❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛
■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
❖ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ❡ ❛s ❈✉r✈❛s ❈✐❝❧♦✐❞❛✐s
♣♦r
▲❡♦♥❛r❞♦ ▼✐r❛♥❞❛ ❞❡ ❈❛str♦
❇r❛sí❧✐❛
❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠át✐❝❛
❖ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ❡ ❛s ❈✉r✈❛s
❈✐❝❧♦✐❞❛✐s
♣♦r
▲❡♦♥❛r❞♦ ▼✐r❛♥❞❛ ❞❡ ❈❛str♦
∗❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡
▼❊❙❚❘❊ ❊▼ ▼❆❚❊▼➪❚■❈❆
❇r❛sí❧✐❛✱ ✷✹ ❞❡ ❥✉♥❤♦ ❞❡ ✷✵✶✹
❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛✿
❉r✳ ❘✐❝❛r❞♦ ❘✉✈✐❛r♦ ✲ ❯♥❇ ✲ ❖r✐❡♥t❛❞♦r
❉r✳ ❑❡❧❧❝✐♦ ❖❧✐✈❡✐r❛ ❆r❛ú❥♦ ✲ ❯♥❇ ✲ ❊①❛♠✐♥❛❞♦r
❉r✳ ❏♦ã♦ P❛❜❧♦ P✐♥❤❡✐r♦ ❞❛ ❙✐❧✈❛ ✲ ❯❋P❆ ✲ ❊①❛♠✐♥❛❞♦r
✧❚♦t❛ P✉❧❝❤r❛ ❡s✱ ▼❛r✐❛✦
❚♦t❛ P✉❧❝❤r❛ ❡s✱ ▼❛r✐❛✦
❊t ♠❛❝✉❧❛ ♦r✐❣✐♥❛❧✐s ♥♦♥ ❡st ✐♥ t❡✦
❊t ♠❛❝✉❧❛ ♦r✐❣✐♥❛❧✐s ♥♦♥ ❡st ✐♥ t❡✦
❚✉✱ ●❧♦r✐❛ ■❡r✉s❛❧❡♠✦
❚✉✱ ▲❛❡t✐t✐❛ ■sr❛❡❧✦
❚✉✱ ❤♦♥♦r✐✜❝❡♥t✐❛ ♣♦♣✉❧✐ ♥♦str✐❧✦
❚✉✱ ❛❞✈♦❝❛t❛ ♣❡❝❝❛t♦r✉♠✦
❖ ▼❛r✐❛✦ ❖ ▼❛r✐❛✦
❱✐r❣♦ Pr✉❞❡♥t✐ss✐♠❛✱
▼❛t❡r ❈❧❡♠❡♥t✐ss✐♠❛✱
❖r❛ ♣r♦ ♥♦❜✐s✳ ■♥t❡r❝❡❞❡ ♣r♦ ♥♦❜✐s
❛❞ ❉♦♠✐♥✉♠ ■❡s✉♠ ❈❤r✐st✉♠✦
❆♥t✐❣♦ ❝â♥t✐❝♦ ❈❛tó❧✐❝♦
❆❣r❛❞❡❝✐♠❡♥t♦s
Pr✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s t♦❞♦ ♣♦❞❡r♦s♦✱ ♣♦r t❡r ♠❡ ❛❜❡♥ç♦❛❞♦ ❛ ❛❧❝❛♥ç❛r ❡st❛ ❣r❛♥❞❡ ✈✐tór✐❛ ❡ ❛ ❙❛♥tíss✐♠❛ ▼ã❡✱ ❛ ❱✐r❣❡♠ ▼❛r✐❛✱ ♣❡❧❛ s✉❛ ♣r♦t❡çã♦ ❡ ✐♥t❡r❝❡ssã♦
❆♦s ♠❡✉s ❢❛♠✐❧✐❛r❡s✱ ♠✐♥❤ã ♠ã❡ q✉❡ é ♠✐♥❤❛ ❣r❛♥❞❡ ✐♥t❡r❝❡ss♦r❛✱ ♠✐♥❤❛s ✐r♠❛s✱ ❆♥❞ré✐❛ ❡ ❱❛♥❡ss❛✱ ♣❡❧♦ ❝❛r✐♥❤♦ s❡♠♣r❡ ♠♦str❛❞♦✱ ♠✐♥❤❛ s♦❣r❛ ❆✐❞❛ q✉❡ é ✈❡r❞❛❞❡✐r❛♠❡♥t❡ ✉♠❛ s❡❣✉♥❞❛ ♠ã❡✱ ♠❛s ❞❡ ✉♠❛ ❢♦r♠❛ t♦❞❛ ❡s♣❡❝✐❛❧ ❛ ♠✐♥❤❛ ❡s♣♦s❛ tã♦ ❛♠❛❞❛ ❡ tã♦ q✉❡r✐❞❛✱ ●❛❜r✐❡❧❛✱ q✉❡ s✉♣♦rt♦✉ ♠✐♥❤❛s ❛✉sê♥❝✐❛s✱ ♥❡❝❡ssár✐❛ ♣❛r❛ ❝♦♠♣❧❡t❛r ♠❡✉s ❡st✉❞♦s ❡ ♣♦r r❡❛❧✐③❛r ✉♠ ❞❡ ♠❡✉s ♠❛✐♦r❡s s♦♥❤♦s✿ ♦ ❞❡ s❡r ♣❛✐✳ ❚❡ ❛♠♦ ♣r✐♥❝❡s❛✦
❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❘✐❝❛r❞♦ ❘✉✈✐❛r♦✱ ♣♦r t❡r s❡ t♦r♥❛❞♦ ♠❛✐s q✉❡ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ♠❛s ✉♠ ♣❛r❝❡✐r♦ ♥❡st❛ ❡t❛♣❛ ✜♥❛❧ ❞❡ ♠❡✉ tr❛❜❛❧❤♦✳ ❆ ❡❧❡ ♠✐♥❤❛ ❡t❡r♥❛ ❣r❛t✐❞ã♦✳ ❊❧❡ s❡rá s❡♠♣r❡ ♣❛r❛ ♠✐♠ ♦ ♠♦❞❡❧♦ ❞❡ ♣r♦❢❡ss♦r q✉❡ ❞❡s❡❥♦ s❡r✳
❖❜r✐❣❛❞♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥❇✱ ❝♦♠ ♦s q✉❛✐s ❝♦♥✈✐✈✐✳ ❊s♣❡❝✐❛❧♠❡♥t❡ ❛♦ ❝♦♦r❞❡♥❛❞♦r ❞♦ ❝✉rs♦ ♣r♦❢✳ ❉r✳ ❘✉✐ ❙❡✐♠❡t③ ❡ ❛♦ ♠♦t✐✈❛❞♦r ❞❡ss❡ tr❛❜❛❧❤♦ ♦ ♣r♦❢✳ ❉r✳ ❘♦❜❡rt♦ ●❛♥❞✉❧❢♦✳
❆❣r❛❞❡ç♦ ❛♦ ❈❆P❊❙ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦ à ❡st❡ tr❛❜❛❧❤♦✳
❆♦s ❛♠✐❣♦s q✉❡ ✜③ ♥❡st❛ ❡t❛♣❛ ❡ q✉❡ ♥✉♥❝❛ ❡sq✉❡❝❡r❡✐✱ ❞❡ ✉♠❛ ❢♦r♠❛ ❡s♣❡❝✐❛❧ ♠❡✉s ❝♦♠♣❛♥❤❡✐r♦s ❊❞s♦♥ ❡ ❊❞♠✉♥❞♦ q✉❡ ♥ã♦ ❞❡✐①❛r❛♠ q✉❡ ❡✉ ❞❡s✐st✐ss❡ ❞♦ s♦♥❤♦ ❞❡ t♦r♥❛r✲ ♠❡ ♠❡str❡✳
❊♥✜♠✱ ❛❣r❛❞❡ç♦ ❛ t♦❞♦s q✉❡ r❡③❛r❛♠ ♣♦r ♠✐♠ ❡ ♣❡ç♦ ❞❡s❝✉❧♣❛s ❛ t♦❞♦s q✉❡ ❢❛❧t❛♠✱ ♠❛s ♥ã♦ ❞❛r ♣❛r❛ ❛❣r❛❞❡❝❡r ❛ ❝❛❞❛ ✉♠✱ ♣♦✐s sã♦ t❛♥t❛s ❛s ♣❡ss♦❛s ❡s♣❡❝✐❛✐s q✉❡ ♠❡✉s ❛❣r❛❞❡❝✐♠❡♥t♦s s❡r✐❛ ❛ ♠❛✐♦r ♣❛rt❡ ❞❡ ♠✐♥❤❛ ❞✐ss❡rt❛çã♦✳
❘❡s✉♠♦
❆♣r❡s❡♥t❛♠♦s ♥❡st❡ tr❛❜❛❧❤♦ ❛s ❝✉r✈❛s ❝✐❝❧♦✐❞❛✐s✿ ❝✐❝❧ó✐❞❡✱ ❡♣✐❝✐❝❧ó✐❞❡ ❡ ❤✐♣♦❝✐❝❧ó✐❞❡✳ ◆♦ ❡♥t❛♥t♦✱ ♣❛r❛ s✉st❡♥t❛r ❛s ❛✜r♠❛çõ❡s q✉❡ s❡rã♦ ❢❡✐t❛s ♥❡st❡ tr❛❜❛❧❤♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ s♦❜r❡ ❛ ❝✐❝❧ó✐❞❡✱ ✐♥✐❝✐❛❧♠❡♥t❡ tr❛t❛r❡♠♦s s♦❜r❡ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧✱ ❛ s✉❛ ❤✐stór✐❛ ❡ s♦❜r❡ ♠❛t❡♠át✐❝♦s ❢❛♠♦s♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ❛♣ós ❞✐s❝♦rr❡r❡♠♦s s♦❜r❡ ♦ ♣r♦❜❧❡♠❛ ❝♦❧♦❝❛❞♦ ♣♦r ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✿ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❇r❛q✉✐stó❝r♦♥❛ q✉❡ ❝♦♥tr✐❜✉✐✉ ❣r❛♥❞❡♠❡♥t❡ ♣❛r❛ ❛s ❞❡s❝♦❜❡rt❛s s♦❜r❡ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ♥♦ ♣♦r✈✐r✳ ❙♦❜r❡ ❛ ❝✐❝❧ó✐❞❡ ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❞✐s❝✉t✐r❡♠♦s s✉❛s ✐♥t❡r❡ss❛♥t❡s ♣r♦♣r✐❡❞❛❞❡s✱ ❛ s❛❜❡r✿ ♦ ❢❛t♦ ❞❡st❛ s❡r t❛✉t♦❝r♦♥❛ ❡ ✐só❝r♦♥❛✳ ❏á ♣❛r❛ ❛ s❡❣✉♥❞❛ ❝✉r✈❛ ❝✐❝❧♦✐❞❛❧✱ ❡♣✐❝✐❝❧ó✐❞❡✱ s❡rá ❛❜♦r❞❛❞♦ ❝♦♠♦ ♣♦r sé❝✉❧♦s ❡st❡ ❢♦✐ ♦ ♠♦❞❡❧♦ ♣❧❛♥❡tár✐♦✱ q✉❡ ❞❡s❝r❡✈✐❛ ♦ ♠♦✈✐♠❡♥t♦ ❞♦s ♣❧❛♥❡t❛s ❡♠ ❡♣✐❝✐❝❧♦s✳ P♦r ✜♠ ❛♥❛❧✐s❛r❡♠♦s ❝♦♠♦ ❛ ❝✐❝❧ó✐❞❡✱ ❡♣✐❝✐❝❧ó✐❞❡ ❡ ❤✐♣♦❝✐❝❧ó✐❞❡ ♣♦❞❡♠ s❡r ❡st✉❞❛❞❛s ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ❝♦rr❡❧❛❝✐♦♥❛♥❞♦ ❛ss✉♥t♦s ❝♦♠♦ ❛str♦♥♦♠✐❛ ❡ ❛rq✉✐t❡t✉r❛ ❡ ❝♦♠♦ ❛ ✉t✐❧✐③❛çã♦ ❞❡ r❡❝✉rs♦s ❝♦♠♣✉t❛❝✐♦♥❛✐s ♣♦❞❡ s❡r ✉t✐❧✐③❛❞❛ ♣❛r❛ ✈✐s✉❛❧✐③❛r ❛s ❢♦r♠❛s ❞❡ss❛s ❝✉r✈❛s ♠❡❞✐❛♥t❡ ❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡✐s ♣ré ❡st❛❜❡❧❡❝✐❞❛s✳
P❛❧❛✈r❛s✲❈❤❛✈❡s✿ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧❀ ❈✐❝❧ó✐❞❡❀ ❇r❛q✉✐stór♦♥❛❀ ❚❛✉tó❝r♦♥❛❀ Pê♥❞✉❧♦ ■só❝r♦♥♦❀ ❊♣✐❝✐❝❧ó✐❞❡❀ ❍✐♣♦❝✐❝❧ó✐❞❡✳
❆❜str❛❝t
❲❡ ♣r❡s❡♥t st✉❞② ✐♥ t❤✐s ✇♦r❦ t❤❡ t❤❡ ❝②❝❧♦✐❞❛❧ ❝✉r✈❡s✿ ❝②❝❧♦✐❞ ❛♥❞ ❤②♣♦❝②❝❧♦✐❞ ❡♣✐❝②❝❧♦✐❞s✳ ❍♦✇❡✈❡r✱ t♦ s✉♣♣♦rt t❤❡ ❝❧❛✐♠s t❤❛t ✇✐❧❧ ❜❡ ♠❛❞❡ ✐♥ t❤✐s ✇♦r❦✱ ♠❛✐♥❧② ♦♥ t❤❡ ❝②❝❧♦✐❞✱ ✐♥✐t✐❛❧❧② ❞❡❛❧ ♦♥ ✈❛r✐❛t✐♦♥❛❧ ❝❛❧❝✉❧✉s✱ ✐ts ❤✐st♦r② ❛♥❞ ❛❜♦✉t ❢❛♠♦✉s ♠❛t❤❡♠❛t✐❝✐❛♥s ✇❤♦ ❝♦♥tr✐❜✉t❡❞ t♦ ✐ts ❞❡✈❡❧♦♣♠❡♥t✱ ❢♦❧❧♦✇✐♥❣ ✇❡ ✇✐❧❧ ❞✐s❝✉ss t❤❡ ♣r♦❜❧❡♠ ♣♦s❡❞ ❜② ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✿ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❇r❛❝❤✐st♦❝❤r♦♥❡ ✇❤✐❝❤ ❝♦rr♦❜♦r❛t❡❞ ❛♥❞ ♠✉❝❤ t♦ t❤❡ ✜♥❞✐♥❣s ♦♥ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ❝❛❧❝✉❧✉s✳ ❆❜♦✉t t❤❡ ❝②❝❧♦✐❞ s♣❡❝✐✜❝❛❧❧② ❞✐s❝✉ss t❤❡✐r ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s✱ ♥❛♠❡❧② t❤❡ ❢❛❝t t❤❛t t❤✐s ✐s t❛✉t♦❝r♦♥❛ ❛♥❞ ✐s♦❝❤r♦♥♦✉s✳ ❆s ❢♦r t❤❡ s❡❝♦♥❞ ❝②❝❧♦✐❞❛❧✱ ❡♣✐❝②❝❧♦✐❞s ❝✉r✈❡✱ ❛s ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ❢♦r ❝❡♥t✉r✐❡s t❤✐s ✇❛s t❤❡ ♣❧❛♥❡t❛r② ♠♦❞❡❧✱ ❞❡s❝r✐❜✐♥❣ t❤❡ ♠♦t✐♦♥ ♦❢ t❤❡ ♣❧❛♥❡ts ♦♥ ❡♣✐❝②❝❧❡s✳ ❋✐♥❛❧❧② ✇❡ ✇✐❧❧ ❛♥❛❧②③❡ ❤♦✇ t❤❡ ❝②❝❧♦✐❞ ❛♥❞ ❤②♣♦❝②❝❧♦✐❞ ❡♣✐❝②❝❧♦✐❞s ❝❛♥ ❜❡ st✉❞✐❡❞ ✐♥ ❤✐❣❤ s❝❤♦♦❧✱ ❝♦rr❡❧❛t✐♥❣ s✉❜❥❡❝ts ❧✐❦❡ ❛str♦♥♦♠② ❛♥❞ ❛r❝❤✐t❡❝t✉r❡ ❛♥❞ ❤♦✇ t❤❡ ✉s❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥❛❧ r❡s♦✉r❝❡s ❝❛♥ ❜❡ ✉s❡❞ t♦ ✈✐s✉❛❧✐③❡ t❤❡ s❤❛♣❡s ♦❢ t❤❡s❡ ❝✉r✈❡s ❜② ❝❤❛♥❣✐♥❣ t❤❡ ♣r❡✲ s❡t ✈❛r✐❛❜❧❡s✳
❑❡②✲❲♦r❞s✿ ❱❛r✐❛t✐♦♥❛❧ ❈❛❧❝✉❧✉s❀ ❈②❝❧♦✐❞❀ ❇r❛❝❤✐st♦❝❤r♦♥❡❀ ❚❛✉t♦❝❤r♦♥❡❀ ■s♦❝❤r♦♥♦✉s P❡♥❞✉❧✉♠❀ ❊♣✐❝②❝❧♦✐❞s❀ ❍②♣♦❝②❝❧♦✐❞✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✶
✶ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r✿ ❯♠❛ ❱❛r✐á✈❡❧ ■♥❞❡♣❡♥❞❡♥t❡ ❡ ✉♠❛ ❉❡♣❡♥❞❡♥t❡ ✸ ✶✳✶ ❖ Pr♦❜❧❡♠❛ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✶✳✶ ❆ ■❞❡♥t✐❞❛❞❡ ❞❡ ❇❡❧tr❛♠✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷ ❆♣❧✐❝❛çã♦ ❞❛ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r✿ ▼❡♥♦r ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ✳ ✳ ✳ ✳ ✶✵
✷ ❆ ❍✐stór✐❛ ❞❛ ❈✉r✈❛ ❞❡ ▼❡♥♦r ❚❡♠♣♦ ✶✷
✸ ❈✐❝❧ó✐❞❡✿ ❆ ❍❡❧❡♥❛ ❞❛ ●❡♦♠❡tr✐❛ ✶✹ ✸✳✶ ❊q✉❛çõ❡s P❛r❛♠étr✐❝❛s ❞❛ ❈✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺
✹ ❆s Pr♦♣r✐❡❞❛❞❡s ❞❛ ❈✐❝❧ó✐❞❡ ✶✼
✹✳✶ ❆ ❇r❛q✉✐stó❝r♦♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✹✳✷ ❆ ❚❛✉tó❝r♦♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✷✳✶ Pr♦✈❛ ▼❛t❡♠át✐❝❛ ❞❡ q✉❡ ❛ ❈✐❝❧ó✐❞❡ é ❚❛✉tó❝r♦♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✹✳✷✳✷ Pr♦✈❛ ❞❛ ❚❛✉t♦❝r♦♥✐❝✐❞❛❞❡ ❞❛ ❈✐❝❧ó✐❞❡✿ ❙♦❧✉çã♦ ❞❡ ▲❛❣r❛♥❣❡ ✳ ✳ ✳ ✳ ✷✸ ✹✳✷✳✸ Pr♦✈❛ ❞❛ ❚❛✉t♦❝r♦♥✐❝✐❞❛❞❡ ❞❛ ❈✐❝❧ó✐❞❡✿ ❯♠ ❈❛♠✐♥❤♦ ❆❧t❡r♥❛t✐✈♦ ✳ ✷✻ ✹✳✷✳✹ ❖ Pê♥❞✉❧♦ ■só❝r♦♥♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✹✳✷✳✺ Pr✐♠❡✐r❛ P❛rt❡ ❞❛ ❉❡♠♦♥str❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✹✳✷✳✻ ❙❡❣✉♥❞❛ P❛rt❡ ❞❛ ❉❡♠♦♥str❛çã♦✿ ❆ ❊✈♦❧✉t❛ ❞❛ ❈✐❝❧ó✐❞❡ é ❛ Pró♣r✐❛
❈✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶
✺ ❚r❛❜❛❧❤❛♥❞♦ ❝♦♠ ❛ ❈✐❝❧ó✐❞❡✱ ❊♣✐❝✐❝❧ó✐❞❡ ❡ ❍✐♣♦❝✐❝❧ó✐❞❡ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✸✺ ✺✳✶ ❆ P❡sq✉✐s❛ ❡ ❈♦♥str✉çã♦ ❞❛ ❇r❛q✉✐stó❝r♦♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✺✳✷ ❖✉tr❛s Pr♦♣r✐❡❞❛❞❡s ■♥t❡r❡ss❛♥t❡s ❞❛ ❈✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✺✳✸ ❆ P❡sq✉✐s❛ ❙♦❜r❡ ❛ ❊♣✐❝✐❝❧ó✐❞❡ ♥❛ ❆str♦♥♦♠✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶
✺✳✸✳✶ ❆ ❊♣✐❝✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✸✳✷ ❖s ❊♣✐❝✐❝❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✺✳✹ ❆ P❡sq✉✐s❛ s♦❜r❡ ❛ ❍✐♣♦❝✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✺✳✹✳✶ ❊✈♦❧✉t❛ ❞❛ ❍✐♣♦❝✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✺✳✹✳✷ ■♥✈♦❧✉t❛ ❞❛ ❍✐♣♦❝✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ✺✳✹✳✸ ❍✐♣♦❝✐❝❧ó✐❞❡ ❊♥❝✉rt❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✹✳✹ ❍✐♣♦❝✐❝❧ó✐❞❡ ❆❧♦♥❣❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✹✳✺ ❊①❡♠♣❧♦s ❞❡ ❍✐♣♦❝✐❝❧ó✐❞❡s ♦♥❞❡ s❡ ✈❛r✐❛ K = R
r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶
❆ ▼♦❞❡❧❛♥❞♦ ❛s ❈✉r✈❛s ❈✐❝❧♦✐❞❛✐s ♥♦ ●❡♦❣❡❜r❛ ❡ ♥♦ ▼❛♣❧❡ ✺✷ ❆✳✶ ❯s❛♥❞♦ ♦ ●❡♦●❡❜r❛ ♣❛r❛ ▼♦❞❡❧❛r ❛ ❈✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ❆✳✷ ❯s❛♥❞♦ ♦ ▼❛♣❧❡ ♣❛r❛ ▼♦❞❡❧❛r ❛ ❈✐❝❧ó✐❞❡✱ ❊♣✐❝✐❝❧ó✐❞❡ ❡ ❛ ❍✐♣♦❝✐❝❧ó✐❞❡ ✳ ✳ ✳ ✺✸ ❆✳✷✳✶ ▼♦❞❡❧❛♥❞♦ ❛ ❈✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ❆✳✷✳✷ ▼♦❞❡❧❛♥❞♦ ❛ ❊♣✐❝✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ❆✳✷✳✸ ▼♦❞❡❧❛♥❞♦ ❛ ❍✐♣♦❝✐❝❧ó✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
■♥tr♦❞✉çã♦
❊st✉❞❛r❡♠♦s ♥❡ss❡ tr❛❜❛❧❤♦ ❛s ❝✉r✈❛s ❝✐❝❧♦✐❞❛✐s✿ ❝✐❝❧ó✐❞❡✱ ❡♣✐❝✐❝❧ó✐❞❡ ❡ ❤✐♣♦❝✐❝❧ó✐❞❡✳ ❚♦❞❛✈✐❛ ♣❛r❛ ❞❛r s✉st❡♥t❛çã♦ ♥❛s ❛✜r♠❛çõ❡s q✉❡ s❡rã♦ ❢❡✐t❛s ♥❡ss❡ tr❛❜❛❧❤♦✱ ♣r✐♥❝✐♣❛❧♠❡♥t❡ s♦❜r❡ ❛ ❝✐❝❧ó✐❞❡✱ ✐♥✐❝✐❛❧♠❡♥t❡ tr❛t❛r❡♠♦s ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧✱ ♣❛ss❛♥❞♦ ♣❡❧❛ s✉❛ ❤✐stór✐❛ ❡ s♦❜r❡ ♦s ❝é❧❡❜r❡s ♠❛t❡♠át✐❝♦s q✉❡ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ s❡✉ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦✱ ♥♦s ❛t❡♥t❛r❡♠♦s ♥♦ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ♣♦r ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✱ q✉❡ ♥♦ sé❝✉❧♦ ❳❱■■ ❞❡s❛✜♦✉ ❛s ♠❡♥t❡s ♠❛✐s ❢❛♥tást✐❝❛s ❞♦ s❡✉ t❡♠♣♦ ♥❡ss❡s t❡r♠♦s✱ ❝♦♥❢♦r♠❡
[✶]✿
❊✉✱ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✱ ♠❡ ❞✐r✐❥♦ ❛♦s ♠❛t❡♠át✐❝♦s ♠❛✐s ❜r✐❧❤❛♥t❡s ❞♦ ♠✉♥❞♦✳ ◆❛❞❛ é ♠❛✐s ❛tr❛❡♥t❡ às ♣❡ss♦❛s ✐♥t❡❧✐❣❡♥t❡s ❞♦ q✉❡ ✉♠ ♣r♦❜❧❡♠❛ ❞❡s❛✜❛❞♦r✱ ❤♦♥❡st♦✱ ❝✉❥❛s s♦❧✉çõ❡s ♣♦ssí✈❡✐s ❞❛rã♦ ❢❛♠❛ ❡ ♣❡r♠❛♥❡❝❡rã♦ ❝♦♠♦ ✉♠ ❞✉r❛❞♦✉r♦ ♠♦♥✉♠❡♥t♦✳ ❙❡❣✉✐♥❞♦ ♦ ❡①❡♠♣❧♦ ❡st❛❜❡❧❡❝✐❞♦ ♣♦r P❛s❝❛❧✱ ❋❡r♠❛t✱ ❡t❝✳✱ ❊✉ ❡s♣❡r♦ ❣❛♥❤❛r ❛ ❣r❛t✐❞ã♦ ❞❡ t♦❞❛ ❛ ❝♦♠✉♥✐❞❛❞❡ ❝✐❡♥tí✜❝❛ ♣♦r ❛♣r❡s❡♥t❛r ❞✐❛♥t❡ ❞♦s ♠❡❧❤♦r❡s ♠❛t❡♠át✐❝♦s ❞❡ ♥♦ss♦ t❡♠♣♦ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ t❡st❛rá s❡✉s ♠ét♦❞♦s ❡ ♦ ♣♦❞❡r ❞❡ s❡✉s ✐♥t❡❧❡❝t♦s✳ ❈❛s♦ ❛❧❣✉é♠ ♠❡ ❝♦♠✉♥✐q✉❡ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦✱ ❊✉ ♦ ❞❡❝❧❛r❛r❡✐ ♣✉❜❧✐❝❛♠❡♥t❡ ♠❡r❡❝❡❞♦r ❞❡ ❡❧♦❣✐♦✳
■♥tr♦❞✉çã♦ ✷
❛ ❡ss❡ ❣✐❣❛♥t❡ ❞❛ ♠❛t❡♠át✐❝❛✳
❈❛♣ít✉❧♦
1
❊q✉❛çã♦ ❞❡ ❊✉❧❡r✿ ❯♠❛ ❱❛r✐á✈❡❧
■♥❞❡♣❡♥❞❡♥t❡ ❡ ✉♠❛ ❉❡♣❡♥❞❡♥t❡
◆❡st❡ ❝❛♣ít✉❧♦ ❛❜♦r❞❛r❡♠♦s ❛ ❤✐stór✐❛ ❡ ❛ ✐❞❡✐❛ ♣♦r tr❛s ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧✱ ♣♦✐s ❡st❡ t❡♠ s✐❞♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ❜ás✐❝❛ ♥♦ ❡st✉❞♦ ❞❡ ✈ár✐♦s ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s ❡ ❞❛s ♠❛✐s ✈❛r✐❛❞❛s ár❡❛s ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❝♦♠♦✿ ❋ís✐❝❛✲▼❛t❡♠át✐❝❛✱ ❊♥❣❡♥❤❛r✐❛✱ ❋ís✐❝❛ ▼♦❞❡r♥❛✱ ▼❛t❡♠át✐❝❛✱ ❡♥tr❡ ♦✉tr❛s✳ ❆s ✐❞❡✐❛s ♣r❡❝✉rs♦r❛s ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ sã♦ ❛♥t✐❣❛s✳ ❍♦❥❡ ❝♦♠ ♦ ✉s♦ ❞❡ ❢♦r♠✉❧❛çõ❡s ✈❛r✐❛❝✐♦♥❛✐s ♣❛r❛ ❛s ❧❡✐s ❞❛ ❋ís✐❝❛✱ t♦r♥❛✲s❡ ♣♦ssí✈❡❧ ❝♦♥❝❡♥tr❛r ❡♠ ✉♠ ú♥✐❝♦ ❢✉♥❝✐♦♥❛❧ t♦❞♦s ♦s ❛s♣❡❝t♦s ✐♥trí♥s❡❝♦s ❞♦ ♣r♦❜❧❡♠❛ ❡♠ q✉❡stã♦✳ ❋♦r♠✉❧❛çõ❡s ✈❛r✐❛❝✐♦♥❛✐s ♣♦❞❡♠ s❡r✈✐r ♥ã♦ ❛♣❡♥❛s ♣❛r❛ ✉♥✐✜❝❛r ❞✐✈❡rs♦s ❝❛♠♣♦s✱ ♠❛s t❛♠❜é♠ ♣❛r❛ s✉❣❡r✐r ♥♦✈❛s t❡♦r✐❛s ❡ ❢♦r♥❡❝❡r ♠❛♥❡✐r❛s ♣♦❞❡r♦s❛s ❞❡ ❡st✉❞❛r ❛ ❡①✐stê♥❝✐❛ ❡ s♦❧✉çã♦ ❞❡ ❞✐✈❡rs❛s ❡q✉❛çõ❡s ❞✐❢❡r❡♥❝✐❛✐s ♣❛r❝✐❛✐s✳ ❆ ❞✐❢❡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 ✭♦t✐♠✐③❛❞♦s✮✳ ❊♥q✉❛♥t♦ ♦ ❝á❧❝✉❧♦ ❞✐❢❡r❡♥❝✐❛❧ ♣r♦❝✉r❛ ♥ú♠❡r♦s ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ♦t✐♠✐③❛❞♦r❛s✱ ♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ♣r♦❝✉r❛ ❡♥❝♦♥tr❛r ❢✉♥çõ❡s ❝♦♠ ♣r♦♣r✐❡❞❛❞❡s ♦t✐♠✐③❛❞♦r❛s✳
❍✐st♦r✐❝❛♠❡♥t❡✱ ❛ ✐❞❡✐❛ ❝❡♥tr❛❧ ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ r❡♠♦♥t❛ à ●ré❝✐❛ ❛♥t✐❣❛✳ ❉❡s❞❡ ❛ ❆♥t✐❣✉✐❞❛❞❡ ❢♦r❛♠ ❢♦r♠✉❧❛❞♦s ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ♦t✐♠✐③❛çã♦✳ ❆s ✐❞❡✐❛s ♠❛✐s ♣r✐♠✐t✐✈❛s ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞❛s ♣♦r ❆r✐stót❡❧❡s ✭384−322✮ ❛✳❈✱ ♦♥❞❡
❝♦♥st❛♠ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ r❡❢❡rê♥❝✐❛s ❛ ✈❡❧♦❝✐❞❛❞❡s ✈✐rt✉❛✐s✱ ❝♦♥❝❡✐t♦ ✉s❛❞♦ ❡♠ ❛❧❣✉♠❛s ❛❜♦r❞❛❣❡♥s ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ♠á①✐♠♦s ❡ ♠í♥✐♠♦s✳ P♦ré♠✱ ❛ ♣r✐♠❡✐r❛ ❛♣❧✐❝❛çã♦ ❞❡ ✉♠ ♣r✐♥❝í♣✐♦ ❞❡ ♠✐♥✐♠✐③❛çã♦ ❢♦✐ ❢❡✐t❛ ♣♦r ❍❡rã♦ ❞❡ ❆❧❡①❛♥❞r✐❛ ✭20−62✮ ❛✳❈✳ ❍❡rã♦ ♣♦st✉❧♦✉
✹
❱✐r❣í❧✐♦✱ q✉❡ ✈✐✈❡✉ ❡♠ 70 ❛✳❈✳ ❊♠ s✉❛ ♦❜r❛ ❡♥❝♦♥tr❛✲s❡ ❛ s❡❣✉✐♥t❡ ❝✐t❛çã♦✿ ❉✐❞♦✱ ✉♠❛
❢❡♥í❝✐❛✱ ♣❡rs✉❛❞✐✉ ✉♠ ❝❤❡❢❡ ❛❢r✐❝❛♥♦ ❛ ❞❛r✲❧❤❡ t❛♥t❛ t❡rr❛ q✉❛♥t♦ ❡❧❛ ♣✉❞❡ss❡ ❝❡r❝❛r ❝♦♠ ♦ ❝♦✉r♦ ❞❡ ✉♠ t♦✉r♦✳ Pr✐♠❡✐r♦ ❡❧❛ ❝♦rt♦✉ ♦ ❝♦✉r♦ ❡♠ ❝❡♥t❡♥❛s ❞❡ t✐r❛s ❜❡♠ ✜♥❛s✳ ❉❡♣♦✐s ✉♥✐✉✲❛s✱ ❡ tr❛ç♦✉ ✉♠ s❡♠✐❝ír❝✉❧♦ ♥♦ ❝❤ã♦✱ ❛ ❜❡✐r❛ ❞♦ ♠❛r ▼❡❞✐t❡rrâ♥❡♦✳ ❊r❛ ❛ ♠á①✐♠❛ ár❡❛ ❝♦st❡✐r❛ q✉❡ ❡❧❛ ♣♦❞❡r✐❛ ❡♥✈♦❧✈❡r✳ ◆❡st❡ ❧✉❣❛r ❡❧❛ ❝♦♥str✉✐✉ ❛ ❝✐❞❛❞❡ ❞❡ ❈❛rt❛❣♦✳ ▼❡s♠♦ s❡♥❞♦ ❧✐t❡rár✐♦✱ ♦ r❡❧❛t♦ ❞❡♠♦♥str❛ q✉❡ ♦s ♣♦✈♦s ❞❛ ❛♥t✐❣✉✐❞❛❞❡ ♣♦ss✉í❛♠ ❝♦♥❤❡❝✐♠❡♥t♦s ❛ r❡s♣❡✐t♦ ❞❡ ár❡❛s ❡ ❝♦♠♣r✐♠❡♥t♦s ♦t✐♠✐③❛❞♦s✳ ❙❛❜✐❛♠ q✉❡✱ ❞❡♥tr❡ ❛s ✜❣✉r❛s ❞❡ ✐❣✉❛❧ ♣❡rí♠❡tr♦✱ ♦ ❝ír❝✉❧♦ é ❛q✉❡❧❛ ❝♦♠ ♠❛✐♦r ár❡❛✳ ❆❝r❡❞✐t❛✲s❡ q✉❡ ❝❤❡❣❛r❛♠ ❛ ❡st❛s ❝♦♥❝❧✉sõ❡s ❛ ♣❛rt✐r ❞❡ ❝á❧❝✉❧♦s ❞❡ t❡♥t❛t✐✈❛ ❡ ❡rr♦✳
❆s ✐♥❞❛❣❛çõ❡s s♦❜r❡ s♦❧✉çõ❡s ót✐♠❛s ✜❝❛♠ ❝♦♠♦ q✉❡ ❡sq✉❡❝✐❞❛s ✈♦❧t❛♥❞♦ ❝♦♠ ♥♦✈♦ ✈✐❣♦r ♥♦ sé❝✉❧♦ XV II✱ ♦♥❞❡ P✐❡rr❡ ❞❡ ❋❡r♠❛t r❡s♦❧✈❡✉ ✉♠ ♣r♦❜❧❡♠❛✱ ❡♠ ót✐❝❛✱ q✉❡
❛❝❛❜♦✉ ❧❡✈❛♥❞♦ ♦ s❡✉ ♥♦♠❡✱ ✜❝❛♥❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦✿ Pr✐♥❝í♣✐♦ ❞❡ ❋❡r♠❛t✱ q✉❡ ❞✐③✐❛ ✲ ❛ tr❛❥❡tór✐❛ ♣❡r❝♦rr✐❞❛ ♣❡❧❛ ❧✉③ ❛♦ s❡ ♣r♦♣❛❣❛r ❞❡ ✉♠ ♣♦♥t♦ ❛ ♦✉tr♦ é t❛❧ q✉❡ ♦ t❡♠♣♦ ❣❛st♦ ❡♠ ♣❡r❝♦rrê✲❧❛ é ✉♠ ♠í♥✐♠♦✳
❆✐♥❞❛ s♦❜ ❛ ❞❡s❝♦❜❡rt❛ ❞❡ ❋❡r♠❛t✱ ❙♥❡❡❧ ❡ ❉❡s❝❛rt❡s✱ ❡♠ 1630✱ ❝♦♥❝❧✉ír❛♠
❡①♣❡r✐♠❡♥t❛❧♠❡♥t❡ q✉❡ q✉❛♥❞♦ ❛ ❧✉③ r❡✢❡t❡ ❡♠ ✉♠ ❡s♣❡❧❤♦✱ ♦ â♥❣✉❧♦ ❞❡ r❡✢❡①ã♦ ✭q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r r✮ é ✐❣✉❛❧ ❛♦ â♥❣✉❧♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ ✭q✉❡ ❞❡♥♦t❛r❡♠♦s ♣♦r i✮ ❡ q✉❡ ♥❛
r❡❢r❛çã♦ ❞❛ ❧✉③✱ ♣r♦✈❡♥✐❡♥t❡ ❞♦ ♠❡✐♦ 1 ✭✈❡❧♦❝✐❞❛❞❡ V1✮ ♣❛r❛ ✉♠ ♠❡✐♦ 2 ✭✈❡❧♦❝✐❞❛❞❡ V2✮✱
♦ s❡♥♦ ❞♦ â♥❣✉❧♦ ❞❡ ✐♥❝✐❞ê♥❝✐❛ ❞✐✈✐❞✐❞♦ ♣❡❧♦ s❡♥♦ ❞♦ â♥❣✉❧♦ ❞❡ r❡❢r❛çã♦ é ✉♠❛ ❝♦♥st❛♥t❡ ✐❣✉❛❧ ❛✿
sen(i)
sen(r) =
V1
V2
.
▼✉✐t♦s ♠❛t❡♠át✐❝♦s ❞❛q✉❡❧❛ é♣♦❝❛ q✉❡r✐❛♠ s❛❜❡r ♦ q✉❡ ❧❡✈❛✈❛♠ ❛ ❡ss❛s ❧❡✐s✱ ❡♥tã♦✱ ❋❡r♠❛t s✉❣❡r✐✉ s❡ ♥ã♦ s❡r✐❛✱ ❡ss❛s ♠❛♥✐❢❡st❛çõ❡s ❞❛ ♥❛t✉r❡③❛✱ r❡s✉❧t❛❞♦s ❞❡ ✉♠❛ ❜✉s❝❛ ❞❡ ♠í♥✐♠♦s ❡ ♠á①✐♠♦s✳ ■s❛❛❝ ◆❡✇t♦♥✱ t❛♠❜é♠ s❡ ♦❝✉♣❛✈❛ ❝♦♠ ♣r♦❜❧❡♠❛s ❡♥✈♦❧✈❡♥❞♦ ♣r✐♥❝í♣✐♦s ❞❡ ♦t✐♠✐③❛çã♦✳ ❊❧❡ q✉❡r✐❛ s❛❜❡r q✉❛❧ ❡r❛ ❛ ❢♦r♠❛ ❞❡ ✉♠ tú♥❡❧ q✉❡ ❧✐❣❛ ❞♦✐s ♣♦♥t♦s ♥❛ s✉♣❡r❢í❝✐❡ ❞❛ ❚❡rr❛✱ ❞❡ ♠♦❞♦ q✉❡ ♣❡r♠✐t❛ ❛ ✉♠ ❝♦r♣♦ ❞❡ ♠❛ss❛ ♠ ❞❡s❧♦❝❛r✲s❡ ❡♥tr❡ ♦s ❞♦✐s ♣♦♥t♦s ♥♦ ♠❡♥♦r t❡♠♣♦✱ ❝♦♠♦ r❡s♣♦st❛ ■s❛❛❝ ◆❡✇t♦♥ ♦❜t❡✈❡ ❝♦♠♦ r❡s♣♦st❛ ❛ ❤✐♣♦❝✐❝❧ó✐❞❡✳
❆♣❡s❛r ❞❡ ❋❡r♠❛t✱ ◆❡✇t♦♥ ❡ ♦✉tr♦s t❡r❡♠ s❡ ❞❡st❛❝❛❞♦ ♥♦ ❡st✉❞♦ ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧✱ sã♦ ♦s ✐r♠ã♦s ❏❛❝q✉❡s ✭1654 −1705✮ ❡ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ✭1667−1748✮✱ ♦s
❝♦♥s✐❞❡r❛❞♦s ♣❛✐s ❞♦ ❈á❧❝✉❧♦ ❞❡ ❱❛r✐❛çõ❡s✳ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ♣♦r t❡r ♣r♦♣♦st♦ ❡♠ 1696 ♦
✶✳✶ ❖ Pr♦❜❧❡♠❛ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ✺
♠á①✐♠❛✮✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ❉✐❞♦✱ ❛❝✐♠❛ ❝✐t❛❞♦✱ é ✉♠ ♣r♦❜❧❡♠❛ ✐s♦♣❡r✐♠étr✐❝♦✳
❱ár✐♦s ♠❛t❡♠át✐❝♦s s❡ ✐♥t❡r❡ss❛r❛♠ ♣♦r ❡st✉❞❛r ❡ ❞❡s❡♥✈♦❧✈❡r ♦ ❝á❧❝✉❧♦ ❞❡ ✈❛r✐❛çõ❡s✱ ♠❛s ❛ ❞❡ s❡ ❞❡st❛❝❛r ♦ ♠❛t❡♠át✐❝♦ ■t❛❧✐❛♥♦✱ ❏♦s❡♣❤✲▲♦✉✐s ▲❛❣r❛♥❣❡ ✭1736 − 1813✮✳
▲❛❣r❛♥❣❡ é ❡♠ ❣❡r❛❧ ♦ ♠❛✐s ♥♦tá✈❡❧ ♠❛t❡♠át✐❝♦ ❞♦ sé❝✉❧♦ ❳❱■■■✱ s❡♥❞♦ s♦♠❡♥t❡ ❊✉❧❡r ✭1701−1783✮ ✉♠ sér✐♦ r✐✈❛❧✱ ❛❧✐❛s ✈❛❧❡ r❡ss❛❧t❛r q✉❡ ♦ ♥♦♠❡ ❈á❧❝✉❧♦ ❞❛s ❱❛r✐❛çõ❡s ❢♦✐ ❞❛❞♦
♣♦r ❊✉❧❡r ♥♦ tr❛❜❛❧❤♦ ✐♥t✐t✉❧❛❞♦ ❊❧❡♠❡♥t❛ ❈❛❧❝✉❧✐ ❱❛r✐❛t✐♦♥✉♠ ✭❊❧❡♠❡♥t♦ ❞♦ ❈á❧❝✉❧♦ ❞❛s ❱❛r✐❛çõ❡s✮✱ ❛♣r❡s❡♥t❛❞♦ à ❆❝❛❞❡♠✐❛ ❞❡ ❇❡r❧✐♠✱ ❡♠1756✱ ❡ ♣✉❜❧✐❝❛❞♦ ❡♠1766✳ ❆ ♣r✐♠❡✐r❛
❡ t❛❧✈❡③ ♠❛✐♦r ❝♦♥tr✐❜✉✐çã♦ ❞❡ ▲❛❣r❛♥❣❡ ♣❛r❛ ❛ ▼❛t❡♠át✐❝❛ ❢♦✐ ❡♠ ❈á❧❝✉❧♦ ❞❡ ❱❛r✐❛çõ❡s✳ ❊ss❡ ❡r❛ ✉♠ r❛♠♦ ♥♦✈♦ ❞❛ ▼❛t❡♠át✐❝❛✱ ❝✉❥♦ ♥♦♠❡ s❡ ♦r✐❣✐♥♦✉ ❞❛s ♥♦t❛çõ❡s ✉s❛❞❛s ♣♦r ▲❛❣r❛♥❣❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ ❛ ♣❛rt✐r ❞❡ 1760✳ ❊♠ 1755 ▲❛❣r❛♥❣❡ ❤❛✈✐❛ ❡s❝r✐t♦ ❛ ❊✉❧❡r
s♦❜r❡ ♦s ♠ét♦❞♦s ❣❡r❛✐s q✉❡ t✐♥❤❛ ❞❡s❡♥✈♦❧✈✐❞♦ ♣❛r❛ tr❛t❛r ❞❡ ♣r♦❜❧❡♠❛s ❞❡ ✐s♦♣❡r✐♠❡tr✐❛ ❡ ❞❡ ♠❛✐s rá♣✐❞❛ q✉❡❞❛✱ ❡ ❊✉❧❡r ❣❡♥❡r♦s❛♠❡♥t❡ r❡t❛r❞♦✉ ❛ ♣✉❜❧✐❝❛çã♦ ❞❡ ✉♠ tr❛❜❛❧❤♦ s❡✉ s♦❜r❡ t❡♠❛ s❡♠❡❧❤❛♥t❡✱ ❛ ✜♠ ❞❡ q✉❡ ♦ ❛✉t♦r ♠❛✐s ❥♦✈❡♠ r❡❝❡❜❡ss❡ t♦❞♦ ♦ ❝ré❞✐t♦ ♣❡❧♦s ♥♦✈♦s ♠ét♦❞♦s q✉❡ ❊✉❧❡r ❝♦♥s✐❞❡r❛✈❛ s✉♣❡r✐♦r❡s ✳
❖✉tr♦s ♠❛t❡♠át✐❝♦s t❛✐s ❝♦♠♦✿ ❈❛r❧ ●✉st❛✈ ❏❛❝♦❜✐ ✭1804−1851✮ ❡ ❉❛✈✐❞ ❍✐❧❜❡rt
✭1862−1943✮✱ ❆❞r✐❡♥✲▼❛r✐❡ ▲❡❣❡♥❞r❡ ✭1752−1833✮✱ ❑❛r❧ ❲✐❧❤❡❧♠ ❚❤❡♦❞♦r ❲❡✐❡rstr❛ss
✭1815−1897✮✱ ❈❛r❧ ❋r❡❡♥❞r✐❝❤ ●❛✉ss ✭1777−1855✮ ❡ ❲✐❧❧✐❛♠ ❘♦✇❛♥ ❍❛♠✐❧t♦♥ ✭1805− 1865✮ t❛♠❜é♠ ❝♦♥tr✐❜✉ír❛♠ ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❈á❧❝✉❧♦ ❞❡ ❱❛r✐❛çõ❡s✳ ❖✉tr♦s
♣r♦❜❧❡♠❛s ❡s♣❡❝í✜❝♦s ❢♦r❛♠ r❡s♦❧✈✐❞♦s ❡ ✉♠❛ t❡♦r✐❛ ❣❡r❛❧ ❞❡s❡♥✈♦❧✈✐❞❛ ❛♦ ❧♦♥❣♦ ❞♦s ❛♥♦s✳ ❆s ♣r✐♠❡✐r❛s ❛♣❧✐❝❛çõ❡s ❞❡ ❈á❧❝✉❧♦ ❞❡ ❱❛r✐❛çõ❡s ❡♠ ❊❝♦♥♦♠✐❛ s✉r❣✐r❛♠ ♥♦ ✜♥❛❧ ❞❡1920
❡ ✐♥í❝✐♦ ❞❡ 1930 ♣♦r ❘♦♦s✱ ❊✈❛♥s✱ ❍♦t❡❧❧✐♥❣ ❡ ❘❛♠s❡②✱ ❝♦♠ ♦✉tr❛s ❛♣❧✐❝❛çõ❡s ♣✉❜❧✐❝❛❞❛s
♠❛✐s t❛r❞❡✳ ❯♠❛ ♥♦✈❛ ❡r❛ ❝♦♠❡ç♦✉ ♥♦ ✐♥í❝✐♦ ❞❡ 1960 ❝♦♠ ❣r✉♣♦s ❞❡ ❡❝♦♥♦♠✐st❛s ❡
❝✐❡♥t✐st❛s ❞♦ ❣❡r❡♥❝✐❛♠❡♥t♦ ✐♥t❡r❡ss❛❞♦s ❡♠ ❝❡rt♦s ♣r♦❜❧❡♠❛s ❞✐♥â♠✐❝♦s✳ ❆ ❚❡♦r✐❛ ❞♦ ❈♦♥tr♦❧❡ Ót✐♠♦✱ ❞❡s❡♥✈♦❧✈✐❞❛ ♥❛ ❘úss✐❛ ♣♦r P♦♥tr②❛❣✐♥ ❡ s❡✉s ❝♦❧❛❜♦r❛❞♦r❡s ♥♦ ✜♥❛❧ ❞❡ 1950 ❡ ♣✉❜❧✐❝❛❞❛ ♥❛ ❧í♥❣✉❛ ✐♥❣❧❡s❛ ❡♠ 1962✱ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞♦ ❈á❧❝✉❧♦ ❞❡
❱❛r✐❛çõ❡s✱ q✉❡ ❛♠♣❧✐❛ ❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ♠❛t❡♠át✐❝❛✳ ❈♦♠ ✉♠ ♣♦✉❝♦ ❞❛ ❤✐stór✐❛ ❞♦ ❝á❧❝✉❧♦ ✈❛r❛✐❛❝✐♦♥❛❧✱ ♣❛ss❛♠♦s ❛❣♦r❛ ❛ ♥♦s ❛t❡♥t❛r ♥♦ ♦❜❥❡t♦ ❞❡ ♣❡sq✉✐s❛ ❞❡st❛ ár❡❛ ❞❛ ♠❛t❡♠át✐❝❛✳
✶✳✶ ❖ Pr♦❜❧❡♠❛ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧
❖ ♣r♦❜❧❡♠❛ ❝❡♥tr❛❧ ❞♦ ❝á❧❝✉❧♦ ✈❛r✐❛❝✐♦♥❛❧✱ ❝♦♠♦ ❡♠ ❬✸❪✱ ♣♦❞❡ s❡r ❡①♣r❡ss♦ ♥❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ❞❡s❡❥❛♠♦s ❡♥❝♦♥tr❛r ✉♠❛ ❢✉♥çã♦ y(x) q✉❡ ♣♦ss✉✐ ✈❛❧♦r❡s ✜①♦s ♥♦s ♣♦♥t♦sx =x1
❡ x =x2✱ t❛❧ q✉❡ ❛ ✐♥t❡❣r❛❧ ❞❡ ❧✐♥❤❛ ✭✐♥t❡❣r❛❧ ❝❛❧❝✉❧❛❞❛ ❛♦ ❧♦♥❣♦ ❞❛ ❝✉r✈❛✮ ❞❡ ✉♠❛ ❞❛❞❛
❢✉♥çã♦ f
y,dy dx, x
✱ é t❛❧ q✉❡✱ J =
Z x2
x1
f
y, dy dx, x
dx✱ s❡❥❛ ✉♠ ❡①tr❡♠♦ ✭♠á①✐♠♦✱
✶✳✶ ❖ Pr♦❜❧❡♠❛ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ✻
❝rít✐❝♦✱ q✉❡ é ✉♠ ♣♦♥t♦ ♥♦ ❞♦♠í♥✐♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ♦♥❞❡ ❛ ♣r✐♠❡✐r❛ ❞❡r✐✈❛❞❛ é ♥✉❧❛✱ ❛ss✐♠ q✉❡r❡♠♦s ❡♥❝♦♥tr❛r y(x) ❝♦♠ ✈❛❧♦r❡s ✜①♦s y1 = f(x1) ❡ y2 = f(x2) t❛❧ q✉❡ ❛ ✐♥t❡❣r❛❧ J
s❡❥❛ ❡st❛❝✐♦♥ár✐❛✳
❍á✱ ♥❛t✉r❛❧♠❡♥t❡✱ ✐♥✜♥✐t❛s ❢✉♥çõ❡s ❝♦♠ ✈❛❧♦r❡s ✜①♦s ❡♠ (x1, y1) ❡ (x2, y2)✱
❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❛ ✐♥t❡❣r❛❧ J ❛ss✉♠❡ ✈❛❧♦r❡s ❞✐❢❡r❡♥t❡s ♣❛r❛ ❝❛❞❛ ✉♠✳ ◆♦ ♣❧❛♥♦
❝❛rt❡s✐❛♥♦✱ ✐ss♦ ❡q✉✐✈❛❧❡ ❛ ❞✐③❡r q✉❡ ❡①✐st❡♠ ✐♥✜♥✐t♦s ❝❛♠✐♥❤♦s ❧✐❣❛♥❞♦ ♦s ♣♦♥t♦s ✜①♦s✱ ♠❛s ♣❛r❛ s♦♠❡♥t❡ ✉♠ ❞❡❧❡sJ é ✉♠ ❡①tr❡♠♦✳ P♦❞❡♠♦s r♦t✉❧❛r t♦❞♦s ♦s ❝❛♠✐♥❤♦s ♣♦ssí✈❡✐s
❡♥tr❡ ♦s ♣♦♥t♦s (x1, y1) ❡ (x2, y2) ♣♦r ♠❡✐♦ ❞❡ ✉♠ ♣❛râ♠❡tr♦ ✈❛r✐❛❝✐♦♥❛❧ α✱ ❞❡ ♠♦❞♦ q✉❡
❝❛❞❛ ❝❛♠✐♥❤♦ s❡❥❛ ❝❛r❛❝t❡r✐③❛❞♦ ♣♦r y(x, α) ♣❛r❛ ✉♠ ❞❛❞♦ ✈❛❧♦r ❞❡ α✱ ❝♦♠♦ α = 0 ♦
❝❛♠✐♥❤♦ ót✐♠♦ ❝♦rr❡s♣♦♥❞❡♥t❡✱ ❞❡♥♦t❛❞♦ ♣♦r y(x,0)t♦r♥❛ J ❡st❛❝✐♦♥ár✐❛✳
P❛r❛ ❝♦♥✜r♠❛r ♥♦ss❛ ❤✐♣ót❡s❡✱ s✉♣♦♥❤❛♠♦s q✉❡ ❝❛❞❛ ❝❛♠✐♥❤♦ s❡❥❛ ✉♠❛ ❞❡❢♦r♠❛çã♦ ❝♦♥tí♥✉❛ ❞♦ ❝❛♠✐♥❤♦ ót✐♠♦ ♥♦ s❡♥t✐❞♦ ❞❡ q✉❡ ♣♦❞❡♠♦s ❡s❝r❡✈❡ry(x, α) = y(x,0) +αη(x)✱
♦♥❞❡ η(x) r❡♣r❡s❡♥t❛ ❛ ❞❡❢♦r♠❛çã♦✱ ♣♦rt❛♥t♦ ❞❡✈❡ s❡r ✉♠❛ ❢✉♥çã♦ ❝♦♥t✐♥✉❛♠❡♥t❡
❞✐❢❡r❡♥❝✐á✈❡❧ ❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞♦ ✐♥t❡r✈❛❧♦ ❞❡ x1 < x < x2✱ ❛♥✉❧❛♥❞♦✲s❡ ♥♦s s❡✉s
❡①tr❡♠♦s✿ η(x1) = η(x2) = 0.
❈♦♠♦ ✉♠ ❡①❡♠♣❧♦ ❞❡ ♣❛r❛♠❡tr✐③❛çã♦ ❝♦♥s✐❞❡r❡♠♦s ♦s ♣♦♥t♦s ✜①♦s ♥♦ ♣❧❛♥♦(x1, y1) =
(0,0) ❡ (x2, y2) = (1,0)✱ ♦♥❞❡ ♦ ❝❛♠✐♥❤♦ ót✐♠♦ s❡❥❛ ♦ s❡❣♠❡♥t♦ ❞❡ r❡t❛ q✉❡ ♦s ✉♥❡✿
y(x,0) = {(x, y)|y= 0,0≤x≤1}✳ ❯♠❛ ❢❛♠í❧✐❛ ❞❡ ❝✉r✈❛s s✉❛✈❡s ♣❛r❛♠❡tr✐③❛❞❛s ♣♦r α
q✉❡ ❝♦♥❡❝t❛♠ ♦s ♣♦♥t♦s ✜①♦s é (α ∈R)t❛❧ q✉❡✿
y(x, α) =αx(1−x),
♦♥❞❡ α = 0 ❢♦r♥❡❝❡ ♦ ❝❛♠✐♥❤♦ ót✐♠♦✿ y(x,0) = 0✳ ▲♦❣♦ η(x) = x(1− x)✱ s❛t✐s❢❛③
η(0) =η(1) = 0.
❈♦♠ t♦❞❛ ❡ss❛ ♠♦t✐✈❛çã♦ t❡♠♦s q✉❡ ❛ ❡q✉❛çã♦ J =
Z x2
x1
f
y,dy dx, x
dx ♣♦❞❡ s❡r
❡s❝r✐t❛ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦ ♣❛râ♠❡tr♦ ❢✉♥❝✐♦♥❛❧ α✱ ❛ss✐♠✿
J =
Z x2
x1
f
y(x, α),dy
dx(x, α), x
dx.
❆ ❝♦♥❞✐çã♦ ✐♠♣♦st❛ ❛♦ ❝❛♠✐♥❤♦ ót✐♠♦ y(x,0) ❞❡ q✉❡ t♦r♥❡ ♦ ❢✉♥❝✐♦♥❛❧ ❛❝✐♠❛
❡st❛❝✐♦♥ár✐♦ ✐♠♣❧✐❝❛ ❡♠ ∂J
∂α = 0.
❉✐❢❡r❡♥❝✐❛♥❞♦J ❡♠ ❢✉♥çã♦ ❡ α ✜❝❛♠♦s ❝♦♠✿
∂J ∂α =
Z x2
✶✳✶ ❖ Pr♦❜❧❡♠❛ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ✼
■♥t❡❣r❛♥❞♦ ♣♦r ♣❛rt❡s ❡ ❝❤❛♠❛♥❞♦ ❞❡
u= ∂f
∂yx
❡ ❞❡
dv= ∂yx
∂αdx
t❡♠♦s
Z x2
x1
∂f ∂yx
∂yx ∂αdx=
∂f ∂yx ∂y ∂α x2 x1 −
Z x2
x1 ∂y ∂α d dx ∂f ∂yx dx.
❈♦♠♦ t♦❞❛s ❛s ❝✉r✈❛s ♣❛r❛♠❡tr✐③❛❞❛s ♣♦r α ❞❡✈❡♠ ♣❛ss❛r ♣❡❧♦s ♣♦♥t♦s ✜①♦s✱ ❝♦♠♦
❡①♣❧✐❝✐t❛❞♦ ♥♦ ✐♥í❝✐♦ ❞❡st❡ tó♣✐❝♦✱ t❡♠♦s✿
∂y ∂α x1 = 0 ❡ ∂y ∂α x2 = 0
t❛❧ q✉❡ ❛ ♣r✐♠❡✐r❛ ♣❛r❝❡❧❛ r❡s✉❧t❛♥t❡ ❞❛ ✐♥t❡❣r❛çã♦ ♣♦r ♣❛rt❡s é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧❛✱ r❡s✉❧t❛♥❞♦✱❡♥tã♦
∂J ∂α =
Z x2
x1 ∂f ∂y − d dx ∂f ∂yx ∂y ∂αdx.
❖❧❤❛♥❞♦ ♣❛r❛ ❛ ❞✐❢❡r❡♥❝✐❛❧ dα ❡ ❝❛❧❝✉❧❛♥❞♦ ❛s ❞❡r✐✈❛❞❛s ❡♠ r❡❧❛çã♦ ❛ α ♣❛r❛ ♦ ❝❛♠✐♥❤♦
ót✐♠♦ α= 0 t❡r❡♠♦s
∂J ∂α α=0 dα=
Z x2
x1 ∂f ∂y − d dx ∂f ∂yx ∂y ∂αdαdx.
❱❛♠♦s ❞❡♥♦♠✐♥❛r ♣♦r ✈❛r✐❛çã♦ ❞❛ ✐♥t❡❣r❛❧ J ❛ s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✿
δJ ≡ ∂J ∂α α=0 dα
❛ss✐♠ ❝♦♠♦ ❛♥❛❧♦❣❛♠❡♥t❡ ❛ ✈❛r✐❛çã♦ ❞❡ y s❡rá
✶✳✶ ❖ Pr♦❜❧❡♠❛ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ✽ ❆❣♦r❛ ❡s❝r❡✈❡♠♦s ∂J ∂α α=0 dα=
Z x2
x1 ∂f ∂y − d dx ∂f ∂yx ∂y ∂αdαdx ❝♦♠♦ δJ =
Z x2
x1 ∂f ∂y − d dx ∂f ∂yx δydx.
❆ss✐♠ ❛ ❝♦♥❞✐çã♦ ∂J
∂α = 0 ♣❛r❛ q✉❡ ❛ ✐♥t❡❣r❛❧ J s❡❥❛ ❡st❛❝✐♦♥ár✐❛ é✱ ♣♦rt❛♥t♦✱
s✐♠♣❧✐s♠❡♥t❡ δJ = 0✳ ■♠♣♦♥❞♦ ❡ss❛ ❝♦♥❞✐çã♦ ❡♠
δJ =
Z x2
x1 ∂f ∂y − d dx ∂f ∂yx δydx
❝♦♠♦ δy é ❛r❜✐trár✐♦✱ ❝♦♥❝❧✉í♠♦s q✉❡✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ♦ t❡r♠♦ ❡♥tr❡ ❝♦❧❝❤❡t❡s ❞❡✈❡
❛♥✉❧❛r✲s❡✱ ♦ q✉❡ ❢♦r♥❡❝❡ ❛ ❡q✉❛çã♦ ❝❤❛♠❛❞❛ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r✳
∂f ∂y − d dx ∂f ∂yx
= 0.
▲❡♦♥❛r❞ ❊✉❧❡r ❝❤❡❣♦✉ ❛ ❡q✉❛çã♦ ❛❝✐♠❛ ❡♠ 1744✱ ♥♦ s❡✉ tr❛❜❛❧❤♦ ▼ét♦❞♦ ♣❛r❛ ❛❝❤❛r
❝✉r✈❛s ♣❧❛♥❛s q✉❡ ♠♦str❛♠ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ♠á①✐♠♦s ❡ ♠í♥✐♠♦s✳ P♦st❡r✐♦r♠❡♥t❡✱ ❡♠ 1760✱ ❏♦s❡♣❤ ▲♦✉✐s ▲❛❣r❛♥❣❡ ❛♣r♦❢✉♥❞♦✉ ❛ ❛♥á❧✐s❡ ♣ré✈✐❛ ❞❡ ❊✉❧❡r ♥♦ s❡✉ tr❛❜❛❧❤♦
❊♥s❛✐♦ s♦❜r❡ ✉♠ ♥♦✈♦ ♠ét♦❞♦ ♣❛r❛ ❞❡t❡r♠✐♥❛r ♦s ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❞❡ ❢ór♠✉❧❛s ✐♥t❡❣r❛✐s ✐♥❞❡✜♥✐❞❛s✳ P♦r ❡ss❡ ♠♦t✐✈♦✱ ❞❡♥tr♦ ❞♦ ❝♦♥t❡①t♦ ❞❛ ♠❡❝â♥✐❝❛✱ ❛ ❡①♣r❡ssã♦ ❛❝✐♠❛ é t❛♠❜é♠ ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡✳ ❆ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r é ✉♠❛ ❝♦♥❞✐çã♦ ♥❡❝❡ssár✐❛✱ ♣♦ré♠ ♥ã♦ s✉✜❝✐❡♥t❡✱ ♣❛r❛ q✉❡ ❛ ✐♥t❡❣r❛❧ J s❡❥❛ ❡st❛❝✐♦♥ár✐❛✳ ❆❧é♠ ❞✐ss♦✱ ♥❛
❞❡❞✉çã♦ ❛❝✐♠❛ ♠♦str❛❞❛ ❢♦✐ ❢❡✐t❛ ❛ s✉♣♦s✐çã♦ ✐♠♣❧í❝✐t❛ ❞❡ q✉❡ ❛ s♦❧✉çã♦ ♣r♦❝✉r❛❞❛ y(x)
s❡❥❛ ❛♦ ♠❡♥♦s ❞✉❛s ✈❡③❡s ❞✐❢❡r❡♥❝✐á✈❡❧✳ ❍á s✐t✉❛çõ❡s ❡♠ q✉❡ s♦❧✉çõ❡s ♥ã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s ❞♦ ♣r♦❜❧❡♠❛ ✈❛r✐❛❝✐♦♥❛❧ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s✱ ❡ q✉❡ ♥ã♦ sã♦ s♦❧✉çõ❡s ❞❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✳
✶✳✶✳✶ ❆ ■❞❡♥t✐❞❛❞❡ ❞❡ ❇❡❧tr❛♠✐
◗✉❛♥❞♦ ❛ ❢✉♥çã♦ f ♥♦ ❢✉♥❝✐♦♥❛❧ ✐♥t❡❣r❛❧ J ♥ã♦ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❛ ✈❛r✐á✈❡❧
✐♥❞❡♣❡♥❞❡♥t❡ x✱ é ♣♦ssí✈❡❧ r❡❞✉③✐r ❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ à s❡❣✉✐♥t❡ ✐❞❡♥t✐❞❛❞❡✱
❝♦♠♦ ❛♣r❡s❡♥t❛❞❛ ❡♠ ❬✶✵❪✿
f −yx ∂f ∂yx
✶✳✶ ❖ Pr♦❜❧❡♠❛ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧ ✾
❝♦♠ C ❝♦♥st❛♥t❡✳ P❛r❛ ❞❡❞✉③✐r ❡ss❛ ✐❞❡♥t✐❞❛❞❡✱ ❝♦♥s✐❞❡r❡♠♦s ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❞❡r✐✈❛❞❛
t♦t❛❧ ❞❛ ❢✉♥çã♦✱ q✉❡ ♥❛❞❛ ♠❛✐s é q✉❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❞❡ ❞✐❢❡r❡♥❝✐❛✐s ❝✉❥♦s ❝♦♠♣♦♥❡♥t❡s ✭❝♦❡✜❝✐❡♥t❡s✮ sã♦ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ ❢✉♥çã♦ f(y, yx, x)✿
df dx = ∂f ∂y dy dx + ∂f ∂yx dyx dx + ∂f ∂x ✐st♦ é✱ df dx = ∂f ∂yyx+
∂f ∂yx
yxx+ ∂f ∂x
♦♥❞❡ ♣♦❞❡♠♦s ✐s♦❧❛r
∂f ∂yyx =
df dx −
∂f ∂yx
yxx− ∂f ∂x
♠✉❧t✐♣❧✐❝❛♥❞♦ ❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✱ ❝♦♠♦ ❞❛❞❛ ❛❜❛✐①♦
∂f ∂y − d dx ∂f ∂yx = 0
♣♦r yx✱ ♦❜t❡♠♦s
yx ∂f ∂y −yx
d dx ∂f ∂yx
= 0. ✭✶✳✶✮
❙✉❜st✐t✉✐♥❞♦
∂f ∂yyx =
df dx −
∂f ∂yx
yxx− ∂f ∂x
♥❛ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧ ❡♠ (1.1)✜❝❛♠♦s✿
df dx −
∂f ∂yx
yxx− ∂f ∂x −yx
d dx ∂f ∂yx
= 0.
❈♦♠♦ d dx yx ∂f ∂yx
=yxx
∂f ∂yx
+yx
d dx ∂f ∂yx ,
q✉❡ ♥❛❞❛ ♠❛✐s é q✉❡ ❛ ❞❡r✐✈❛❞❛ ❞❡ ✉♠ ♣r♦❞✉t♦✱ t❡♠♦s ❝♦♠ ✉♠ ♣❡q✉❡♥♦ r❡❛rr❛♥❥♦✱ ✉♠❛ ❢♦r♠❛ ❛❧t❡r♥❛t✐✈❛ ❞❛ ❡q✉❛çã♦ ❞❡ ❊✉❧❡r✿
−∂f
∂x + d dx
f −yx
∂f ∂yx
= 0.
❈♦♠♦ ♦❜s❡r✈❛❞♦ ❛❝✐♠❛✱ ❝❛s♦f ♥ã♦ ❞❡♣❡♥❞❛ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❡x✱ ❡♥tã♦ ∂f
∂x = 0✱ ❛ss✐♠
❛ ❡q✉❛çã♦ ❛❝✐♠❛ r❡❞✉③✲s❡ ❛✿
d dx
f −yx
∂f ∂yx
= 0,
✶✳✷ ❆♣❧✐❝❛çã♦ ❞❛ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r✿ ▼❡♥♦r ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ✶✵
✶✳✷ ❆♣❧✐❝❛çã♦ ❞❛ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r✿ ▼❡♥♦r ❉✐stâ♥❝✐❛
❊♥tr❡ ❉♦✐s P♦♥t♦s
❖ ❡❧❡♠❡♥t♦ ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ❛r❝♦ ♥♦ ♣❧❛♥♦ é ❞❛❞♦ ♣♦rds=pdx2+dy2✱ ❝♦❧♦❝❛♥❞♦
dx2 ❡♠ ❡✈✐❞ê♥❝✐❛ t❡♠♦s q✉❡
ds = s 1 + dy dx 2
dx✱ ♦ ❝♦♠♣r✐♠❡♥t♦ t♦t❛❧ ❞❛ ❝✉r✈❛ ♥♦
♣❧❛♥♦ ❧✐❣❛♥❞♦ ♦s ♣♦♥t♦s ❞❡ ❝♦♦r❞❡♥❛❞❛s P = (x1, y1) ❡ Q = (x2, y2) é L =
Z Q
P
ds =
Z x2
x1 s 1 + dy dx 2
dx✱ q✉❡r❡♠♦s q✉❡ ❛ ❝✉r✈❛ q✉❡ ✉♥❡ ♦s ♣♦♥t♦sP ❡Q t❡♥❤❛ ❝♦♠♣r✐♠❡♥t♦
♠í♥✐♠♦✱ ♣❛r❛ t❛❧y(x)t❡♠ q✉❡ s❡r ♠í♥✐♠♦✱ ❡♥tã♦✱ ❝❤❛♠❛♥❞♦ ❞❡f =
s 1 + dy dx 2 ✳ ❈♦♠♦
f ♥ã♦ ❞❡♣❡♥❞❡ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞❡y✱ t❡♠♦s q✉❡✿
d dx dy dx s 1 + dy dx
2 = 0,
❛ss✐♠✱ t❡♠✲s❡✿ dy dx s 1 + dy dx
2 =C,
❡❧❡✈❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ❛♦ q✉❛❞r❛❞♦ ✜❝❛♠♦s ❝♦♠✿
dy dx
2
=C2
" 1 + dy dx 2# ❡♥tã♦ dy dx 2
−C2
dy dx
2
=C2
❛ss✐♠ ❝♦❧♦❝❛♥❞♦
dy dx
2
❡♠ ❡✈✐❞ê♥❝✐❛ ❡ ♦ ✐s♦❧❛♥❞♦ ♥❛ ❡q✉❛çã♦ ✜❝❛♠♦s ❝♦♠✿
dy dx =±
r r
1−r =a,
♦♥❞❡ a ∈R✱r =C2 ❡
✶✳✷ ❆♣❧✐❝❛çã♦ ❞❛ ❊q✉❛çã♦ ❞❡ ❊✉❧❡r✿ ▼❡♥♦r ❉✐stâ♥❝✐❛ ❊♥tr❡ ❉♦✐s P♦♥t♦s ✶✶
❆ ❡q✉❛çã♦ ❞✐❢❡r❡♥❝✐❛❧✿ dy
dx = a ❢♦r♥❡❝❡ ❛ s♦❧✉çã♦ ❣❡r❛❧ ♣❛r❛ y(x) = ax+b✱ ♦♥❞❡ b
é ✉♠❛ ❝♦♥st❛♥t❡ ❞❡ ✐♥t❡❣r❛çã♦✳ ❈♦♠♦ ❛ ❝✉r✈❛ y(x) ❞❡✈❡ ♣❛ss❛r ♣❡❧♦s ♣♦♥t♦s ❞✐st✐♥t♦s (x1, y1) ❡ (x2, y2) ♦♥❞❡ ❛s ❝♦♥st❛♥t❡s a ❡ b sã♦ ❞❡t❡r♠✐♥❛❞❛s ♣❡❧❛ r❡s♦❧✉çã♦ ❞♦ s❡❣✉✐♥t❡
s✐st❡♠❛ ❞❡ ❡q✉❛çõ❡s ❧✐♥❡❛r❡s✿
ax1+b=y1 ❡ ax2+b=y2,
✉t✐❧✐③❛♥❞♦ ♦ ♠ét♦❞♦ ❞❛ ❛❞✐çã♦ ♣❛r❛ ❡♥❝♦♥tr❛r♠♦s ♦s ❝♦❡✜❝✐❡♥t❡s a ❡ b t❡♠♦s q✉❡✿
a= y1−y2
x1−x2
❡ b= y2x1−y1x2
x1−x2
❡
y(x) = y1 −y2
x1 −x2
x+ y2x1−y1x2
x1−x2
❈❛♣ít✉❧♦
2
❆ ❍✐stór✐❛ ❞❛ ❈✉r✈❛ ❞❡ ▼❡♥♦r ❚❡♠♣♦
❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ t❡✈❡ ❣r❛♥❞❡ ✐♥✢✉ê♥❝✐❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧✳ ❊❧❡ ♥❛s❝❡✉ ❡♠ ❇❛s✐❧❡✐❛ ❡♠ 1667✱ ✜❧❤♦ ❞❡ ◆✐❝♦❧❛✉s ❇❡r♥♦✉❧❧✐✱ ✉♠ ❜♦t✐❝ár✐♦✱ ❡ s✉❛ ❡s♣♦s❛✱
▼❛r❣❛r❡t❤❛ ❙❝❤ö♥❛✉❡r✱ ♦♥❞❡ ❝♦♠❡ç♦✉ ❛ ❡st✉❞❛r ♠❡❞✐❝✐♥❛ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇❛s❡❧✳ ❙❡✉ ♣❛✐ ❞❡s❡❥❛✈❛ q✉❡ s❡✉s ❡st✉❞♦s ♦ t♦r♥❛ss❡ ❛♣t♦ ♣❛r❛ q✉❡ ❡❧❡ ❛ss✉♠✐ss❡ ♦ ❝♦♠ér❝✐♦ ❞❡ ❡s♣❡❝✐❛r✐❛s ❞❛ ❢❛♠í❧✐❛✱ ♠❛s ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ♥ã♦ ❣♦st❛✈❛ ❞❡ ♥❡❣ó❝✐♦s ❡ ❝♦♥✈❡♥❝❡✉ ♦ ♣❛✐ ❛ ❞❡✐①á✲❧♦ ❡st✉❞❛r ♠❡❞✐❝✐♥❛✳ ◆♦ ❡♥t❛♥t♦✱ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✱ t❛♠❜é♠ ♥ã♦ s❡ ✐♥t❡r❡ss♦✉ ♣❡❧♦s ❡st✉❞♦s ❞❡ ♠❡❞✐❝✐♥❛ ❡ ❝♦♠❡ç♦✉ ❛ ❡st✉❞❛r ♠❛t❡♠át✐❝❛ ❝♦♠ s❡✉ ✐r♠ã♦ ♠❛✐s ✈❡❧❤♦ ❏❛❝q✉❡s✳ ❆♦ ❧♦♥❣♦ ❞❛ ❡❞✉❝❛çã♦ ❞❡ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ ♥❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇❛s❡❧✱ ♦s ✐r♠ã♦s ❇❡r♥♦✉❧❧✐ tr❛❜❛❧❤❛r❛♠ ❥✉♥t♦s ❡ ❣❛st❛r❛♠ ♠✉✐t♦ ❞♦ s❡✉ t❡♠♣♦ ❛ ❡st✉❞❛r ♦ r❡❝é♠✲❞❡s❝♦❜❡rt♦ ❝á❧❝✉❧♦ ✐♥✜♥✐t❡s✐♠❛❧✳ ❊❧❡s ❡st❛✈❛♠ ❡♥tr❡ ♦s ♣r✐♠❡✐r♦s ♠❛t❡♠át✐❝♦s✱ ♥ã♦ só ♣♦r ❡st✉❞❛r ❡ ❝♦♠♣r❡❡♥❞❡r ♦ ❝á❧❝✉❧♦✱ ♠❛s ♣♦r ❛♣❧✐❝á✲❧♦ ❛ ✈ár✐♦s ♣r♦❜❧❡♠❛s✳ ❯♠ ❞❡st❡s ♣r♦❜❧❡♠❛s✱ q✉❡ ♦ ❢❡③ s❡r ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ♣❛✐ ❞♦ ❈á❧❝✉❧♦ ❱❛r✐❛❝✐♦♥❛❧✱ ❢♦✐ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❇r❛q✉✐stó❝r♦♥❛✳
❊♠ ❥✉♥❤♦ ❞❡ 1696✱ ❇❡r♥♦✉❧❧✐ ❞❡s❛✜♦✉ ❛s ♠❡♥t❡s ♠❛✐s ❜r✐❧❤❛♥t❡s ❞❡ s✉❛ é♣♦❝❛ ❛
r❡s♦❧✈❡r ✉♠ ♣r♦❜❧❡♠❛ q✉❡ ❡❧❡ ❛♣r❡s❡♥t♦✉ ♥❛ r❡✈✐st❛ ❝✐❡♥tí✜❝❛ ❆❝t❛ ❊r✉❞✐t♦r✉♠ ✭r❡✈✐st❛ ❞♦s ❡r✉❞✐t♦s✮✱ q✉❡ ❢♦✐ ✉♠❛ r❡✈✐st❛ ❝✐❡♥tí✜❝❛ ♠❡♥s❛❧ ❛❧❡♠ã ♣✉❜❧✐❝❛❞❛ ❡♥tr❡ ✶✻✽✷ ❡ ✶✼✽✷✱ ♠❛♥t✐❞❛ ♣♦r ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐③✱ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ ❡❧❡ ❥á ❤❛✈✐❛ r❡s♦❧✈✐❞♦✳ ❊✐s ❛ ♠♦t✐✈❛çã♦ ❞❡ ❇❡r♥♦✉❧❧✐ ❛♦s ❣ê♥✐♦s ❞❡ s❡✉ t❡♠♣♦✱ ❞❛❞❛ ❡♠ [✶]✿
✶✸
❝♦♠✉♥✐q✉❡ ❛ s♦❧✉❝ã♦ ❞♦ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦✱ ❊✉ ♦ ❞❡❝❧❛r❛r❡✐ ♣✉❜❧✐❝❛♠❡♥t❡ ♠❡r❡❝❡❞♦r ❞❡ ❡❧♦❣✐♦✳
❖ ❞❡s❛✜♦ ♣r♦♣♦st♦ ♣♦r ❇❡r♥♦✉❧❧✐ tr❛t❛✈❛✲s❡ ❞❡ ❡♥❝♦♥tr❛r q✉❛❧ ❞❡✈❡r✐❛ s❡r ❛ ❢♦r♠❛ ❞❡ ✉♠❛ r❛♠♣❛ ♣❛r❛ q✉❡ ✉♠❛ ♣❛rtí❝✉❧❛✱ ❞❡s❧✐③❛♥❞♦ ♣♦r ❡❧❛ ❛ ♣❛rt✐r ❞♦ r❡♣♦✉s♦ ❡ s♦❜ ❛ ❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡✱ ❣❛st❡ ♦ ♠❡♥♦r t❡♠♣♦ ♣♦ssí✈❡❧ ♣❛r❛ ❛t✐♥❣✐r ♦✉tr♦ ♣♦♥t♦ ♠❛✐s ❜❛✐①♦ ❞❛ tr❛❥❡tór✐❛✳ ❖ ❝♦♥t❡ú❞♦ ❞❛ ♣✉❜❧✐❝❛çã♦ ♥❛ ❧í♥❣✉❛ ❧❛t✐♥❛ ❡r❛✱ ❝♦♠ ❡♠ [✶]✳
❉❛t✐s ✐♥ ♣❧❛♥♦ ✈❡rt✐❝❛❧✐ ❞✉♦❜✉s ♣✉♥❝t✐s A ❡t B ❛ss✐❣♥❛r❡ ♠♦❜✐❧✐ M✱ ✈✐❛♠ AM B ♣❡r
q✉❛♠ ❣r❛✈✐t❛t❡ s✉❛ ❞❡s❝❡♥❞s ❡t ♠♦✈❡r✐ ✐♥❝✐♣✐❡♥s ❛ ♣✉♥❝t♦A✱ ❜r❡✈✐ss✐♠♦ t❡♠♣♦r❡ ♣❡r✈❡♥✐❛t
❛❞ ❛❧t❡r✉♠ ♣✉♥❝t✉♠B✳ ✭❙❡❥❛♠A❡B ❞♦✐s ♣♦♥t♦s ❞❡ ✉♠ ♣❧❛♥♦ ✈❡rt✐❝❛❧✳ ❊♥❝♦♥tr❡ ❛ ❝✉r✈❛
♥❛ q✉❛❧ ✉♠❛ ♣❛rtí❝✉❧❛ M s✉❥❡✐t❛ s♦♠❡♥t❡ à ❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡✱ ❞❡s❝r❡✈❡ ❛ tr❛❥❡tór✐❛ ♠❛✐s
rá♣✐❞❛ ❡♥tr❡ ♦s ♣♦♥t♦s A ❡B✮✳
❊ss❡ ❞❡s❛✜♦ ❢♦✐ ❡♥❝❛♠✐♥❤❛❞♦ ♣♦r ❝❛rt❛ às ♠❡♥t❡s ♠❛✐s ❜r✐❧❤❛♥t❡s ❞♦ ♠✉♥❞♦ ❞❛ é♣♦❝❛ ❞❛♥❞♦✲❧❤❡s ✉♠ ♣r❛③♦ ❞❡ s❡✐s ♠❡s❡s✱ ❞❡♣♦✐s ♣r♦rr♦❣❛❞♦s ♣♦r ♠❛✐s q✉❛tr♦ ♠❡s❡s ♣❛r❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❢♦ss❡ s♦❧✉❝✐♦♥❛❞♦✳ ❆❧é♠ ❞♦ ♣ró♣r✐♦ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐✱ ♦✉tr♦s ❝✐♥❝♦ ♠❛t❡♠át✐❝♦s ❛♣r❡s❡♥t❛r❛♠ s♦❧✉çõ❡s ♦r✐❣✐♥❛✐s ♣❛r❛ ♦ ♣r♦❜❧❡♠❛✿ ❏♦❤❛♥♥ ❇❡r♥♦✉❧❧✐ (1667−1748)❀ ❙✐r
■s❛❛❝ ◆❡✇t♦♥ (1643−1727)❀ ❏❛❝q✉❡s ❇❡r♥♦✉❧❧✐ (1654−1705)❀ ●♦tt❢r✐❡❞ ❲✐❧❤❡❧♠ ▲❡✐❜♥✐③
(1646−1716)❀ ❊❤r❡♥❢r✐❡❞ ❲❛❧t❤❡r ✈♦♥ ❚s❝❤✐r♥❤❛✉s(1651−1708)❡ ●✉✐❧❧❛✉♠❡ ❞❡ ▲✬❍ô♣✐t❛❧
(1661−1704).
❋✐❣✉r❛ ✷✳✶✿ ■♠❛❣❡♥s✳
❈❛♣ít✉❧♦
3
❈✐❝❧ó✐❞❡✿ ❆ ❍❡❧❡♥❛ ❞❛ ●❡♦♠❡tr✐❛
❆ ❝✐❝❧ó✐❞❡ ❢♦✐ ♣❡r❝❡❜✐❞❛ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣❡❧♦ ❢r❛♥❝ês✱ ❈❤❛r❧❡s ❇♦✈❡❧❧❡s ✭1479−1566✮✱
q✉❡ ♥✉♠ tr❛❜❛❧❤♦ ❞❡ ❣❡♦♠❡tr✐❛ ♣✉❜❧✐❝❛❞♦ ❡♠ P❛r✐s✱ ❡♠1501✱ s❡ r❡❢❡r❡ ❛ ❡ss❛ ❝✉r✈❛ ❧✐❣❛♥❞♦✲
❛ ❝♦♠ ♦ ♣r♦❜❧❡♠❛ ❞❛ q✉❛❞r❛t✉r❛ ❞♦ ❝ír❝✉❧♦✳ ❖s ♣r✐♠❡✐r♦s ❡st✉❞♦s r✐❣♦r♦s♦s q✉❡ s❡ t❡♠ ❝♦♥❤❡❝✐♠❡♥t♦ sã♦ ❞❡✈✐❞♦s ❛ ●✐❧❡s P❡rs♦♥ ❞❡ ❘♦❜❡r✈❛❧ ✭1602 −1675✮ q✉❡ ❛ ❝❤❛♠♦✉ ❞❡
tr♦❝❤ó✐❞❡ ✭r♦❞❛ ❡♠ ❣r❡❣♦✮ ❡ ❛ ❊✈❛♥❣❡❧✐st❛ ❚♦r✐❝❡❧❧✐ ✭1608−1647✮✱ ✉♠ ❞✐s❝í♣✉❧♦ ❞❡ ●❛❧✐❧❡✉
●❛❧✐❧❡✐ ✭1564−1642✮✳ ❖ ♣ró♣r✐♦ ●❛❧✐❧❡✉ ●❛❧✐❧❡✐ t❛♠❜❡♠ ❡st✉❞♦✉ ❛ ❝✉r✈❛ t❡♥❞♦ ✐♥❝❧✉s✐✈❡
❛ ❝❤❛♠❛❞♦ ❞❡ ❝✐❝❧ó✐❞❡ ❡ r❡❢❡r✐✉✲s❡ ❛ s✉❛ ❢♦r♠❛ ❣r❛❝✐♦s❛✱ ❛♣♦♥t❛♥❞♦✲❛ ❝♦♠♦ s✉❣❡stã♦ ♣❛r❛ ♦ ♣❡r✜❧ ❞♦s ❛r❝♦s ❞❡ ❝♦♥str✉❝õ❡s ❡♠ ❛rq✉✐t❡t✉r❛ ❡ ❛ ❇❧❛✐s❡ P❛s❝❛❧ ✭1623−1662✮ q✉❡ ❛
❝❤❛♠♦✉ ❞❡ r♦✉❧❡tt❡ ❛✜r♠❛♥❞♦ s♦❜r❡ ❡❧❛ ❡♠ ❬✷❪✿
✏❆ ❝✐❝❧ó✐❞❡ é ✉♠❛ ❝✉r✈❛ tã♦ ✉s✉❛❧ ❡ ❝♦rr❡♥t❡ q✉❡ ❞❡♣♦✐s ❞❛ r❡t❛ ❡ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ♥❡♥❤✉♠❛ ♦✉tr❛ ❝✉r✈❛ é tã♦ ❝♦♠✉♠❡♥t❡ ❡♥❝♦♥tr❛❞❛✳ ➱ ❞❡s❝r✐t❛ tã♦ ❢r❡q✉❡♥t❡♠❡♥t❡ ❞✐❛♥t❡ ❞❡ ♥♦ss♦s ♦❧❤♦s q✉❡ é s✉r♣r❡❡♥❞❡♥t❡ q✉❡ ♥ã♦ t❡♥❤❛ s✐❞♦ ❝♦♥s✐❞❡r❛❞❛ ♣❡❧♦s ❛♥t✐❣♦s✑✳
◆❛ é♣♦❝❛✱ ❤❛✈✐❛ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♥♦✈❛s ❝✉r✈❛s✱ ♣❛r❛ t❡st❛r ❛ ❡✜❝✐ê♥❝✐❛ ❞❡ ♥♦✈♦s ♠ét♦❞♦s✱ ❛s ❝✉r✈❛s ❝✐❝❧ó✐❞❛✐s✱ ❡♥tã♦✱ q✉❡ sã♦ ❛q✉❡❧❛s ❣❡r❛❞❛s ♣♦r ✉♠ ♣♦♥t♦ ❞❡ ✉♠ ❝ír❝✉❧♦ q✉❡ r♦❞❛ s❡♠ r❡s✈❛❧❛r s♦❜r❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ❝❤❛♠❛❞❛ ❝✐❝❧ó✐❞❡ q✉❛♥❞♦ ♦ ❝ír❝✉❧♦ ❣❡r❛❞♦r r♦❞❛ s♦❜r❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ ✐♥✜♥✐t♦✱ ✐st♦ é ✉♠ r❡t❛✱ ❡♣✐❝✐❝❧ó✐❞❡✱ q✉❛♥❞♦ ♦ ❝ír❝✉❧♦ ❣❡r❛❞♦r r♦❞❛ s♦❜r❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ ✜♥✐t♦✱ ❡①t❡r✐♦r ❛ ❡❧❛ ❡ ❤✐♣♦❝✐❝❧ó✐❞❡✱ q✉❛♥❞♦ ♦ ❝ír❝✉❧♦ ❣❡r❛❞♦r r♦❞❛ s♦❜r❡ ✉♠❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❞❡ r❛✐♦ ✜♥✐t♦✱ ✐♥t❡r✐♦r ❛ ❡❧❛✳
❆s ❝✉r✈❛s ❝✐❝❧♦✐❞❛✐s✱ ❡♥tã♦✱ ❧♦❣♦ s❡ t♦r♥❛r❛♠ ♣♦♣✉❧❛r ❡♥tr❡ ♦s ♠❛t❡♠át✐❝♦s✱ s❡♥❞♦ ✐♠❡♥s❛♠❡♥t❡ ❡st✉❞❛ ♣♦r ❝é❧❡❜r❡ ♠❛t❡♠át✐❝♦s ❝♦♠♦✿ ▼❡rs❡♥♥❡✱ ❈❤r✐st♦♣❤❡r ❲r❡♥✱ P❛s❝❛❧✱ ❍✉②❣❡♥s✱ ♦s ✐r♠ã♦s ❇❡r♥♦✉❧❧✐✱ ◆❡✇t♦♥✱ ▲❡✐❜♥✐③✱ ❞❡♥tr❡ ♦✉tr♦s ❛❧é♠ ❞♦s ❛ ❝✐♠❛ ❝✐t❛❞♦s✳
❉❡s❛✜♦s ❢♦r❛♠ ❢❡✐t♦s às ❣r❛♥❞❡s ♠❡♥t❡s ❞♦ sé❝✉❧♦ XV II s♦❜r❡ ❛s ❝✉r✈❛s ❝✐❝❧♦✐❞❛✐s✱
✸✳✶ ❊q✉❛çõ❡s P❛r❛♠étr✐❝❛s ❞❛ ❈✐❝❧ó✐❞❡ ✶✺
❢ís✐❝❛s ❡ ❣❡♦♠étr✐❝❛s✱ ❢♦✐ ❡♠ ❞❡❝♦rrê♥❝✐❛ ❞❡st❛s ❞✐s♣✉t❛s✱ q✉❡ ❛s ❝✉r✈❛s ❝✐❝❧♦✐❞❛s ❡ ❡♠ ❡s♣❡❝✐❛❧ ❛ ❝✉r✈❛ ❝✐❝❧ó✐❞❡ ❢♦✐ ❝❤❛♠❛❞❛ ❞❡ ❛✿ ❍❡❧❡♥❛ ❞❛ ❣❡♦♠❡tr✐❛✳
✸✳✶ ❊q✉❛çõ❡s P❛r❛♠étr✐❝❛s ❞❛ ❈✐❝❧ó✐❞❡
❆ ❝✐❝❧ó✐❞❡ é ❢♦r♠❛❞❛ ♣❡❧❛ ❝✉r✈❛ tr❛ç❛❞❛ ♣♦r ✉♠ ♣♦♥t♦ q✉❛❧q✉❡r ❞❛ ❜♦r❞❛ ❞❡ ✉♠❛ r♦❞❛ q✉❡ r♦❧❛ s❡♠ ❞❡s❧✐③❛r ♣♦r ✉♠ ♣❧❛♥♦ ❤♦r✐③♦♥t❛❧✳
❋✐❣✉r❛ ✸✳✶✿ ❈✐❝❧ó✐❞❡✳
❖❜s❡r✈❛♥❞♦ ❛ ✜❣✉r❛ ❛❝✐♠❛✱ ♣❡r❝❡❜❡♠♦s q✉❡ ♦ s❡❣♠❡♥t♦OT ≡rθé ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦
❛r❝♦ P T✳ ❉❡s❡❥❛♠♦s ❛❣♦r❛ ❞❡t❡r♠✐♥❛r ✉♠❛ ❡q✉❛çã♦ q✉❡ ❞❡s❝r❡✈❛ ❝♦rr❡t❛♠❡♥t❡ ❛ ♣♦s✐çã♦
❞♦ ♣♦♥t♦ P✱ ❝♦♠♦ ❡♠ ❬✺❪✱ ♣❛r❛ ✐ss♦ ✈❛♠♦s ❞❡t❡r♠✐♥❛r ❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❞❡st❡
♣♦♥t♦ ❡♠ r❡❧❛çã♦ ❛ x❡ ❡♠ r❡❧❛çã♦ ❛ y✱ ❛♠❜❛s ❡♠ ❢✉♥çã♦ ❞❡r ❡ ❞❡ θ✳
P❡❧❛ ✜❣✉r❛ t❡♠♦s q✉❡✿
rθ=x+rsen(θ),
❛ss✐♠✱
x=rθ−rsen(θ),
❧♦❣♦
✸✳✶ ❊q✉❛çõ❡s P❛r❛♠étr✐❝❛s ❞❛ ❈✐❝❧ó✐❞❡ ✶✻
❛ss✐♠ ❞❡t❡r♠✐♥❛♠♦s ❛ ❡q✉❛çã♦ ♣❛r❛♠étr✐❝❛ ❞❛ ❡q✉❛çã♦ ❞❡s❡❥❛❞❛ ❡♠ r❡❧❛çã♦ ❛ x✳ ❆❣♦r❛
♣❛r❛ y t❡♠♦s✿
r=CQ+y,
♦♥❞❡ CQ=rcos(θ)✱ ❧♦❣♦
r =rcos(θ) +y,
✐st♦ é✱
y=r−rcos(θ),
♣♦rt❛♥t♦✱
y=r(1−cos(θ)).
❆ss✐♠ ❞❡t❡r♠✐♥❛♠♦s ❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s q✉❡ ❞❡t❡r♠✐♥❛♠ ♦ ♣♦♥t♦ P✱ ❞❡st❛
❢♦r♠❛ ❛ ❝✉r✈❛ q✉❡ ❞❡t❡♠ t♦❞♦s ♦s ♣♦♥t♦s ❞❡ P q✉❡ t❡♠ ❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❛❝✐♠❛
♠♦str❛❞❛s é ❛ ❝✉r✈❛ ❝✐❝❧ó✐❞❡✳
❈❛♣ít✉❧♦
4
❆s Pr♦♣r✐❡❞❛❞❡s ❞❛ ❈✐❝❧ó✐❞❡
❙ã♦ ✐♠♣r❡ss✐♦♥❛♥t❡s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❢ís✐❝❛s q✉❡ ❡ss❛ ❝✉r✈❛ ❛♣r❡s❡♥t❛✱ ❞❡♥tr❡ ❡❧❛s ❞❡st❛❝❛♠♦s✿ ❛ ❇r❛q✉✐stó❝r♦♥❛ ❡ ❛ ❚❛✉tó❝r♦♥❛ q✉❡ r❡s♦❧✈❡✉ ♦ ♣r♦❜❧❡♠❛ ❞♦ ♣ê♥❞✉❧♦ ✐só❝r♦♥♦✳
✹✳✶ ❆ ❇r❛q✉✐stó❝r♦♥❛
❖ ♣r♦❜❧❡♠❛ ❞❛ ❜r❛q✉✐stó❝r♦♥❛ ❣✐r❛ ❡♠ t♦r♥♦ ❞❡ s❡ ❡♥❝♦♥tr❛r ✉♠❛ ❝✉r✈❛ q✉❡ ❢❛ç❛ ❝♦♠ q✉❡ ✉♠ ♦❜❥❡t♦ ❞❡ ♠❛ss❛ m s✉❥❡✐t♦ ❛♣❡♥❛s à ❛❝❡❧❡r❛çã♦ ❣r❛✈✐t❛❝✐♦♥❛❧ ♣❡r❝♦rr❛ ❛ ❞✐stâ♥❝✐❛
❡♥tr❡ ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ♥♦ ♠❡♥♦r t❡♠♣♦ ♣♦ssí✈❡❧ ♣❛rt✐♥❞♦ ❞♦ r❡♣♦✉s♦✳ ❙❡❥❛ ❛ ❢✉♥çã♦
y(x) q✉❡ ❞❡✜♥❡ ❛ ❝✉r✈❛ ❞❡s❡❥❛❞❛ ❡ s❡❥❛♠ ♦s ♣♦♥t♦s P = (0,0) ❡ Q = (a, b) ❞✐st✐♥t♦s
♣❡rt❡♥❝❡♥t❡s ❛ ❡ss❛ ❝✉r✈❛✳ P❡❧♦ ❢❛t♦ ❞❡ s❡r ✉♠ s✐st❡♠❛ ❝♦♥s❡r✈❛t✐✈♦ ✭s❡♠ ♥❡♥❤✉♠❛ ♣❡r❞❛ ❞❡ ❡♥❡r❣✐❛✮ t❡♠♦s q✉❡ ❛ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛ é ❝♦♥s❡r✈❛❞❛✱ ❛ss✐♠✿
mv2
2 =mgy,
❛ss✐♠✱ v(x) = p2gy✱ ♣♦✐s ❛ ✈❡❧♦❝✐❞❛❞❡ ❞❛ ♣❛rtí❝✉❧❛ ❞❡♣❡♥❞❡ ❞❛ ❝♦♦r❞❡♥❛❞❛ x✳
❙❛❜❡♠♦s t❛♠❜é♠ q✉❡ ♦ t❡♠♣♦ ❞❡ q✉❡❞❛ ❞❛ ♣❛rtí❝✉❧❛ é ❞❛❞❛ ♣♦r✿
T[y(x)] =
Z ds
v , ♦♥❞❡ds
2
=dx2
+dy2
,
❧♦❣♦✱ t❡♠♦s
T[y(x)] =
Z sdx2+dy2
✹✳✶ ❆ ❇r❛q✉✐stó❝r♦♥❛ ✶✽
❛ss✐♠✱ ❝♦❧♦❝❛♥❞♦ dx2 ❡♠ ❡✈✐❞ê♥❝✐❛ ♥♦ ✐♥t❡❣r❛♥❞♦ ✜❝❛♠♦s ❝♦♠
T[y(x)] =
Z b
0
s
1 +dxdy2
2gy(x) dx.
❈❤❛♠❡♠♦s ❞❡ w=
s
1 +dydx2
2gy(x) q✉❡ é ❛ ❢✉♥çã♦ ❛ s❡r ✈❛r✐❛❞❛✳
❯t✐❧✐③❛r❡♠♦s ♣❛r❛ r❡s♦❧✈❡r ❡ss❡ ❢✉♥❝✐♦♥❛❧ ❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❇❡❧tr❛♠✐ ❡ ❢❛r❡♠♦s ❛ s❡❣✉✐♥t❡ s✐♣❧✐✜❝❛çã♦ ❞❡ ♥♦t❛çã♦ dy
dx ❝❤❛♠❛r❡♠♦s ❞❡y
′ ❛ss✐♠ t❡♠♦s✿
∂w ∂y′ =
1
2 1 +y
′2−12
2y′(2gy)−12 ,
❛ss✐♠ ❛♣❧✐❝❛♥❞♦ ♥❛ ✐❞❡♥t✐❞❛❞❡ ❞❡ ❇❡❧tr❛♠✐ ♣❛r❛ ∂w
∂y′ ✜❝❛♠♦s ❝♦♠
p
1 +y′2
√
2gy −
y′2
p
1 +y′2
1
√
2gy =C,
♦ ♣r✐♠❡✐r♦ ♠❡♠❜r♦ ❞❛ ❡q✉❛çã♦ t❡♠ ❝♦♠♦ ♠í♥✐♠♦ ♠ú❧t✐♣❧♦ ❝♦♠✉♠ (M.M.C)♦ t❡r♠♦
p
1 +y′2p2gy
❧♦❣♦ ❛ ❡q✉❛çã♦ ✜❝❛✿
1 +y′2
−y′2
p
(1 +y′2
) (2gy) =C,
✐st♦ é✱
1
p
(1 +y′2
) (2gy) =C
❡❧❡✈❛♥❞♦ ❛♠❜♦s ♦s ♠❡♠❜r♦s ❞❛ ❡q✉❛çã♦ ❛♦ q✉❛❞r❛❞♦ ✜❝❛♠♦s✿
y 1 +y′2
= 1
2gC2
❝❤❛♠❛♥❞♦ ❞❡ k = 1
2gC2 ✜❝❛♠♦s ❝♦♠
y 1 +y′2
=k,
✐st♦ é✱
y′2
= k−y
✹✳✶ ❆ ❇r❛q✉✐stó❝r♦♥❛ ✶✾
❛ss✐♠✱
y′ =
s k−y
y ,
✈♦❧t❛♥❞♦ ❛ ❡s❝r❡✈❡r y′ = dy
dx ❡ ✐s♦❧❛♥❞♦ dx t❡♠♦s
dx=
r y
k−ydy
❢❛ç❛♠♦s ✉♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ ❡ ❝❤❛♠❡♠♦s y = ksen2
(θ) ❝♦♠ ❞❡r✐✈❛❞❛ dy = 2ksen(θ)cos(θ)dθ
dx=
s
ksen2
(θ)
k−ksen2(θ)2ksen(θ)cos(θ)dθ
❝♦❧♦❝❛♥❞♦ ♥♦ ❞❡♥♦♠✐♥❞❛❞♦r ❞❛ ❢r❛çã♦ k ❡♠ ❡✈✐❞ê♥❝✐❛ ❡ ✉t✐❧✐③❛♥❞♦ ❛ r❡❧❛çã♦ ❢✉♥❞❛♠❡♥t❛❧
❞❛ tr✐❣♦♥♦♠❡tr✐❛✱ ❛ s❛❜❡r✿ sen2
(θ) +cos2
(θ) = 1 ❛ ❡q✉❛çã♦ ✜❝❛ ❡s❝r✐t❛ ❝♦♠♦
dx= 2ksen2
(θ)dθ
❛ss✐♠✱
x=
Z
2ksen2
(θ)dθ.
❯s❛r❡♠♦s ❛❣♦r❛ ♣❛r❛ r❡s♦❧✈❡r ❡st❛ ✐♥t❡❣r❛❧ ❛ s❡❣✉✐♥t❡ ✐❞❡♥t✐❞❛❞❡ tr✐❣♦♥♦♠étr✐❝❛
sen2
(θ) = 1−cos(2θ)
2
❡♥tã♦
x= 2k Z 1
−cos(2θ)
2 dθ
❡ ❛ss✐♠ ♦❜t❡♠♦s
x= 2k
θ
2−
1
4sen(2θ)
+C.
❙❡ ❝❤❛♠❛r♠♦s ❞❡
2θ =φ,
❡♥tã♦
x= 2k
φ
4 −
1
4sen(φ)
✹✳✷ ❆ ❚❛✉tó❝r♦♥❛ ✷✵
❆ss✐♠ ❝♦♥❝❧✉í♠♦s q✉❡
x= k
2(φ−sen(φ)) +C,
♦♥❞❡ C ∈R, s❡ ✜③❡r♠♦s C = 0, ❡♥tã♦✿
x= k
2(φ−sen(φ)).
P❛r❛ ❞❡t❡r♠✐♥❛r♠♦s ♦ ✈❛❧♦r ❞❡ y ❜❛st❛ ❝❛❧❝✉❧❛r ❛ ✐♥t❡❣r❛❧ ❞♦ ✐♥t❡❣r❛♥❞♦ s
k−y y
✉t✐❧✐③❛♥❞♦ ❛ ♠❡s♠❛ ♠✉❞❛♥ç❛ ❞❡ ✈❛r✐á✈❡❧ y=ksen2
(θ) ❡ dx= 2ksen2
(θ)dθ ❛ss✐♠
y =
Z sk
−ksen2
(θ)
ksen2
(θ) 2ksen
2
(θ)dθ,
❧♦❣♦
y =k Z
sen(2θ)dθ
❛ss✐♠✱
y=−k
2cos(2θ) +C,
s❡ ✜③❡r♠♦s C = k
2 ❡ 2θ=φ, ❡♥tã♦✿
y= k
2(1−cos(φ)),
q✉❡ sã♦ ❛s ❡q✉❛çõ❡s ♣❛r❛♠étr✐❝❛s ❞❛ ❝✐❝❧ó✐❞❡ ❡ ❝♦♠♦ q✉❡rí❛♠♦s ❞❡♠♦♥str❛r ❛ ❝✐❝❧ó✐❞❡ é ❜r❛q✉✐stó❝r♦♥❛✳
✹✳✷ ❆ ❚❛✉tó❝r♦♥❛
❯♠❛ ♦✉tr❛ ♣r♦♣r✐❡❞❛❞❡ ✐♠♣r❡ss✐♦♥❛♥t❡ ❞❛ ❝✐❝❧ó✐❞❡ é ♦ ❢❛t♦ ❞♦ t❡♠♣♦ ♣❛r❛ q✉❡ ✉♠ ❝♦r♣♦ ❞❡sç❛ ♣♦r ❡❧❛ ✭❝✐❝❧ó✐❞❡ ✐♥✈❡rt✐❞❛✮✱ ✐♥❞❡♣❡♥❞❛ ❞❛ ❛❧t✉r❛ q✉❡ ❢♦✐ ❧❛♥ç❛❞❛✱ ♠❛s ❛♣❡♥❛s ❞♦ r❛✐♦ ❞❛ ♠❡s♠❛✱ q✉❡ é ❞✉❛s ✈❡③❡s ♦ r❛✐♦ ❞♦ ❞✐s❝♦ q✉❡ ❞❡✉ ♦r✐❣❡♠ à ❝✐❝❧ó✐❞❡✱ ♣♦r ❛♣r❡s❡♥t❛r ❡ss❛ ♣r♦♣r✐❡❞❛❞❡ ❛ ❝✐❝❧♦í❞❡ ✐♥✈❡rt✐❞❛ é ❝❤❛♠❛❞❛ ❞❡ t❛✉t♦❝r♦♥❛ ✭❝✉r✈❛ ❞❡ ♠❡s♠♦ t❡♠♣♦✮✳
✹✳✷✳✶ Pr♦✈❛ ▼❛t❡♠át✐❝❛ ❞❡ q✉❡ ❛ ❈✐❝❧ó✐❞❡ é ❚❛✉tó❝r♦♥❛
✹✳✷ ❆ ❚❛✉tó❝r♦♥❛ ✷✶
❝♦♠ ❛✜♥❝♦ ✉♠❛ ✈❡rsã♦ ❣❡r❛❧ ❞♦ ♣r♦❜❧❡♠❛ ❞❛ t❛✉t♦❝rô♥✐❝❛ ✭Pr♦❜❧❡♠❛ ▼❡❝â♥✐❝♦ ❞❡ ❆❜❡❧✮✱ ♥♦♠✐♥❛❧♠❡♥t❡✱ ❞❛❞♦ à ❢✉♥çã♦ T(y) q✉❡ ❡s♣❡❝✐✜❝❛ ♦ t❡♠♣♦ t♦t❛❧ ❞❡ ❞❡s❝✐❞❛ ♣❛r❛ ✉♠❛
❞❛❞❛ ❛❧t✉r❛ ✐♥✐❝✐❛❧✱ ❡♥❝♦♥tr❛♥❞♦✲s❡ ✉♠❛ ❡q✉❛çã♦ q✉❡ ♥♦s ❢♦r♥❡❝❡ ❛ s♦❧✉çã♦✳ ❖ ♣r♦❜❧❡♠❛ ❞❛ ❚❛✉t♦❝rô♥✐❝❛ é ✉♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❞♦ ♣r♦❜❧❡♠❛ ♠❡❝â♥✐❝♦ ❞❡ ❆❜❡❧✱ q✉❛♥❞♦ T(y) é ✉♠❛
❝♦♥st❛♥t❡✳ ❆ s♦❧✉çã♦ ❞❡ ❆❜❡❧ ❝♦♠❡ç❛ ❝♦♠ ♦ Pr✐♥❝í♣✐♦ ❞❛ ❈♦♥s❡r✈❛çã♦ ❞❡ ❊♥❡r❣✐❛✳
❋✐❣✉r❛ ✹✳✶✿ ❚r❛❥❡tór✐❛✳
◆❛ ❞❛❞❛ ❝✉r✈❛ ❛❞♦t❛r❡♠♦s ❛ ♣♦s✐çã♦ ♠❛✐s ❜❛✐①♦ ❞❛ ♠❡s♠❛ ❝♦♠♦ s❡♥❞♦ O✱ σ ♦ ❛r❝♦ AP ❡ P ✉♠ ♣♦♥t♦ ❛r❜✐trár✐♦ ❡♥tr❡ AO✳ ❈♦♥s✐❞❡r❛♥❞♦ ✉♠ ❝♦r♣♦ ❞❡ ♠❛ss❛ m ♦♥❞❡ ❛t✉❡
❛♣❡♥❛s ❛ ❛❝❡❧❡r❛çã♦ ❞❛ ❣r❛✈✐❞❛❞❡ ♥❛ tr❛❥❡tór✐❛ P O✱ ❝♦♠♦ ❡♠ q✉❛❧q✉❡r ♦✉tr❛ t❡r❡♠♦s
❛ ❝♦♥s❡r✈❛çã♦ ❞❛ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛✱ ❥á q✉❡ ♦ s✐st❡♠❛ é ❝♦♥s❡r✈❛t✐✈♦✳ ❊♥tã♦ ❛ ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛ ♥♦ ♣♦♥t♦ P é ✐❣✉❛❧ à ❡♥❡r❣✐❛ ♠❡❝â♥✐❝❛ ♥♦ ♣♦♥t♦ O✱ ❛ss✐♠
mgy0 =mgy+
1
2m
dσ
dt 2
,
✈✐st♦ q✉❡✱ v = dσ
dt✳ ❆ss✐♠ dσ
dt 2
= 2 (gy0−gy)
❧♦❣♦✱ s❡❣✉❡ q✉❡
dσ dt
= dσ
dy dy dt =−
p
2g(y0−y),
