ❛ r❡t❛ s✳ ◆♦t❡
❛✐♥❞❛ q✉❡
sen(t) = |02M|
r
❡ q✉❡
cos(t) = |02N|
r
r❡❧❛çõ❡s ❢❛❝✐❧♠❡♥t❡ ♦❜s❡r✈❛❞❛s ♥♦s tr✐â♥❣✉❧♦s M P O2 ❡P O2N✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳
❈♦♠♦ t é ❛ ♠❡❞✐❞❛ ❞❡ AO\2P✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❛r❝♦ ❞❡ C2 ❞❡ A ❛ P q✉❡ ❥á ❢❡③
❝♦♥t❛t♦ ❝♦♠ s é rt✳ ▲♦❣♦✱|AO|=rt✳
❆ss✐♠✱
x=|OQ|=|OA| ± |QA|=|OA| ± |O2M|=rt±r|sen(t)|
y=|OT|=|OO1| ± |T O1|=r± |O2N|=rt±r|cos(t)|
♦♥❞❡ ♦ s✐♥❛❧ ❞❡♣❡♥❞❡ ❞❛ ♣♦s✐çã♦ ❞❡ Q♥❛ s❡♠✐rr❡t❛−→OA ❡ ❞❛ ♣♦s✐çã♦ ❞❡T ♥❛ s❡♠✐rr❡t❛
−−→
OO1✱ q✉❡✱ ♣♦r s✉❛ ✈❡③✱ ✈❛r✐❛♠ ❝♦♠ ❛ ♠❡❞✐❞❛t ❞♦ â♥❣✉❧♦ AO\2P✳
❆♥❛❧✐s❛♥❞♦ ♦ s✐♥❛❧ ❞❡sen(t)❡cos(t)♥♦s ✐♥t❡r✈❛❧♦sh0,π
2 i
,hπ
2, π i
,
π,3π
2
,
3π
2 ,2π
✱ ♦❜t❡♠♦s ❛s s❡❣✉✐♥t❡s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❞❛ ❝✐❝❧♦✐❞❡✿
(
x=r(t−sen(t))
✷✹ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛
❱❡❥❛ ❝♦♠♦ é ❢❡✐t♦ ♦ ♠♦✈✐♠❡♥t♦ ♥❛ s❡q✉ê♥❝✐❛ ❞❡ ✜❣✉r❛s ✷✳✺✱ ✷✳✻✱ ✷✳✼✱ ✷✳✽ ❡ ✷✳✾✳
❋✐❣✉r❛ ✷✳✺✿ t= 2π
3 ❬✷❪✳ ❋✐❣✉r❛ ✷✳✻✿ t=π ❬✷❪✳
❋✐❣✉r❛ ✷✳✼✿ t= 3π
2 ❬✷❪✳ ❋✐❣✉r❛ ✷✳✽✿ t = 2π ❬✷❪✳
✸ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦
✸✳✶ ❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧
◆♦ ❝♦♥t❡①t♦ ❞✐s♣✉t❛❞♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❇r❛q✉✐stó❝r♦♥❛ ❢♦✐ ♣r♦❞✉③✐❞♦ ♠❛t❡r✐❛❧ s✐❣✲ ♥✐✜❝❛t✐✈♦ ♣❛r❛ ❡①♣❧♦r❛r ✉♠❛ ♥♦✈❛ ár❡❛ ♥❛ ♠❛t❡♠át✐❝❛✿ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧✳ ❊st❛ t❡♦r✐❛ é ✉t✐❧✐③❛❞❛ ♥❛ r❡s♦❧✉çã♦ ❛♣r❡s❡♥t❛❞❛ ♥❡st❛ s❡çã♦✳
❋✐❣✉r❛ ✸✳✶✿ ❊s❝♦❧❤❛ ❞♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❇r❛q✉✐stó❝r♦♥❛✳ ❙❡❣✉♥❞♦ ♦ ♣r✐♥❝í♣✐♦ ❣❡r❛❧ ❞❛ ❝♦♥s❡r✈❛çã♦ ❞❡ ❡♥❡r❣✐❛✱ ❛ ❡♥❡r❣✐❛ t♦t❛❧ ❞❡ ✉♠ s✐st❡♠❛ ✐s♦❧❛❞♦ é s❡♠♣r❡ ❝♦♥st❛♥t❡✱ ♦✉ s❡❥❛✱ ❛ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛ Emec ❞❡ ✉♠ s✐st❡♠❛ ♥♦ q✉❛❧
❛❣❡♠ s♦♠❡♥t❡ ❢♦rç❛s ❝♦♥s❡r✈❛t✐✈❛s ♥ã♦ s❡ ❛❧t❡r❛ ❝♦♠ ♦ ♣❛ss❛r ❞♦ t❡♠♣♦✳ ❚❡♠♦s ❡♥tã♦ q✉❡ ❛ s♦♠❛ ❞❛s ❡♥❡r❣✐❛s ❝✐♥ét✐❝❛ K ❡ ♣♦t❡♥❝✐❛❧U é ❝♦♥st❛♥t❡ ♣❛r❛ q✉❛❧q✉❡r ✐♥t❡r✈❛❧♦
❞❡ t❡♠♣♦✳
❙❡♥❞♦ ❛ss✐♠✱ ❞❛❞♦s q✉❛✐sq✉❡r ♣♦♥t♦s A ❡ B✱ Emec = KA +UA = KB +UB =
constante✳
❈♦♥s✐❞❡r❡♠♦s✱ t❛❧ ❝♦♠♦ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✶✶❪✱ q✉❡ ❛ ♣❛rtí❝✉❧❛ ❞❡ ♠❛ss❛m t❡♠ ✉♠
✷✻ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦
❞❡s❧♦❝❛♠❡♥t♦ ✈❡rt✐❝❛❧ y ❡ v é ♦ ♠ó❞✉❧♦ ❞❛ ✈❡❧♦❝✐❞❛❞❡ ❡♠ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ✐♥st❛♥t❡✳
◆♦t❡ q✉❡ ♦ ❡✐①♦ Y ❢♦✐ ♦r✐❡♥t❛❞♦ ♥♦ s❡♥t✐❞♦ ♦♣♦st♦ ❛♦ ✉s✉❛❧✳ ❚❛❧ ❡s❝♦❧❤❛ é ❝♦♥✈❡♥✐❡♥t❡✱
♣♦✐s✱ ♥❡ss❡ ❝❛s♦✱ ❛ ❢♦rç❛ ❡①❡r❝✐❞❛ ♣❡❧❛ ❣r❛✈✐❞❛❞❡ ✜❝❛ ♦r✐❡♥t❛❞❛ ♥♦ s❡♥t✐❞♦ ♣♦s✐t✐✈♦✳ ❆❧é♠ ❞✐ss♦✱ ♦ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s ❢♦✐ ❡s❝♦❧❤✐❞♦ ❞❡ ♠♦❞♦ q✉❡ ♦ ♣♦♥t♦ ✐♥✐❝✐❛❧ ✜q✉❡ ❧♦❝❛❧✐③❛❞♦ ♥❛ ♦r✐❣❡♠✳
❚❡♠♦s q✉❡ ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❣r❛✈✐t❛❝✐♦♥❛❧ ❛ss♦❝✐❛❞❛ ❛ ✉♠ s✐st❡♠❛ ♣❛rtí❝✉❧❛✲❚❡rr❛ ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❛ ❛❧t✉r❛ ✭♣♦s✐çã♦ ✈❡rt✐❝❛❧✮ ❞❛ ♣❛rtí❝✉❧❛ ❡♠ r❡❧❛çã♦ à ♣♦s✐çã♦ ❞❡ r❡✲ ❢❡rê♥❝✐❛✱ ❡ ♥ã♦ ❞❛ ♣♦s✐çã♦ ❤♦r✐③♦♥t❛❧✳ ❆ ✈❛r✐❛çã♦ ❞❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞♦ ♣♦♥t♦ ❞❡ r❡❢❡rê♥❝✐❛✱ ♠❛s ❛♣❡♥❛s ❞❛ ✈❛r✐❛çã♦ ❞❡ ❛❧t✉r❛✳ ◆♦ ♣♦♥t♦ A✱ ❛
❡s❢❡r❛ ♣♦ss✉✐ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ ❞❛❞❛ ♣♦r Ugravitacional = mgy ❡ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛ ♥✉❧❛✱
♣♦✐s ♣❛rt❡ ❞♦ r❡♣♦✉s♦✳ ❏á ♥♦ ♣♦♥t♦ B✱ ❛ ❡♥❡r❣✐❛ ♣♦t❡♥❝✐❛❧ é ♥✉❧❛ ❡ ❛ ❡♥❡r❣✐❛ ❝✐♥ét✐❝❛
é ❞❛❞❛ ♣♦r mv2 2 ✳
❉❡ss❛ ❢♦r♠❛✱ ♣♦❞❡♠♦s ♦❜t❡r ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ❡♠ q✉❡❞❛ ❧✐✈r❡✱ ❛ ♣❛rt✐r ❞♦ r❡♣♦✉s♦✱ ❛ q✉❛❧q✉❡r ♠♦♠❡♥t♦
mgy = mv
2
2
⇒v =p2gy
P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ✈❡❧♦❝✐❞❛❞❡✱
v = ds
dt
⇒ dsdt = 1
v
⇒ dsdtdxds = 1
v ds dx
❙✉❜st✐t✉✐♥❞♦ v ♣♦r √2gy ❡ ds dx ♣♦r
p
1 +y′2 t❡r❡♠♦s ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦ ♣❛r❛ ❛
❞❡r✐✈❛❞❛ ❞♦ t❡♠♣♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ❞❡s❧♦❝❛♠❡♥t♦ ❤♦r✐③♦♥t❛❧
dt dx = 1 √ 2gy p 1 +y′2
❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ✷✼
❆ss✐♠✱ ✐♥t❡❣r❛♥❞♦ ❝♦♠ r❡❧❛çã♦ ❛ x ♣♦❞❡✲s❡ ❞❡t❡r♠✐♥❛r ♦ t❡♠♣♦ t♦t❛❧ ♣❛r❛ s❡ ❞❡s✲
❧♦❝❛r ❞❡ A ♣❛r❛ B
t= Z x1
0
s
1 +y′2
2gy dx
P♦rt❛♥t♦ ♦ t❡♠♣♦ t♦t❛❧ ❣❛st♦ ♣❛r❛ t❛❧ ❞❡s❧♦❝❛♠❡♥t♦ é
t = √1
2g
Z x1 0
s 1 +y′2
y dx ✭✸✳✶✮
◗✉❡r❡♠♦s ❡♥❝♦♥tr❛r y = f(x) t❛❧ q✉❡ t s❡❥❛ ♠í♥✐♠♦✳ ❆✐♥❞❛ t❡♠♦s y(0) = 0 ❡
y(x1) = y1✱ ❝♦♠♦ ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛✳
❆ ❞❡s❝r✐çã♦ ♠❛t❡♠át✐❝❛ ❞❡ s✐st❡♠❛s r❡❧❛❝✐♦♥❛❞♦s ❛ ❧❡✐s ❞❛ ❢ís✐❝❛ ❝♦♠✉♠❡♥t❡ ❡♥✈♦❧✈❡ ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❡ ❡①✐st❡ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❢❡rr❛♠❡♥t❛s ❞❡s❡♥✈♦❧✈✐❞❛s ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ s♦❧✉❝✐♦♥❛r ♣r♦❜❧❡♠❛s ❞❡ss❡ t✐♣♦✳ ❖ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ é ✉♠❛ ❢❡rr❛♠❡♥t❛ ✉t✐❧✐③❛❞❛ ♣❛r❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛çã♦✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❇r❛q✉✐stó❝r♦♥❛✳
❆ ❞✐❢❡r❡♥ç❛ ❡♥tr❡ ♦s ❝á❧❝✉❧♦s ✈❛r✐❛❝✐♦♥❛❧ ❡ ❞✐❢❡r❡♥❝✐❛❧ é ❛ ♥❛t✉r❡③❛ ❞♦s r❡s♣❡❝t✐✈♦s ♦❜❥❡t♦s ❛ s❡r❡♠ ♠❛①✐♠✐③❛❞♦s ♦✉ ♠✐♥✐♠✐③❛❞♦s✿ ❡♥q✉❛♥t♦ ♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ♣r♦❝✉r❛ ♥ú♠❡r♦s q✉❡ t❡♥❤❛♠ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ♦t✐♠✐③❛r✱ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ♣r♦❝✉r❛ ❢✉♥çõ❡s ❝♦♠ t❛❧ ♣r♦♣r✐❡❞❛❞❡✳
❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ t❡♦r✐❛ ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ✉t✐❧✐③❛❞❛ ❛ s❡❣✉✐r ❡♥❝♦♥tr❛✲s❡ ❡♠ ❬✶✷❪✱ ♣á❣✐♥❛s ✶✸✲✶✼✱ ✷✶✲✷✷✳
❉♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧✱ s✉♣♦♥❞♦ q✉❡ ❡①✐st❛ ✉♠❛ ❢✉♥çã♦ ❡s❝❛❧❛r y(x) ❞❡ ❝❧❛ss❡ C1✱
s❛t✐s❢❛③❡♥❞♦ ❛s ❝♦♥❞✐çõ❡s ❞❡ ❢r♦♥t❡✐r❛ y(x0) = y0 ❡ y(x1) = y1✱ ❡ q✉❡ s❡❥❛ ✉♠ ❡①tr❡♠♦
♣❛r❛ ♦ ❢✉♥❝✐♦♥❛❧ v[y(x)] =Rx1
x0 F (x, y(x), y
′(x))dx ♦♥❞❡ F é ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡C2✱
t❡♠✲s❡ q✉❡ t❛❧ ❢✉♥çã♦ ❡①tr❡♠❛❧ ❞❡✈❡ s❛t✐s❢❛③❡r ❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ s❡❣✉♥❞❛ ♦r❞❡♠ ❞❛❞❛ ♣♦r
Fy−
d
dxFy′ = 0 ✭✸✳✷✮
❞❡♥♦♠✐♥❛❞❛ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r✳
❊♠ ♣❛rt✐❝✉❧❛r✱ F ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞❡ y ❡ y′✱ ❧♦❣♦ é ♣♦ssí✈❡❧ r❡❞✉③✐r ❛ ❡q✉❛çã♦ ❞❡
❊✉❧❡r à ✐❞❡♥t✐❞❛❞❡ F −y′F
✷✽ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦
❉❡ ❢❛t♦✱ ♣❛r❛ ✈❡r✐✜❝❛r ❛ ✈❛❧✐❞❛❞❡ ❞❡st❛ ✐❞❡♥t✐❞❛❞❡✱ ♠✉❧t✐♣❧✐❝❛r❡♠♦s ❛ ❡q✉❛çã♦ ✸✳✷ ♣♦r y′
y′∂F
∂y −y
′ d
dx ∂F
∂y′ = 0 ✭✸✳✸✮
P♦r ♦✉tr♦ ❧❛❞♦✱ ❛ ❞❡r✐✈❛❞❛ t♦t❛❧ ♥♦s ❢♦r♥❡❝❡ ❛ s❡❣✉✐♥t❡ r❡❧❛çã♦
dF dx = ∂F ∂x + ∂F ∂yy
′ +∂F
∂y′y ′′
♦✉ ❛✐♥❞❛✱
∂F ∂yy
′ = dF
dx − ∂F
∂x − ∂F ∂y′y
′′
❈♦♠♦ F ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ✈❛r✐á✈❡❧x✱ ❡♥tã♦ ∂F
∂x = 0✳ ❙✉❜st✐t✉✐♥❞♦ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛
❡♠ ✸✳✸
dF dx −
∂F ∂y′y
′′−y′ d
dx ∂F
∂y′ = 0 ✭✸✳✹✮
❈♦♠♦✱ ♣❡❧❛ r❡❣r❛ ❞♦ ♣r♦❞✉t♦✱ d dx
y′∂F ∂y′
=y′′∂F ∂y′ +y′
d dx
∂F
∂y′✱ ❡♥tã♦
y′ d
dx ∂F ∂y′ =
d dx
y′∂F
∂y′
−y′′∂F
∂y′
❙✉❜st✐t✉✐♥❞♦ ♥❛ ❡q✉❛çã♦ ✸✳✹✱ t❡r❡♠♦s
dF dx −
∂F ∂y′y
′′− d
dx
y′∂F
∂y′
+y′′∂F
∂y′ = 0
❆ ❡①♣r❡ssã♦ ❢♦r♥❡❝❡
d dx
F −y′∂F
∂y′
= 0
q✉❡✱ ♣♦r ✐♥t❡❣r❛çã♦✱ r❡s✉❧t❛ ♥❛ ■❞❡♥t✐❞❛❞❡ ❞❡ ❇❡❧tr❛♠✐✿
F −y′F
❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ✷✾
❱❛♠♦s ✉t✐❧✐③❛r ❡st❛ ✐❞❡♥t✐❞❛❞❡ ♥❛ r❡s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛✳ ❙❡❥❛
F(y, y′) = q1+y′2
y ✱ ❡♥tã♦ Fy′ s❡rá ❞❛❞❛ ♣♦r
Fy′(y, y′) =
∂
1
√yp1 +y′2
∂y′ =
1
√y
dp1 +y′2
dy′
♣❛r❛ ❡♥❝♦♥tr❛r ♦ r❡s✉❧t❛❞♦ ❞❡s❡❥❛❞♦ ❢❛③❡♠♦s u= 1 +y′2✱ ❧♦❣♦ du
dy′ = 2y′✳ ❆ss✐♠✱
Fy′ = 1
√yd √
u du
du dy′ =
1
√y2√1
u2y
′ = y′
p
y(1 +y′2)
❙❡❣✉❡ q✉❡
F −y′F
y′ =
s 1 +y′2
y −y
′ · y′
p
y(1 +y′2)
!
=
p
1 +y′22
p
y(1 +y′2) −
y′2
p
y(1 +y′2)
= 1 +y
′2 −y′2
p
y(1 +y′2)
= p 1
y(1 +y′2)
❖✉ s❡❥❛✱F−y′F
y′é ❡q✉✐✈❛❧❡♥t❡ ❛ √ 1
y(1+y′2
)✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ t❡♠♦s ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡
✸✳✺
1 p
y(1 +y′2) =C
q✉❡ ♣♦❞❡ s❡r r❡❡s❝r✐t❛✱ ❞❡ ❢♦r♠❛ ♠❛✐s s✐♠♣❧❡s✱ ❝♦♠♦
y(1 +y′2) = 1
C2 =k, com k >0 ✭✸✳✻✮
❆ss✐♠✱ t❡♠♦s ✉♠❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠ ♥ã♦ ❧✐♥❡❛r✳ P❛r❛ r❡s♦❧✈ê✲ ❧❛✱ ❝♦♥s✐❞❡r❡♠♦s ❛ s✉❜st✐t✉✐çã♦y′(x(t)) = cotg(t)t❡♥❞♦ ❡♠ ✈✐st❛ ❛ ✐❞❡♥t✐❞❛❞❡csc2(t) =
1 + cotg2(t)✳ ❚❡♠♦s
y= k
1 + cotg2(t) =
k
csc2(t) =ksen
✸✵ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦
❉❡r✐✈❛♥❞♦ y ❝♦♠ r❡❧❛çã♦ ❛ t✱ ✉t✐❧✐③❛♥❞♦ ❛ r❡❣r❛ ❞❛ ❝❛❞❡✐❛✱ dy
dt =k·2 sen(t) cos(t)
❈♦♠♦ y′(x) = dy
dx✱ ❡♥tã♦ dx dy =
1
y′(x)✱ ❛ss✐♠
dx dt = dx dy dy dt = 1
cotg(t) ·2ksen(t) cos(t) = 2ksen
2(t)
■♥t❡❣r❛♥❞♦ dx
dt = 2ksen
2(t)♦❜t❡r❡♠♦s ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ x(t)✳ P❛r❛ t❛❧ ✐♥t❡❣r❛çã♦
✉t✐❧✐③❛♠♦s ❛ ✐❞❡♥t✐❞❛❞❡ sen2(t) = 1
2 −
1
2cos(2t) q✉❡ ♣♦❞❡ s❡r ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛❞❛
✉t✐❧✐③❛♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❝♦ss❡♥♦ ❞❡ ❛r❝♦ ❞✉♣❧♦ ❡ ❛ ✐❞❡♥t✐❞❛❞❡ tr✐❣♦♥♦♠étr✐❝❛ ❢✉♥❞❛♠❡♥t❛❧
x(t) = Z
2ksen2(t)dt = 2k
Z 1 2 −
1
2cos(2t)
dt=kt− ksen(2t)
2 +k2 ♦✉ s❡❥❛✱
x(t) = k
2(2t−sen(2t)) +k2
❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ♣❡❧❛ ❡①♣r❡ssã♦ ✸✳✼✱
y(t) = k
2(1−cos(2t))
❆ss✐♠✱ ❢❛③❡♥❞♦ 2t =θ ❡ s❡♥❞♦k2 = 0✱ ♣♦✐sx(0) = 0✱ ♦❜t❡♠♦s
x= k
2(θ−sen(θ)) ✭✸✳✽✮
❡
y= k
2(1−cos(θ)) ✭✸✳✾✮
❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ r❡❢r❛çã♦ ✸✶
❋✐❣✉r❛ ✸✳✷✿ P❛râ♠❡tr♦s ♥♦ ♣r♦❜❧❡♠❛✳
❋✐❣✉r❛ ✸✳✸✿ ❋❡✐①❡ ❞❡ ❝✐❝❧♦✐❞❡s ❬✸❪✳
❖ ❣rá✜❝♦ ❝♦♥té♠ ♦ ♣♦♥t♦ A(0,0)✱ ❥á q✉❡ s❛t✐s❢❛③ ♦ s✐st❡♠❛✳ P♦❞❡♠♦s ❡s❝♦❧❤❡r ❛ ❝♦♥st❛♥t❡ r ❞❡ ♠♦❞♦ q✉❡ ❛ ❝✉r✈❛ ♣❛ss❡ t❛♠❜é♠ ♣❡❧♦ ♣♦♥t♦ B✳
❖ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ❝♦♥s✐❞❡r❛ ❝♦♥❞✐çõ❡s ✐❞❡❛✐s✱ ♣♦ré♠ ❤á ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ t♦r♥á✲ ❧♦ ♠❛✐s ♣ró①✐♠♦ ❞❛ s✐t✉❛çã♦ r❡❛❧✿ ❝♦♥s✐❞❡r❛r ♦ ❛tr✐t♦✳ ❍á ❛❜♦r❞❛❣❡♠ ❞❛ ❜r❛q✉✐stó❝r♦♥❛ ❝♦♠ ❛tr✐t♦ ❡♠ ❬✹❪✳
✸✳✷ ❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ r❡❢r❛çã♦
✸✷ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦
P❡❧♦ ♣r✐♥❝í♣✐♦ ❞♦ t❡♠♣♦ ♠í♥✐♠♦ ❞❡ ❋❡r♠❛t✱ s❛❜❡♠♦s q✉❡ ❛ tr❛❥❡tór✐❛ r❡❛❧ ♣❡r❝♦r✲ r✐❞❛ ♣♦r ✉♠ r❛✐♦ ❞❡ ❧✉③ ❞❡ ❆ ♣❛r❛ ❇ é ❛ q✉❡ ♠✐♥✐♠✐③❛ ♦ t❡♠♣♦ t♦t❛❧ ❞❡ ♣❡r❝✉rs♦✳
❙❛❜❡♠♦s t❛♠❜é♠✱ ♣❡❧♦ ❢❡♥ô♠❡♥♦ ❞❛ r❡❢r❛çã♦✶✱ q✉❡ s❡ t✐✈❡r♠♦s ❞♦✐s ♠❡✐♦s ❞✐st✐♥t♦s ❛ ❧❡✐ ❞❛ r❡❢r❛çã♦ ❞❡ ❙♥❡❧❧ ♥♦s ❢♦r♥❡❝❡ ❛ r❡❧❛çã♦
senµ1
senµ2
= v1
v2
♦♥❞❡ ♦ â♥❣✉❧♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ éµ1✱ ♦ â♥❣✉❧♦ ❞❡ r❡❢r❛çã♦ éµ2✱ ❛♠❜♦s ♠❡❞✐❞♦s ❝♦♠ r❡❧❛çã♦
à ♥♦r♠❛❧✱ ❡ v1 ❡ v2 sã♦ ❛s ✈❡❧♦❝✐❞❛❞❡s ❞❛ ❧✉③ ♥♦s ♠❡✐♦s 1 ❡ 2✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆ ❧✉③
t❡♠ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡✱ ❡♠ ❣❡r❛❧✱ ❞✐❢❡r❡♥t❡ ❝♦♥❢♦r♠❡ ♦ ♠❡✐♦ ❡♠ q✉❡ s❡ ♣r♦♣❛❣❛✳
❈♦♥s✐❞❡r❛♥❞♦ ✉♠ r❛✐♦ ❞❡ ❧✉③ q✉❡ ✈❛✐ ❞❡ A❛ P ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛v1
❡ s❡❣✉❡ ❞❡P ❛ B ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ ❝♦♥st❛♥t❡v2 t❡♠♦s ♥❛ ✜❣✉r❛ ✸✳✹ ✉♠ ❡sq✉❡♠❛ ♣❛r❛ ♦
❢❡♥ô♠❡♥♦ ❞❛ r❡❢r❛çã♦ ❞❡ ✉♠ r❛✐♦ ❞❡ ❧✉③✳
❋✐❣✉r❛ ✸✳✹✿ ❊sq✉❡♠❛ ♣❛r❛ ♦ ❢❡♥ô♠❡♥♦ ❞❛ r❡❢r❛çã♦ ❞❡ ✉♠ r❛✐♦ ❞❡ ❧✉③✳ ❆✐♥❞❛ t❡♠♦s q✉❡
v1
v2
=
✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ♥♦ ✈á❝✉♦
n1
✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ♥♦ ✈á❝✉♦
n2
= n1
n2
=cte=K
♦♥❞❡ n1 ❡ n2 sã♦ ❝♦♥st❛♥t❡s ❛❞✐♠❡♥s✐♦♥❛✐s✱ ❞❡♥♦♠✐♥❛❞❛s í♥❞✐❝❡s ❞❡ r❡❢r❛çã♦✱ q✉❡ ❞❡✲
♣❡♥❞❡♠ ❞♦ ♠❡✐♦ ♦♥❞❡ ❛ ❧✉③ ❡stá s❡ ♣r♦♣❛❣❛♥❞♦✳
❆ ❧❡✐ ❞❡ ❙♥❡❧❧ ❢♦✐ ❞❡s❝♦❜❡rt❛ ❡♠♣✐r✐❝❛♠❡♥t❡ ♣❡❧♦ ❢ís✐❝♦ ❤♦❧❛♥❞ês ❲✐❧❧❡❜r♦r❞ ✈❛♥ ❘♦✐❥❡♥ ❙♥❡❧❧ ✭✶✺✾✶✲✶✻✷✻✮✱ ❡♠ ✶✻✷✶✱ t❡✈❡ s✉❛ ♣r✐♠❡✐r❛ ♣r♦✈❛ ♠❛t❡♠át✐❝❛ ❢♦r♥❡❝✐❞❛ ♣♦r
❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ r❡❢r❛çã♦ ✸✸
❋❡r♠❛t ❡ t❡♠ ❝♦♠♦ ❜❛s❡ ♦ ♣r✐♥❝í♣✐♦ ❞♦ t❡♠♣♦ ♠í♥✐♠♦✳
❉❡ ❢❛t♦✱ ❝♦♥s✐❞❡r❛♥❞♦ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❡ q✉❡ t❡♠♣♦ ❣❛st♦= ❡s♣❛ç♦ ♣❡r❝♦rr✐❞♦
✈❡❧♦❝✐❞❛❞❡
t❡♠♦s q✉❡ ♦ t❡♠♣♦ ❣❛st♦ ♣❛r❛ ♦ r❛✐♦ ❞❡ ❧✉③ ✐r ❞❡ ❆ ❛té ❇ é ❞❛❞♦ ♣❡❧❛ s♦♠❛ ❡♥tr❡ ♦ t❡♠♣♦ ❣❛st♦ ❞❡ ❆ ❛ P ❡ ♦ t❡♠♣♦ ❣❛st♦ ❞❡ P ❛ ❇✳
T(x) =
√
a2+x2
v1
+ q
b2+ (c−x)2
v2
❆ss✐♠✱ ♥♦ss♦ ♣r♦❜❧❡♠❛ ❡stá r❡❞✉③✐❞♦ ❛ ❝❛❧❝✉❧❛rxq✉❡ ♠✐♥✐♠✐③❡T(x)✱ ❧♦❣♦ t❛❧ ♣♦♥t♦ ❞❡✈❡ s❛t✐s❢❛③❡r T′(x) = 0✳
T′(x) = 1
v1
x
√
a2+x2 −
1
v2
c−x
q
b2+ (c−x)2
= 0
⇒ 1
v1
x
√
a2+x2 =
1
v2
c−x
q
b2+ (c−x)2
✭✸✳✶✵✮
❖❜s❡r✈❡ ❛ ♣❛rt✐r ❞♦s tr✐â♥❣✉❧♦s ◗❆P ❡ ❘P❇✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ q✉❡ x
√
a2+x2 = sen(µ1)
❡ q✉❡ √ c−x b2
+(c−x)2 = sen(µ2)✳ ❙✉❜st✐t✉✐♥❞♦ ❡st❛s ❡①♣r❡ssõ❡s ♥❛ ❡q✉❛çã♦ ✸✳✶✵ ❝❤❡❣❛♠♦s ❛
1
v1
sen(µ1) =
1
v2
sen(µ2)
♦✉ s❡❥❛✱
sen(µ1)
sen(µ2)
= v1
v2
q✉❡ é ❛ r❡❧❛çã♦ ♣r♦❝✉r❛❞❛✳
■♠❛❣✐♥❡♠♦s ❛❣♦r❛ ✉♠ ♠❡✐♦ ó♣t✐❝♦ ❢♦r♠❛❞♦ ♣♦r ❧â♠✐♥❛s l1, l2,· · · , ln ❤♦r✐③♦♥t❛✐s ❡
✜♥❛s t❛✐s q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ❡♠ ❝❛❞❛ ❧â♠✐♥❛ é v1, v2,· · · , vn ❝♦♥❢♦r♠❡ ♠♦str❛ ❛
✜❣✉r❛ ✸✳✺✳ ❊♥tã♦✱ ✉♠ r❛✐♦ ❞❡ ❧✉③ q✉❡ ♣❛rt❡ ❞❡ ❆ ❡ ❝❤❡❣❛ ❛ ❇✱ s❡❣✉✐rá ✉♠❛ tr❛❥❡tór✐❛ ❞❡ ♠♦❞♦ q✉❡
sen(µj)
vj
=K
✸✹ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦
❋✐❣✉r❛ ✸✳✺✿ ▼❡✐♦ ó♣t✐❝♦ ❡ ❛ tr❛❥❡tór✐❛ ❞❡s❝r✐t❛ ♣♦r ✉♠ r❛✐♦ ❞❡ ❧✉③ ♣❛rt✐♥❞♦ ❞❡ ❆ ❡ ❝❤❡❣❛♥❞♦ ❡♠ ❇ ❬✹❪✳
❉✐③❡♠♦s q✉❡ ✉♠ ♠❡✐♦ é ♠❡♥♦s r❡❢r✐♥❣❡♥t❡ q✉❡ ♦✉tr♦ q✉❛♥❞♦ s❡✉ í♥❞✐❝❡ ❞❡ r❡❢r❛çã♦
n é ♠❡♥♦r q✉❡ ♦ ❞♦ ♦✉tr♦✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❛ ❧✉③ s❡ ♣r♦♣❛❣❛ ♣♦r ❡❧❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡
♠❛✐♦r✳ ◆♦ ❡①❡♠♣❧♦ ❛♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ✸✳✺✱ ♦ r❛✐♦ ♣❛ss❛ ♣♦r ♠❡✐♦s ❝❛❞❛ ✈❡③ ♠❡♥♦s r❡❢r✐♥❣❡♥t❡s✱ ♦✉ s❡❥❛✱ ♦♥❞❡ ❛ ✈❡❧♦❝✐❞❛❞❡ é ♠❛✐♦r✳
❋✐❣✉r❛ ✸✳✻✿ ❋♦t♦❣r❛✜❛ q✉❡ ♠♦str❛ ❛ r❡✲ ✢❡①ã♦ ❡ ❛ r❡❢r❛çã♦ ❞❡ ✉♠ ❢❡✐①❡ ❞❡ ❧✉③ ✐♥✲ ❝✐❞❡♥t❡ ❡♠ ✉♠❛ s✉♣❡r❢í❝✐❡ ❞❡ á❣✉❛ ❤♦✲ r✐③♦♥t❛❧ ❬✺❪✳
❋✐❣✉r❛ ✸✳✼✿ ❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ✸✳✻ ❬✺❪✳
◗✉❛♥❞♦ t❡♠♦s ✉♠ ♠❡✐♦ lj+1 ♠❡♥♦s r❡❢r✐♥❣❡♥t❡ q✉❡ ♦ ♠❡✐♦ lj (nj+1 < nj)✱ ♦ r❛✐♦
r❡❢r❛t❛❞♦ s❡ ❛❢❛st❛rá ❞❛ ♥♦r♠❛❧ à s✉♣❡r❢í❝✐❡ ♥♦ ♣♦♥t♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛✳ ■ss♦ s✐❣♥✐✜❝❛ ❞✐③❡r q✉❡ ♦ â♥❣✉❧♦ ❞❡ r❡❢r❛çã♦ é ♠❛✐♦r ❞♦ q✉❡ ♦ â♥❣✉❧♦ ❞♦ r❛✐♦ ✐♥❝✐❞❡♥t❡✳
❉❡ ❢❛t♦✱ ❝♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛ ❞❛ ❧❡✐ ❞❡ ❙♥❡❧❧✱nj+1·sen (µj) = nj·sen(µj+1)✳ ❙❡nj+1 <
❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ r❡❢r❛çã♦ ✸✺
❧❡✐ ❞❡ ❙♥❡❧❧ s❡❥❛ ✐❣✉❛❧ à ♠✉❧t✐♣❧✐❝❛çã♦ ❞♦ s❡❣✉♥❞♦ ♠❡♠❜r♦✳ ❙❡♥❞♦sen(µj+1)>sen(µj)✱
t❡♠♦s q✉❡ µ(j+ 1)> µj✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ ♦ r❛✐♦ s❡ ❛❢❛st❛rá ❞❛ ♥♦r♠❛❧✳
❖ ♣r✐♥❝í♣✐♦ ❞♦ t❡♠♣♦ ♠í♥✐♠♦ ❡ ❛ ❧❡✐ ❞❡ ❙♥❡❧❧ ✈ê♠ ❛♦ ❡♥❝♦♥tr♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛✱ ♣♦ré♠ ❡♥❝♦♥tr❛♠♦s ✉♠❛ ❞✐✜❝✉❧❞❛❞❡ ♠❛✐♦r✿ ❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ ❛ ♣❛rtí❝✉❧❛ s❡ ❞❡s❧♦❝❛ s♦❜r❡ ❛ ❝✉r✈❛ ✈❛r✐❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣♦s✐çã♦ ❡♠ q✉❡ ❡❧❛ s❡ ❡♥❝♦♥tr❛✱ ❡♥q✉❛♥t♦ ♥♦ ❝❛s♦ ❞❛ r❡❢r❛çã♦ ❞♦ r❛✐♦ ❞❡ ❧✉③ ❛ ✈❡❧♦❝✐❞❛❞❡ é ❝♦♥st❛♥t❡ ❡♠ ❝❛❞❛ ♠❡✐♦✳ P❛r❛ tr❛♥s♣♦r ❡st❛ ❞✐✜❝✉❧❞❛❞❡ ❢❛r❡♠♦s ✉s♦ ❞❛ ♥♦çã♦ ❞❡ ❧✐♠✐t❡✳
◗✉❛♥❞♦ ❛ ♣❛rtí❝✉❧❛ t✐✈❡r ❞❡s❝✐❞♦ ✉♠❛ ❛❧t✉r❛ ❤✱ s✉❛ ✈❡❧♦❝✐❞❛❞❡ s❡rá √2gh ✭▲❡✐ ❞❛
q✉❡❞❛ ❧✐✈r❡✮✳ ❊♥tã♦ ♦ ❝❛♠✐♥❤♦ q✉❡ ❢♦r♥❡❝❡rá ♦ t❡♠♣♦ ♠í♥✐♠♦ s❡rá ❛ tr❛❥❡tór✐❛ s❡❣✉✐❞❛ ♣♦r ✉♠ r❛✐♦ ❞❡ ❧✉③ ♥✉♠ ♠❡✐♦ t❛❧ q✉❡ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ❧✉③ ✈❛r✐❡ ❝♦♥t✐♥✉❛♠❡♥t❡ ❝♦♠ ❛ ❞❡s❝✐❞❛ h ❡ s❡❥❛ ♣r❡❝✐s❛♠❡♥t❡ √2gh✳ ❚❡r❡♠♦s
sen(µj)
√
2gh =K ✭✸✳✶✶✮
s❡♥❞♦ µ ♦ â♥❣✉❧♦ q✉❡ t❛❧ ❝❛♠✐♥❤♦ ❢❛③ ❝♦♠ ❛ ✈❡rt✐❝❛❧✱ ❝♦♥❢♦r♠❡ ✜❣✉r❛ ✸✳✽✳
❋✐❣✉r❛ ✸✳✽✿ ➶♥❣✉❧♦ q✉❡ ♦ ❝❛♠✐♥❤♦ ❞❡s❝r✐t♦ ♣❡❧♦ r❛✐♦ ❞❡ ❧✉③ ❡ ❛ ✈❡rt✐❝❛❧ ❬✹❪✳
❱❛♠♦s✱ ❛❣♦r❛✱ ✈❡r✐✜❝❛r q✉❡ ❛ ❝✐❝❧♦✐❞❡ s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✸✳✶✶✳ ❈♦♥❤❡❝❡♥❞♦ s✉❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ♦❜t❡♠♦s
dx
dθ =r(1−cos(θ)) dy
dθ =rsen(θ)
✸✻ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦
tg(µ) = dx
dy = dx dθ
dθ dy =
1−cos(θ) sen(θ)
= 1− cos2 θ 2
−sen2
θ 2 2 sen θ 2 cos θ 2 =
1−cos2
θ
2
+ sen2
θ 2 2 sen θ 2 cos θ 2 = 1−
1−sen2
θ
2
+ sen2
θ 2 2 sen θ 2 cos θ 2 =−
2 sen2
θ 2 2 sen θ 2 cos θ 2 = sen2 θ 2 cos2 θ 2 = tg θ 2 ❆ss✐♠✱
tg(µ) = tg
θ
2
❡✱ ♣♦rt❛♥t♦✱
µ= θ 2
P♦r ♦✉tr♦ ❧❛❞♦✱ ✉t✐❧✐③❛♥❞♦ ❛ ❡①♣r❡ssã♦ ♣❛r❛ ❝♦ss❡♥♦ ❞❡ ❛r❝♦ ❞✉♣❧♦ ❡ ❛ ✐❞❡♥t✐❞❛❞❡ tr✐❣♦♥♦♠étr✐❝❛ ❢✉♥❞❛♠❡♥t❛❧✱
v =p2gy=p2gr(1−cos(θ)) = 2√grsen
θ
2
❆ss✐♠✱ ❞❡ ❢❛t♦✱
sen(µ)
v = sen θ 2 2 sen θ 2
√gr =
1
❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ s❛❧✈❛♠❡♥t♦ ♥❛ ♣r❛✐❛ ✸✼
♦♥❞❡ K ♥ã♦ ❞❡♣❡♥❞❡ ❞❡ θ✳ P♦rt❛♥t♦✱ ❛ ❝✐❝❧♦✐❞❡ t❡♠ ❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ♣r♦❝✉r❛♠♦s
❡ ❛ss✐♠✱ ❡❧❛ é s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❇r❛q✉✐stó❝r♦♥❛✱ ❝♦♠♦ ❥á ❤❛✈í❛♠♦s ♦❜t✐❞♦ ♥❛ r❡s♦❧✉çã♦ ✈❛r✐❛❝✐♦♥❛❧ ❞♦ ♣r♦❜❧❡♠❛✳
✸✳✸ ❆ ❇r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ s❛❧✈❛♠❡♥t♦
♥❛ ♣r❛✐❛
❖s ❝♦♥t❡ú❞♦s ❞❛ ❢ís✐❝❛ ❡ ❞❛ ♠❛t❡♠át✐❝❛ q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ❝♦♠♦ ❜❛s❡ ♥❛ s❡ssã♦ ❛♥t❡r✐♦r ❝♦♥st❛♠ tr❛❞✐❝✐♦♥❛❧♠❡♥t❡ ♥♦ ♣r♦❣r❛♠❛ ♣r❡✈✐st♦ ♣❛r❛ ♦ s❡❣✉♥❞♦ ❛♥♦ ❞♦ ❊♥✲ s✐♥♦ ▼é❞✐♦ ♦ q✉❡ t♦r♥❛ ❛ r❡s♦❧✉çã♦ ❝♦♠♣r❡❡♥sí✈❡❧ ❛ ❛❧✉♥♦s ❛ ♣❛rt✐r ❞❡st❡ ♥í✈❡❧ ❞❡ ❡♥s✐♥♦✳
❖ ♣r♦❜❧❡♠❛ ❞❡ s❛❧✈❛♠❡♥t♦ ♥❛ ♣r❛✐❛✱ ❡①♣♦st♦ ♥❡st❛ s❡çã♦✱ ❡♥✈♦❧✈❡ ❝♦♥❞✐çõ❡s ❛♥á❧♦✲ ❣❛s às ❞♦ ♣r♦❜❧❡♠❛ ❞❡ r❡❢r❛çã♦ ❡✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ às ❞♦ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛ ❡ ♣♦❞❡ ❛✉①✐❧✐❛r ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❛ r❡s♣♦st❛ ❛♦ ♣r♦❜❧❡♠❛ q✉❡ ♥ã♦ é ✐♥t✉✐t✐✈❛ ❛ q✉❛❧q✉❡r ♣❡ss♦❛✳
❙❡ ♣❡r❣✉♥t❛r♠♦s ❛ ❛❧❣✉é♠ q✉❛❧ é ♦ ❝❛♠✐♥❤♦ ♠❛✐s rá♣✐❞♦ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞❡s♥✐✲ ✈❡❧❛❞♦s✱ ♣♦ss✐✈❡❧♠❡♥t❡ r❡s♣♦♥❞❡rá q✉❡ é ❛ r❡t❛ ❛♦ ✐♠❛❣✐♥❛r q✉❡ ♦ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦ é s❡♠♣r❡✱ t❛♠❜é♠✱ ♦ ♠❛✐s rá♣✐❞♦✳
❙❡ ♦s ❞♦✐s ♣♦♥t♦s ❡st✐✈❡r❡♠ s♦❜ ✉♠ ❝❛♠♣♦ ✉♥✐❢♦r♠❡✱ ❡♥tã♦ ❛ tr❛❥❡tór✐❛ ❞❡ ♠❡♥♦r t❡♠♣♦ s❡rá r❡t✐❧í♥❡❛✱ ♣♦ré♠ ♥♦ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛ ❛ ♣❛rtí❝✉❧❛ ❡stá s♦❜ ❛ ❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡✱ ❧♦❣♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ✈❛r✐❛ ❝♦♥❢♦r♠❡ ❛ ❛❧t✉r❛✳
✸✽ ❇✉s❝❛ ♣❡❧❛ s♦❧✉çã♦
q✉❡ ❡stá ♥♦ ♠❛r ♥♦ ♠❡♥♦r t❡♠♣♦ ♣♦ssí✈❡❧✳ ◗✉❛❧ tr❛❥❡tór✐❛ s❡r✐❛ ❡s❝♦❧❤✐❞❛ ♣❛r❛ ❝❤❡❣❛r ♦ ♠❛✐s rá♣✐❞♦ ♣♦ssí✈❡❧ ❛té ❛ ✈ít✐♠❛❄
❙❡❣✉✐r ❛ tr❛❥❡tór✐❛ ✶ ❛té ❛ ✈ít✐♠❛ ♥ã♦ é ❛ ♠❡❧❤♦r ♦♣çã♦ ❞❡ ❝❛♠✐♥❤♦✳ ■ss♦ ❛❝♦♥t❡❝❡ ♣♦rq✉❡ ♥❛ ❛r❡✐❛ ✈♦❝ê t❡♠ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ♠❛✐♦r ❞♦ q✉❡ ♥❛ á❣✉❛✳ ❆ss✐♠✱ é ♠❡❧❤♦r ❝♦rr❡r ✉♠ ♣♦✉❝♦ ♠❛✐s ♥❛ ❛r❡✐❛✱ ♣♦rq✉❡ s✉❛ ✈❡❧♦❝✐❞❛❞❡ s❡rá ♠❛✐♦r ❡ ❞❡✐①❛r ♣❛r❛ ❞❡♣♦✐s ❡♥tr❛r ♥❛ á❣✉❛✱ ♦✉ s❡❥❛✱ é ♠❡❧❤♦r s❡❣✉✐r ❛ tr❛❥❡tór✐❛ ✷✳
❈♦♠ ❛ ❧✉③ ❛❝♦♥t❡❝❡ ❛ ♠❡s♠❛ ❝♦✐s❛✿ ❛ ❧✉③ ♣r♦♣❛❣❛✲s❡ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡s ❞✐❢❡r❡♥t❡s ❡♠ ♠❡✐♦s ❞✐❢❡r❡♥t❡s✱ ❝♦♥❢♦r♠❡ ❡①♣♦st♦ ♥❛ s❡çã♦ ❛♥t❡r✐♦r✳
❆✐♥❞❛ ♣♦❞❡rí❛♠♦s ✐♠❛❣✐♥❛r q✉❡ ♦ s❛❧✈❛✲✈✐❞❛s ❡stá ❡♠ ✉♠ ❝❛❧ç❛❞ã♦✱ ♦♥❞❡ s✉❛ ✈❡✲ ❧♦❝✐❞❛❞❡ s❡r✐❛ ❛✐♥❞❛ ♠❛✐♦r ❝♦♠ r❡❧❛çã♦ à ❛r❡✐❛✱ ♣♦rt❛♥t♦ s❡r✐❛ ❝♦♥✈❡♥✐❡♥t❡ ❝♦rr❡r ✉♠ ♣♦✉❝♦ ♠❛✐s ♥❡❧❡✳
❱♦❧t❛♥❞♦ ❛♦ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛✱ ❝♦♠♦ ❛ ✈❡❧♦❝✐❞❛❞❡ ❝♦♠ q✉❡ ❛ ♣❛rtí❝✉❧❛ s❡ ❞❡s❧♦❝❛ s♦❜r❡ ❛ ❝✉r✈❛ ✈❛r✐❛ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ♣♦s✐çã♦ ❡♠ q✉❡ ❡❧❛ s❡ ❡♥❝♦♥tr❛✱ ❡♥q✉❛♥t♦ ♥♦ ❝❛s♦ ❞❛ r❡❢r❛çã♦ ❞♦ r❛✐♦ ❞❡ ❧✉③ ❛ ✈❡❧♦❝✐❞❛❞❡ é ❝♦♥st❛♥t❡ ❡♠ ❝❛❞❛ ♠❡✐♦✱ t❛♠❜é♠ ❢❛r❡♠♦s ✉s♦ ❞❛ ♥♦çã♦ ❞❡ ❧✐♠✐t❡ ♣❛r❛ tr❛♥s♣♦r ❡st❛ ❞✐✜❝✉❧❞❛❞❡✳ ❊ss❛ s✐t✉❛çã♦ t❛♠❜é♠ ❡stá r❡♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✸✳✺✳
✹ Pr♦♣♦st❛ ❞✐❞át✐❝❛
◆❡st❡ ❝❛♣ít✉❧♦✱ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ r❡❧❛❝✐♦♥❛❞❛ ❛♦ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛ ❡♥✈♦❧✈❡♥❞♦ ❞✐❢❡r❡♥t❡s ❛❜♦r❞❛❣❡♥s✿ ❛♣❧✐❝❛çã♦✱ ✉s♦ ❞❡ ❢❡rr❛♠❡♥t❛ t❡❝♥♦✲ ❧ó❣✐❝❛ ❡ ❡①♣❡r✐♠❡♥t❛çã♦✳
❖ ♦❜❥❡t✐✈♦ ❞❡st❛ ♣r♦♣♦st❛ é ❝♦♥tr✐❜✉✐r ❝♦♠ ❛ ♣rát✐❝❛ ♣❡❞❛❣ó❣✐❝❛ ❞♦ ♣r♦❢❡ss♦r✱ ❛♣r❡✲ s❡♥t❛♥❞♦ ♦ ❝♦♥t❡ú❞♦ ❞❡ ❢♦r♠❛ ❝♦♥t❡①t✉❛❧✐③❛❞❛ ❡ r❡❧❛❝✐♦♥❛♥❞♦ ❛ ♠❛t❡♠át✐❝❛ ❡ ❛ ❢ís✐❝❛✳
❖ ♣r♦❢❡ss♦r t❡♠ ❛✉t♦♥♦♠✐❛ ♣❛r❛ ♦♣t❛r ♣❡❧❛s s✉❣❡stõ❡s q✉❡ ❥✉❧❣❛r ❛❞❡q✉❛❞❛s ❛♦ s❡✉ ♣❧❛♥❡❥❛♠❡♥t♦ ❞✐❞át✐❝♦ ♣♦❞❡♥❞♦ t❛♠❜é♠ ♠♦❞✐✜❝á✲❧❛s ♦✉ ❛✐♥❞❛ ❝r✐❛r ♦✉tr❛s✱ s❡♥❞♦ ✐♠✲ ♣♦rt❛♥t❡ q✉❡ ❛s ❛t✐✈✐❞❛❞❡s ❡st✐♠✉❧❡♠ ❛ r❡✢❡①ã♦ ❡ ❛ ✈✐sã♦ ❝rít✐❝❛ ❞♦s ❛❧✉♥♦s✳ ❖ ♣ú❜❧✐❝♦ ❛❧✈♦ sã♦ ❛❧✉♥♦s ❞♦ s❡❣✉♥❞♦ ❡ t❡r❝❡✐r♦ ❛♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳
✹✳✶ ❆❜♦r❞❛❣❡♠ ❛tr❛✈és ❞❡ ❛♣❧✐❝❛çã♦✿ r❛♠♣❛ ❞❡ s❦❛t❡
❈♦♥❢♦r♠❡ ❛♣r❡s❡♥t❛❞♦ ❡♠ ❬✻❪✱ ♥❛s ❝♦♠♣❡t✐çõ❡s ❞❡ ✈❡rt✐❝❛❧✱ ♦s s❦❛t✐st❛s sã♦ ❛✈❛❧✐❛❞♦s s❡❣✉♥❞♦ ❝r✐tér✐♦s ❞❡ ❝r✐❛t✐✈✐❞❛❞❡ ❡ ❣r❛✉ ❞❡ ❞✐✜❝✉❧❞❛❞❡ ❞❛s ♠❛♥♦❜r❛s✱ q✉❡ ❞❡✈❡♠ s❡r ❡①❡❝✉t❛❞❛s ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ t❡♠♣♦ ♣ré✲❡st❛❜❡❧❡❝✐❞♦✳ ❉❡ss❛ ❢♦r♠❛✱ q✉❛♥t♦ ♠❡♥♦s t❡♠♣♦ ♦ s❦❛t✐st❛ ❣❛st❛r ♣❡r❝♦rr❡♥❞♦ ❛ ❡①t❡♥sã♦ ❞❛ r❛♠♣❛ ❞❡ ✉♠ ❧❛❞♦ ♣❛r❛ ♦ ♦✉tr♦✱ ♠❛✐s t❡♠♣♦ ❧❤❡ s♦❜r❛rá ♣❛r❛ ❡①❡❝✉t❛r ❛s ♠❛♥♦❜r❛s q✉❡ ❝♦♥t❛♠ ♣♦♥t♦s✳
❙❡♥❞♦ ❛ss✐♠✱ é ✐♥t❡r❡ss❛♥t❡ ❡♥❝♦♥tr❛r ✉♠❛ ❝✉r✈❛✱ ♣❛r❛ q✉❡ ♣♦ss❛ s❡r ❝♦♥str✉í❞❛ ✉♠❛ ♣✐st❛ ❞❡ s❦❛t❡✱ q✉❡ ♣♦ss✉❛ ♦ ♠❡♥♦r t❡♠♣♦ ❞❡ ❞❡s❝✐❞❛✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ ♦ s❦❛t✐st❛ t❡♥❤❛ ♠❛✐s t❡♠♣♦ ♣❛r❛ r❡❛❧✐③❛r ♠❛✐s ♠❛♥♦❜r❛s ❞✉r❛♥t❡ ❛ ❝♦♠♣❡t✐çã♦✳
P♦❞❡rí❛♠♦s ♥♦s ♣❡r❣✉♥t❛r s❡ ❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ q✉❡ ❝♦♠♣õ❡ ❛ ❧❛t❡r❛❧ ❞❛ r❛♠♣❛ ❞❛ ✜❣✉r❛ ✹✳✶ é✱ ❞❡ ❢❛t♦✱ ❛ ❝✉r✈❛ ❞❡ t❡♠♣♦ ♠í♥✐♠♦ ❞❡ ❞❡s❝✐❞❛✳
❯♠❛ s✐t✉❛çã♦ s❡♠❡❧❤❛♥t❡ s❡r✐❛ ♣❡r❣✉♥t❛r✿ q✉❛❧ ❞❡✈❡ s❡r ❛ ❢♦r♠❛ ❞♦ ❡s❝♦rr❡❣❛❞♦r ❞❡ ✉♠ ♣❛rq✉❡ ✐♥❢❛♥t✐❧ ♣❛r❛ q✉❡ ♦ t❡♠♣♦ ❞❡ ❞❡s❝✐❞❛ s❡❥❛ ♦ ♠❡♥♦r ♣♦ssí✈❡❧❄ ❉❡s❝♦❜r✐r q✉❛❧ é ❛ ❝✉r✈❛ q✉❡ ♣♦ss✉✐ ♦ t❡♠♣♦ ❞❡ ❞❡s❝✐❞❛ ♠❛✐s ❝✉rt♦ é ♦ ♠❡s♠♦ q✉❡ r❡s♦❧✈❡r ♦
✹✵ Pr♦♣♦st❛ ❞✐❞át✐❝❛
♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛✳
❋✐❣✉r❛ ✹✳✶✿ ❊s❜♦ç♦ ❞❡ ✉♠❛ P✐st❛ ❍❛❧❢ P✐♣❡ ❬✻❪✳
◆❡ss❡ ❝♦♥t❡①t♦✱ ♣♦❞❡ s❡r ✐♥tr♦❞✉③✐❞♦ ♦ ♣r♦❜❧❡♠❛ ❡ ✉♠ ♣♦✉❝♦ ❞❡ s✉❛ ❤✐stór✐❛✱ ♣r❡✲ s❡♥t❡s ♥♦ ❝❛♣ít✉❧♦ ✷✳
◗✉❛♥t♦ à s♦❧✉çã♦✱ ❛♣❡s❛r ❞♦ ❡st✉❞♦ ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧✱ ❞❛ s❡ssã♦ ✸✳✶✱ ♥ã♦ s❡r ❛❝❡ssí✈❡❧ ❛♦s ❛❧✉♥♦s ❞❡st❛ ❢❛✐①❛✱ ♦ ♣r♦❢❡ss♦r ♣♦❞❡ r❡❧❛t❛r q✉❡ ❛ ♠❛t❡♠át✐❝❛ ♦❢❡r❡❝❡ ❝♦♠ r✐❣♦r ❛ r❡s♣♦st❛ ❛♦ ♣r♦❜❧❡♠❛✱ ❛ ♣❛rt✐r ❞❛s ❝♦♥❞✐çõ❡s ❞❛❞❛s ❡ ❝❤❡❣❛ à s♦❧✉çã♦ q✉❡ ❡r❛ ❞❡s❝♦♥❤❡❝✐❞❛✳
❆ r❡❧❛çã♦ ❡♥tr❡ ❛ ❜r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ s❛❧✈❛♠❡♥t♦✱ s❡çã♦ ✸✳✸✱ ♣♦❞❡ ❛✉✲ ①✐❧✐❛r ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞♦ ❢❛t♦ ❞❡ ♦ ❝❛♠✐♥❤♦ ♠❛✐s rá♣✐❞♦ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞❡s♥✐✈❡❧❛❞♦s ♥ã♦ s❡r ❛ r❡t❛ ❛♣❡s❛r ❞❡ s❡r ♦ ❝❛♠✐♥❤♦ ♠❛✐s ❝✉rt♦✳ ❊♠ ❛ss♦❝✐❛çã♦ ❛♦ ❡st✉❞♦ ❞❡ ó♣✲ t✐❝❛✱ ♣r❡s❡♥t❡ ♥❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ❢ís✐❝❛✱ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ❜r❛q✉✐stó❝r♦♥❛ ❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ r❡❢r❛çã♦✱ s❡ssã♦ ✸✳✷✱ ❧❡✈❛rá ♦s ❛❧✉♥♦s ❛ ❝♦♥❝❧✉✐r❡♠ q✉❡ ❛ ❝✐❝❧♦✐❞❡ é ❛ r❡s♣♦st❛ ♣r♦❝✉r❛❞❛✳
❱♦❧t❛♥❞♦ à r❛♠♣❛ ❞❡ s❦❛t❡ ❞❛ ✜❣✉r❛ ✹✳✶✱ s✉❜st✐t✉✐♥❞♦ ♦s ❛r❝♦s ❞❡ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♣♦r ❛r❝♦s ❞❡ ❝✐❝❧♦✐❞❡✱ t❡r❡♠♦s ✉♠❛ r❛♠♣❛ ❧✐❣❛♥❞♦ ✉♠ ♣♦♥t♦ ❞❡ ❛❧t✉r❛ ✶✱✻ ♠❡tr♦s ❡ ♦✉tr♦ ❛ ③❡r♦ ♠❡tr♦✱ q✉❡ ♠❡❧❤♦r❛ ❛ ❡✜❝✐ê♥❝✐❛ ♣❛r❛ ❛s ❝♦♠♣❡t✐çõ❡s ❞❡ ✈❡rt✐❝❛❧✳
❊q✉❛❝✐♦♥❛♥❞♦ ❛ ♥♦✈❛ ♣❧❛♥t❛ ❞❡ r❛♠♣❛ ❡♠ ✉♠ s✐st❡♠❛ ❞❡ ❝♦♦r❞❡♥❛❞❛s✱ ❝♦♠ θ ✭❡♠
r❛❞✐❛♥♦s✮ ♥♦ ❡✐①♦ ❞❛s ❛❜s❝✐ss❛s✱ t❡♠♦s ❛ ❝✉r✈❛ ❛♣r❡s❡♥t❛❞❛ ♥❛ ✜❣✉r❛ ✹✳✷✳
