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❆ ♥❡✇ ❛♣♣r♦❛❝❤ ❢♦r ❙❲❊ s♦✉r❝❡ ◆✉♠❡r✐❝❛❧ ❞✐s❝r❡t✐③❛t✐♦♥s ❢♦r s❤❛❧❧♦✇ ✇❛t❡r t❡r♠s ❆r♥❛✉❞ ❉✉r❛♥ ❡q✉❛t✐♦♥s ✇✐t❤ s♦✉r❝❡ t❡r♠s ♦♥ ✉♥str✉❝t✉r❡❞ ●♦✈❡r♥✐♥❣ ♠❡s❤❡s ❊q✉❛t✐♦♥s ◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ t♦♣♦❣r❛♣❤② Pr❡✲❜❛❧❛♥❝❡❞ ❛♣♣r♦❛❝❤ ❆r♥❛✉❞ ❉✉r❛♥ ❋✐rst ♦r❞❡r s❝❤❡♠❡ ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ◆✉♠❡r✐❝❛❧ ■✸▼✱ ▼♦♥t♣❡❧❧✐❡r✱ ❯▼❘ ❈◆❘❙ ✺✶✹✾✱ ❋r❛♥❝❡ ✈❛❧✐❞❛t✐♦♥s ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✲ ❆❝❝♦✉♥t✐♥❣ ❏♦✐♥t ❲♦r❦ ✇✐t❤ ❋✳▼❛r❝❤❡ ✭■✸▼✱ ▼♦♥t♣❡❧❧✐❡r✱ ❯▼❘ ❈◆❘❙ ✺✶✹✾✱ ❋r❛♥❝❡✮ ❢r✐❝t✐♦♥❛❧ t❡r♠s ◗✳▲✐❛♥❣ ✭❙❈❊● ◆❡✇❝❛st❧❡✱ ❯♥✐t❡❞ ❑✐♥❣❞♦♠✮ ❚❤❡ ✶❉ s❝❤❡♠❡ ❈✳❇❡rt❤♦♥ ❛♥❞ ❘✳❚✉r♣❛✉❧t ✭▲❛❜♦r❛t♦✐r❡ ❞❡ ▼❛t❤❡♠❛t✐q✉❡s ❏❡❛♥ ▲❡r❛②✱ ❯♥✐✈❡rs✐té ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ❞❡ ◆❛♥t❡s✱ ❋r❛♥❝❡✮ ◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥s ❍❨P ✷✵✶✷ ❆r♥❛✉❞ ❉✉r❛♥ ✭■✸▼✮ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ✷✺✴✵✻✴✷✵✶✷ ✶ ✴ ✸✸

●♦✈❡r♥✐♥❣ ❊q✉❛t✐♦♥s ❆ ♥❡✇ ❛♣♣r♦❛❝❤ P❧❛♥ ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ❆r♥❛✉❞ ❉✉r❛♥ ●♦✈❡r♥✐♥❣ ❊q✉❛t✐♦♥s ●♦✈❡r♥✐♥❣ ❊q✉❛t✐♦♥s ✶ ◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ t♦♣♦❣r❛♣❤② Pr❡✲❜❛❧❛♥❝❡❞ ◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ t♦♣♦❣r❛♣❤② ✷ ❛♣♣r♦❛❝❤ ❋✐rst ♦r❞❡r s❝❤❡♠❡ Pr❡✲❜❛❧❛♥❝❡❞ ❛♣♣r♦❛❝❤ ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ❋✐rst ♦r❞❡r s❝❤❡♠❡ ◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥s ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ❆❝❝♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥s ❢r✐❝t✐♦♥❛❧ t❡r♠s ❚❤❡ ✶❉ s❝❤❡♠❡ ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ◆✉♠❡r✐❝❛❧ ❆❝❝♦✉♥t✐♥❣ ❢r✐❝t✐♦♥❛❧ t❡r♠s ✸ ✈❛❧✐❞❛t✐♦♥s ❚❤❡ ✶❉ s❝❤❡♠❡ ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥s ❆r♥❛✉❞ ❉✉r❛♥ ✭■✸▼✮ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ✷✺✴✵✻✴✷✵✶✷ ✷ ✴ ✸✸

●♦✈❡r♥✐♥❣ ❊q✉❛t✐♦♥s ❆ ♥❡✇ ❛♣♣r♦❛❝❤ ✷❉ s❤❛❧❧♦✇ ✇❛t❡r ❡q✉❛t✐♦♥s ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ❆r♥❛✉❞ ❉✉r❛♥ ✉ t + ∇ . H ( ✉ ) = −B ( ✉ , ③ ) − F ( ✉ ) , ✇✐t❤ ●♦✈❡r♥✐♥❣ ❊q✉❛t✐♦♥s  q ① q ②    ❤ q ✷ q ① q ② ◆✉♠❡r✐❝❛❧ ❤ + ✶ ① ✷ ❣❤ ✷ ✉ = q ① H ( ✉ , ③ ) = ( ❋ , ● )( ✉ , ③ ) =   tr❡❛t♠❡♥t ♦❢ t❤❡   ❤   q ✷ t♦♣♦❣r❛♣❤② q ② q ① q ② ❤ + ✶ ② ✷ ❣❤ ✷ Pr❡✲❜❛❧❛♥❝❡❞ ❤ ❛♣♣r♦❛❝❤ ❙♦✉r❝❡ t❡r♠s ✿ ❋✐rst ♦r❞❡r s❝❤❡♠❡ ❙❡❝♦♥❞ ♦r❞❡r � � ❡①t❡♥s✐♦♥ q ✷ ① + q ✷ q ✷ ① + q ✷ ② ② ◆✉♠❡r✐❝❛❧ F ( ✉ ) = t � � B ( ✉ , ③ ) = t ( ✵ , ❣❤ ∂ ① ③ , ❣❤ ∂ ② ③ ) ✵ , κ q ① , κ q ② ✈❛❧✐❞❛t✐♦♥s ❤ γ ❤ γ ❆❝❝♦✉♥t✐♥❣ ❢r✐❝t✐♦♥❛❧ t❡r♠s ❚❤❡ ✶❉ s❝❤❡♠❡ ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ❤ ✿ ✇❛t❡r ❤❡✐❣❤t ◆✉♠❡r✐❝❛❧ ❢r❡❡ s✉r❢❛❝❡ ✈❛❧✐❞❛t✐♦♥s η ✿ ❢r❡❡ s✉r❢❛❝❡ ✻ ✻ ③ ✿ t♦♣♦❣r❛♣❤② η ✲ ❤ ✉ = ( ✉ , ✈ ) ✿ ✈❡❧♦❝✐t② ✉ ✈❡❝t♦r t♦♣♦❣r❛♣❤② ✻ ③ q = ( q ① , q ② ) ✿ ❞✐s❝❤❛r❣❡ ✈❡❝t♦r ❆r♥❛✉❞ ❉✉r❛♥ ✭■✸▼✮ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ✷✺✴✵✻✴✷✵✶✷ ✸ ✴ ✸✸

◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ t♦♣♦❣r❛♣❤② ❆ ♥❡✇ ❛♣♣r♦❛❝❤ P❧❛♥ ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ❆r♥❛✉❞ ❉✉r❛♥ ●♦✈❡r♥✐♥❣ ❊q✉❛t✐♦♥s ✶ ●♦✈❡r♥✐♥❣ ❊q✉❛t✐♦♥s ◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ t♦♣♦❣r❛♣❤② Pr❡✲❜❛❧❛♥❝❡❞ ❛♣♣r♦❛❝❤ ✷ ◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ t♦♣♦❣r❛♣❤② ❋✐rst ♦r❞❡r s❝❤❡♠❡ ❙❡❝♦♥❞ ♦r❞❡r Pr❡✲❜❛❧❛♥❝❡❞ ❛♣♣r♦❛❝❤ ❡①t❡♥s✐♦♥ ❋✐rst ♦r❞❡r s❝❤❡♠❡ ◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥s ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ❆❝❝♦✉♥t✐♥❣ ◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥s ❢r✐❝t✐♦♥❛❧ t❡r♠s ❚❤❡ ✶❉ s❝❤❡♠❡ ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ◆✉♠❡r✐❝❛❧ ✸ ❆❝❝♦✉♥t✐♥❣ ❢r✐❝t✐♦♥❛❧ t❡r♠s ✈❛❧✐❞❛t✐♦♥s ❚❤❡ ✶❉ s❝❤❡♠❡ ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥ ◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥s ❆r♥❛✉❞ ❉✉r❛♥ ✭■✸▼✮ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ✷✺✴✵✻✴✷✵✶✷ ✹ ✴ ✸✸

◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ t♦♣♦❣r❛♣❤② Pr❡✲❜❛❧❛♥❝❡❞ ❛♣♣r♦❛❝❤ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ Pr❡✲❜❛❧❛♥❝❡❞ ❢♦r♠✉❧❛t✐♦♥ ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ❆r♥❛✉❞ ❉✉r❛♥ ◆✉♠❡r✐❝❛❧ r❡q✉✐r❡♠❡♥ts ✿ ●♦✈❡r♥✐♥❣ Pr❡s❡r✈❛t✐♦♥ ♦❢ ♠♦t✐♦♥❧❡ss st❡❛❞② st❛t❡s ✿ ✉ = ✵ ❛♥❞ η = ❝st ✳ ❊q✉❛t✐♦♥s ❤✲♣♦s✐t✐✈✐t② ♣r❡s❡r✈❛t✐♦♥✳ ◆✉♠❡r✐❝❛❧ tr❡❛t♠❡♥t ♦❢ t❤❡ t♦♣♦❣r❛♣❤② ⇒ ✧♣r❡ ✲ ❜❛❧❛♥❝❡❞✧ s❡t ♦❢ ❡q✉❛t✐♦♥s ❬ ◗✳ ▲✐❛♥❣ ❛♥❞ ❋✳ ▼❛r❝❤❡✳ ◆✉♠❡r✐❝❛❧ Pr❡✲❜❛❧❛♥❝❡❞ r❡s♦❧✉t✐♦♥ ♦❢ ✇❡❧❧✲❜❛❧❛♥❝❡❞ s❤❛❧❧♦✇ ✇❛t❡r ❡q✉❛t✐♦♥s ✇✐t❤ ❝♦♠♣❧❡① s♦✉r❝❡ t❡r♠s✳ ❛♣♣r♦❛❝❤ ❋✐rst ♦r❞❡r s❝❤❡♠❡ ❆❞✈❛♥❝❡s ✐♥ ❲❛t❡r ❘❡s♦✉r❝❡s✱ ✷✵✵✾✳❪ ❙❡❝♦♥❞ ♦r❞❡r ▼❛✐♥ ✐❞❡❛ ✿ ❯s❡ η ❛s ❛ ❝♦♥s❡r✈❛t✐✈❡ ✈❛r✐❛❜❧❡ ✭✐♥st❡❛❞ ♦❢ ❤ ✮✳ ❡①t❡♥s✐♦♥ ◆✉♠❡r✐❝❛❧ ✈❛❧✐❞❛t✐♦♥s ❉❡r✐✈❡❞ ❢♦r♠✉❧❛t✐♦♥ ❢♦r t❤❡ ❢r✐❝t✐♦♥❧❡ss ♠♦❞❡❧ ❆❝❝♦✉♥t✐♥❣ ❢r✐❝t✐♦♥❛❧ t❡r♠s ❚❤❡ ✶❉ s❝❤❡♠❡ ❱ t + ∇ · ˜ ❍ ( ❱ , ③ ) = ˜ ❙ ( ❱ , ③ ) , ✇✐t❤ ❙❡❝♦♥❞ ♦r❞❡r ❡①t❡♥s✐♦♥    q ① q ②  η ◆✉♠❡r✐❝❛❧ ✷ ❣ ( η ✷ − ✷ η ③ ) ˜ ✈❛❧✐❞❛t✐♦♥s ✉q ① + ✶ ❱ = ❍ ( ❱ , ③ ) = ✈q ① q ①     ✷ ❣ ( η ✷ − ✷ η ③ ) ✈q ② + ✶ q ② ✉q ②   ✵ ˜  . ❙ ( ❱ , ③ ) = − ❣ η ③ ①  − ❣ η ③ ② ❆r♥❛✉❞ ❉✉r❛♥ ✭■✸▼✮ ❆ ♥❡✇ ❛♣♣r♦❛❝❤ ❢♦r ❙❲❊ s♦✉r❝❡ t❡r♠s ✷✺✴✵✻✴✷✵✶✷ ✺ ✴ ✸✸