ϋϛϧτϯɾϠίϏํఔࣜͷղͷੑ࣭ͱ෺ཧతҙຯ E. Nakai 2026 ೥ 4 ݄ 5 ೔ 1 ͸͡Ίʹ ຊߘͰ͸ɺղੳྗֶͷجૅΛֶͼऴ͑ͨํΛର৅ͱͯ͠ɺϋϛϧτϯɾϠίϏํఔࣜͷղ͕ຬͨ ͢ੑ࣭Λࣔ͢͜ͱͰɺͦͷ෺ཧతͳҙຯΛ໌Β͔ʹ͠·͢ɻඞཁͳલఏ஌ࣝ͸࣍ͷͱ͓ΓͰ͢ɻ 1. ࠷খ࡞༻ͷݪཧΛ༻͍ͯ࡞༻ੵ෼͔ΒϥάϥϯδϡͷӡಈํఔࣜΛಋ͘ྲྀΕ 2. ϥάϥϯδϡܗࣜΛϋϛϧτχΞϯܗࣜʹม׵ͯ͠ϋϛϧτϯͷӡಈํఔࣜΛಋ͘ྲྀΕ 3. ϋϛϧτϯܗࣜʹ͓͚Δਖ਼४ม׵ͷੑ࣭ͱ฼ؔ਺ͷ໾ׂ ຊߘͰѻ͏෺ཧܥ͸ɺҰൠԽ࠲ඪ (q1 , · · · , qN ) Λ࣋ͭࣗ༝౓ N ͷܥͱ͠·͕͢ɺؔ਺ͷҾ਺ͱ͠ ͯදه͢Δࡍ͸ɺN ݸͷม਺ (q1 , · · · , qN ) Λ·ͱΊͯ q ͱද͠·͢ɻҰൠԽӡಈྔ (p1 , · · · , pN ) ʹ ͍ͭͯ΋ಉ༷Ͱ͢ɻͨͱ͑͹ɺϋϛϧτχΞϯ H(q, p, t) ͸ɺ࣮ࡍʹ͸ɺH(q1 , · · · , qN , p1 , . . . , pN , t) Λද͢΋ͷͱߟ͍͑ͯͩ͘͞ɻ 2 ϋϛϧτϯํఔࣜʹΑΔ࣌ؒൃలͱਖ਼४ม׵ͷؔ܎ ϋϛϧτχΞϯ H(q, p, t) ͕༩͑ΒΕͨ࣌ʹɺdt ΛҰ࣍ͷඍগྔͱͯ͠ɺୈೋछͷ฼ؔ਺ G(q, P, t) Λ࣍ࣜͰఆٛ͠·͢ɻ ʢ͜͜Ͱ͸ɺt ͸ಛఆͷ࣌ࠁʹݻఆͯ͠ߟ͑·͢ɻ ʣ G(q, P, t) = i qi Pi + H(q, P, t)dt ͜ͷ฼ؔ਺ʹΑΔਖ਼४ม׵͸ɺ࣍Ͱ༩͑ΒΕ·͢ɻ pi = ∂G ∂qi = Pi + ∂H(q, p, t) ∂qi p=P × dt Qi = ∂G ∂Pi = qi + ∂H(q, p, t) ∂pi p=P × dt 1 

͜ΕΛ (Q, P) ʹ͍ͭͯղ͍ͨ΋ͷΛ Q(q, p, t), P(q, p, t) ͱ͢Δͱɺ࣍ͷ߃౳͕ࣜ੒Γཱͪ·͢ɻ pi = Pi (q, p, t) + ∂H(q, p, t) ∂qi p=P (q,p,t) × dt (1) Qi (q, p, t) = qi + ∂H(q, p, t) ∂pi p=P (q,p,t) × dt (2) (1) ΑΓ Pi (q, p, t) = pi + O(dt) ͳͷͰɺ্ࣜ͸ dt ͷҰ࣍ͷൣғͰ࣍ͷΑ͏ʹॻ͖׵͑ΒΕ·͢ɻ pi = Pi (q, p, t) + ∂H(q, p, t) ∂qi dt (3) Qi (q, p, t) = qi + ∂H(q, p, t) ∂pi dt (4) ैͬͯɺ dpi = Pi (q, p, t) − pi = − ∂H(q, p, t) ∂qi dt dqi = Qi (q, p, t) − qi = ∂H(q, p, t) ∂pi dt Ͱ͋Γɺ͜ΕΒ͸ɺϋϛϧτϯͷӡಈํఔࣜ ˙ pi = − ∂H(q, p, t) ∂qi ˙ qi = ∂H(q, p, t) ∂pi ʹैͬͨ࣌ؒൃలʹҰக͠·͢ɻਖ਼४ม׵ͷ߹੒͸ਖ਼४ม׵ʹͳΔͷͰɺ༗ݶͷ࣌ؒมԽ΋ɺ΍͸ Γਖ਼४ม׵ͱͯ͠දݱͰ͖Δ͜ͱ͕Θ͔Γ·͢ɻ 3 ॳظঢ়ଶʹҾ͖໭͢ਖ਼४ม׵ͷߏ੒ લઅͰࣔͨ͠ࣄ࣮ΑΓɺॳظঢ়ଶ (α, β) ͔Βൃలͨ࣌͠ࠁ t ͷঢ়ଶΛ (q(α, β, t), p(α, β, t)) ͱ͢ Δͱɺ(α, β) Λ (q(α, β, t), p(α, β, t)) ʹஔ͖׵͑Δਖ਼४ม׵͕ଘࡏ͠·͢ɻ͜ͷਖ਼४ม׵ͷୈҰछ ͷ฼ؔ਺Λ −S(α, q, t) ͱ͢Δͱɺූ߸Λม͑ͨ S(α, q, t) ͸ɺٯม׵ (q(α, β, t), p(α, β, t)) → (α, β) ͷୈҰछͷ฼ؔ਺ͱͳΓɺ࣍ͷม׵͕ࣜ੒Γཱͪ·͢ɻ pi = ∂S(α, q, t) ∂qi (5) βi = − ∂S(α, q, t) ∂αi (6) (q, p) Λ (α, β) Ͱද͍ͨ͠৔߹ɺ(6) ͔Β q = q(α, β, t) ͕ܾ·Γɺ͜ΕΛ (5) ʹ୅ೖ͢Δͱɺ p = p(α, β, t) ͕ܾ·Γ·͢ɻैͬͯɺ࣍ͷ߃౳͕ࣜ੒Γཱͪ·͢ɻ pi (α, β, t) = ∂S(α, q, t) ∂qi q=q(α,β,t) (7) βi = − ∂S(α, q, t) ∂αi q=q(α,β,t) 2 

·ͨɺม਺ (α, β) ʹର͢ΔϋϛϧτχΞϯ͸࣍Ͱ༩͑ΒΕ·͢ɻ K(α, β, t) = H(q, p, t)| q=q(α,β,t),p=p(α,β,t) + ∂S(α, q, t) ∂t q=q(α,β,t) = H (q, p(α, β, t), t)| q=q(α,β,t) + ∂S(α, q, t) ∂t q=q(α,β,t) = H q, ∂S(α, q, t) ∂q , t q=q(α,β,t) + ∂S(α, q, t) ∂t q=q(α,β,t) (∵ (7)) (8) Ұํɺม਺ (α, β) ͸࣌ؒʹґଘ͠ͳ͍ఆ਺ͳͷͰɺ͜ΕΒʹର͢ΔӡಈํఔࣜΛಋ͘ϋϛϧτχ Ξϯ K(α, β, t) ͸߃౳తʹ 0 ʹͳΓ·͢ɻ ʢҰൠʹ͸ఆ਺ C ͱͳΔ΂͖Ͱ͕͢ɺK(α, β, t) = C Λ ຬͨ͢฼ؔ਺ S(α, q, t) ʹରͯ͠ɺ৽͍͠฼ؔ਺Λ S(α, q, t) − Ct Ͱఆٛ͢Δͱɺ͜Ε͸ಉ͡ਖ਼४ ม׵Λಋ͖ɺ͔ͭɺK(α, β, t) = 0 Λຬͨ͠·͢ɻ ʣ ैͬͯɺ(8) ΑΓɺ࣍ͷؔ܎͕੒Γཱͪ·͢ɻ H q, ∂S(α, q, t) ∂q , t + ∂S(α, q, t) ∂t q=q(α,β,t) = 0 ͜ͷࣜ͸ɺ࣮ܦ࿏্ͷ఺ q = q(α, β, t) ʹ͓͍ͯ੒ཱ͢Δ΋ͷͰ͕͢ɺ͜͜Ͱɺॳظ࠲ඪ α ͱ࣌ ࠁ t Λݻఆͨ͠··ɺॳظӡಈྔ β Λ͞·͟·ͳ஋ʹม͑Δ͜ͱΛߟ͑·͢ɻ͜ΕʹԠͯ͡ɺ࣌ ࠁ t ʹ͓͚Δ࠲ඪ q ΋͞·͟·ʹมԽ͢ΔͷͰɺ ʢ෺ཧతʹ౸ୡՄೳͳൣғʹ q ͷఆٛҬΛݶఆ͢ Δͱ͍͏ཧղͷݩʹʣ୅ೖ৚݅Λ֎ͨؔ͠਺ͦͷ΋ͷʹର͢Δ߃౳ࣜͱͯ͠ɺ࣍ͷؔ܎͕੒Γཱͪ ·͢ɻ H q, ∂S(α, q, t) ∂q , t + ∂S(α, q, t) ∂t = 0 (9) ͜ͷภඍ෼ํఔࣜΛϋϛϧτϯɾϠίϏํఔࣜͱݺͼ·͢ɻ 4 ॳظঢ়ଶ͔Βͷ࣌ؒൃలΛҾ͖ى͜͢ਖ਼४ม׵ͷߏ੒ ͜͜Ͱɺٞ࿦ͷॱ൪Λٯʹͯ͠Έ·͢ɻࠓɺϋϛϧτχΞϯ K(α, β, t) = 0 Λ࣋ͭܥΛߟ͑Δ ͱɺӡಈํఔࣜͷղ͸ࣗ໌ʹఆ਺ (α, β) Ͱ༩͑ΒΕ·͢ɻ͜ͷ࣌ɺϋϛϧτϯɾϠίϏํఔࣜ (9) Λ߃౳తʹຬͨؔ͢਺ S(α, q, t) ͕ଘࡏͨ͠ͱͯ͠ɺ−S(α, q, t) ΛୈҰछͷ฼ؔ਺ͱ͢Δਖ਼४ม׵ (α, β) → (q, p) Λߟ͑Δͱɺม׵ࣜ͸ઌ΄Ͳͷ (5)(6) ʹҰக͠·͢ɻ͜ΕΒΛ (α, β) ʹ͍ͭͯղ ͍ͨ΋ͷΛ α(q, p, t), β(q, p, t) ͱ͢Δͱɺ(5) ͔Β α = α(q, p, t) ͕ܾ·Γɺ͜ΕΛ (6) ʹ୅ೖͯ͠ β = β(q, p, t) ͕ܾ·ΔͷͰɺ࣍ͷ߃౳͕ࣜ੒Γཱͪ·͢ɻ pi = ∂S(α, q, t) ∂qi α=α(q,p,t) (10) βi (q, p, t) = − ∂S(α, q, t) ∂αi α=α(q,p,t) 3 

ͦͯ͠ɺม਺ (q(t), p(t)) ʹର͢ΔϋϛϧτχΞϯ ˜ K(q, p, t) ͸ɺ࣍ͷΑ͏ʹܭࢉ͞Ε·͢ɻ ˜ K(q, p, t) = K(α, β, t) − ∂S(α, q, t) ∂t α=α(q,p,t) = H q, ∂S(α, q, t) ∂q , t α=α(q,p,t) (∵ (9)) = H(q, p, t) (∵ (10)) (11) ͭ·Γɺม਺ (q(t), p(t)) ͸ϋϛϧτχΞϯ H(q, p, t) ʹैͬͯӡಈ͢Δܥʹͳ͓ͬͯΓɺ H(q, p, t) ʹΑΔӡಈํఔࣜͷղʹͳΓ·͢ɻݴ͍׵͑ΔͱɺϋϛϧτϯɾϠίϏํఔࣜ (9) ͷղ S(α, q, t) ΛٻΊͨޙʹɺ−S(α, q, t) ΛୈҰछͷ฼ؔ਺ͱ͢Δਖ਼४ม׵ (5)(6) Λ༻͍ͯ q = q(α, β, t), p = p(α, β, t) Λܭࢉ͢Ε͹ɺ ʢॳظ৚݅Λ (α, β) ͱͨ͠৔߹ͷʣϋϛϧτχΞϯ H(q, p, t) ʹΑΔӡಈํఔࣜͷղ͕ಘΒΕ·͢ɻ લઅͷٞ࿦ͱ͋ΘͤΔͱɺؔ਺ S(α, q, t) ʹର͢Δ࣍ͷ 2 ͭͷ৚݅͸ಉ஋ʹͳΓ·͢ɻ 1. −S(α, q, t) ͸ɺӡಈํఔࣜʹै͏࣌ؒൃల (α, β) → (q(t), p(t)) Λಋ͘ਖ਼४ม׵ʹର͢ΔୈҰछ ͷ฼ؔ਺Ͱ͋Δ 2. S(α, q, t) ͸ɺϋϛϧτϯɾϠίϏํఔࣜͷղͰ͋Δ ӡಈํఔࣜʹ൐͏࣌ؒൃల͕ਖ਼४ม׵Ͱ͋Δ͜ͱ͸อূ͞Ε͍ͯΔͷͰɺͦͷ฼ؔ਺͸ඞͣଘࡏ ͠·͢ɻ͕ͨͬͯ͠ɺϋϛϧτϯɾϠίϏํఔࣜʹ͸ɺඞͣղ͕ଘࡏ͢Δͱݴ͑·͢ɻ 5 ϋϛϧτϯɾϠίϏํఔࣜͷղͷੑ࣭ લઅͷ࠷ޙʹࣔͨ͠৚݅Λຬͨؔ͢਺ S(α, q, t) ʹ͍ͭͯɺ͞Βʹͦͷੑ࣭Λਂ۷Γ͠·͢ɻ· ͣɺલఏͱͯ͠ɺؔ਺ −S(α, q, t) ͕ୈҰछͷ฼ؔ਺ͱͯ͠Ҿ͖ى͜͢ਖ਼४ม׵ͷ۩ମతͳม׵ࣜ ͸ (5)(6) Ͱ༩͑ΒΕͯɺ͜ͷม׵͸߃౳ࣜ (7)(10) Λຬͨ͠·͢ɻ ͦͯ͠ɺ্هͷ৚݅Λຬͨؔ͢਺ S(α, q, t) Λ༻͍ͯɺ৽͍ؔ͠਺ ˜ S(α, β, t) = S(α, q(α, β, t), t) Λߟ͑Δͱɺ࣍ͷܭࢉ͕੒Γཱͪ·͢ɻ d dt ˜ S(α, β, t) = i ∂S(α, q, t) ∂qi q=q(α,β,t) × ˙ qi (α, β, t) + ∂S(α, q, t) ∂t q=q(α,β,t) = i ∂S(α, q, t) ∂qi q=q(α,β,t) × ˙ qi (α, β, t) − H q, ∂S(α, q, t) ∂q , t q=q(α,β,t) (∵ (9)) = i pi (α, β, t) ˙ qi (α, β, t) − H (q(α, β, t), p(α, β, t), t) (∵ (7)) ͜ͷ࠷ޙͷදࣜ͸ɺ࣌ࠁ t ʹ͓͚Δ࣮ܦ࿏ q(t) = q(α, β, t) ্ͰͷϥάϥϯδΞϯ L(q, ˙ q, t) ͷ஋ ʹҰக͍ͯ͠·͢ɻैͬͯɺ্ࣜΛ࣌ࠁ t′ = 0 ∼ t ͷൣғͰੵ෼͢Δͱ࣍ͷؔ܎͕ಘΒΕ·͢ɻ S(α, q(α, β, t), t) − S(α, q(α, β, 0), 0) = t 0 L(q(α, β, t′), ˙ q(α, β, t′), t′) dt′ 4 

࣌ࠁ t = 0 ʹ͓͚Δ࠲ඪ͸ q(α, β, 0) = α Ͱ͋Δ͜ͱΛ༻͍Δͱɺ্ࣜͷࠨลୈೋ߲͸ɺ S(α, q(α, β, 0), 0) = S(α, α, 0) ͱͳΓ·͕͢ɺຊઅͷ࠷ޙʹࣔ͢Α͏ʹɺS(α, α, 0) = 0 ͱͳΔͷͰɺ ͕࣍੒Γཱͪ·͢ɻ S(α, q(α, β, t), t) = t 0 L(q(α, β, t′), ˙ q(α, β, t′), t′) dt′ (12) ͜͜ͰɺҰൠʹɺ(α, β, t) ΛܾΊΔͱ࣮ܦ࿏্ͷ q = q(α, β, t) ͕Ұҙʹܾ·Δ͜ͱʹ஫ҙ͠· ͢ɻͭ·Γɺ(α, β, q, t) (q ͸࣌ࠁ t ʹ͓͚Δ࣮ܦ࿏্ͷ࠲ඪ q(α, β, t)) ͷதͰಠཱͳม਺͸ 3 ͭͰ ͋Γɺͨͱ͑͹ɺ(α, q, t) ΛܾΊΔͱ β = β(α, q, t) (13) ͕Ұҙʹܾ·Γ·͢ɻ͋Δ͍͸ɺ(β, q, t) ΛܾΊΔͱ α = α(β, q, t) (14) ͕Ұҙʹܾ·Γ·͢ɻ͜͜Ͱɺ(13) ͱ (14) ͸ɺ(q, t) Λݻఆͯ͠ɺα ͱ β Λ૬ޓม׵͢Δ΋ͷͳ ͷͰɺ β(α, q, t)|α=α(β,q,t) = β ͕߃౳తʹ੒Γཱͪ·͢ɻಉ༷ʹߟ͑Δͱɺq = q(α, β, t) ͱ (13) ͸ɺ(α, t) Λݻఆͯ͠ɺq ͱ β Λ ૬ޓม׵͢Δؔ܎ʹͳ͓ͬͯΓɺ q(α, β, t)| β=β(α,q,t) = q ͕੒Γཱͪ·͢ɻͦ͜Ͱɺ(12) ʹ β = β(α, q, t) Λ୅ೖ͢Δͱ͕࣍ಘΒΕ·͢ɻ S(α, q, t) = t 0 L(q(α, β, t′), ˙ q(α, β, t′), t′) dt′ β=β(α,q,t) ͜͜Ͱɺq(α, β, t′)| β=β(α,q,t) ͸ɺڥք৚݅ q(0) = α ͱ q(t) = q Ͱܾ·Δ࣮ܦ࿏ͷ࣌ࠁ t′ ʹ͓͚ Δ࠲ඪ q(t′) Ͱ͋Δ͜ͱʹ஫ҙͯ͠ɺ͜ΕΛ q(α, q, t′) ͱදه͢Δͱɺ࣍ͷΑ͏ʹॻ͖௚ͤ·͢ɻ S(α, q, t) = t 0 L(q(α, q, t′), ˙ q(α, q, t′), t′) dt′ (15) Ҏ্ʹΑΓɺ ʮϋϛϧτϯɾϠίϏํఔࣜͷղ S(α, q, t) ͸ɺڥք৚݅ q(0) = α, q(t) = q Λຬͨ ࣮͢ܦ࿏ q(α, q, t′) ʹԊͬͨ࡞༻ੵ෼ W(α, q, t) ʹҰக͢Δʯͱ͍͏ஶ͍݁͠Ռ͕ಘΒΕ·ͨ͠ɻ ͜͜Ͱ࠷ޙʹɺS(α, α, 0) = 0 Λ͓͖ࣔͯ͠·͢ɻ࣌ࠁ t = 0 ʹ͓͍ͯɺਖ਼४ม׵ (α, β) → (q(α, β, t), p(α, β, t)) ͸߃౳ม׵ʢq(α, β, 0) = α, p(α, β, 0) = βʣʹͳΔͷͰɺ(5)(6) ΑΓ͕࣍੒Γ ཱͪ·͢ɻ ∂S(α, q, 0) ∂qi = pi (α, β, 0) = βi ∂S(α, q, 0) ∂αi = −βi 5 

্ࣜͷӈล͸ q ʹґଘ͠ͳ͍ͷͰɺࠨลʹ q = α Λ୅ೖ͢Δͱɺ͕࣍ಘΒΕ·͢ɻ ∂S(α, q, 0) ∂qi q=α = βi ∂S(α, q, 0) ∂αi q=α = −βi ͜͜Ͱɺؔ਺ S(α, α, 0) ͸ɺ߃౳ؔ਺ id(x) = x Λ༻͍ͯɺS(α, id(α), 0) ͱॻ͚Δ͜ͱʹ஫ҙ͠ ͯɺ͜ΕΛ αi Ͱภඍ෼͢Δͱɺ࣍ͷܭࢉ͕੒Γཱͪ·͢ɻ ∂ ∂αi S(α, α, 0) = ∂S(α, q, 0) ∂αi q=id(α) + ∂S(α, q, 0) ∂qi q=id(α) × d id(α) dα = −βi + βi = 0 ͜Ε͸ɺS(α, α, 0) ͕ม਺ α ʹґଘ͠ͳ͍ఆ਺ C Ͱ͋Δ͜ͱΛҙຯ͠·͢ɻ฼ؔ਺ʹ͸೚ҙͷ ఆ਺ΛՃ͑Δࣗ༝౓͕͋ΔͷͰɺ͜ͷఆ਺Λ 0 ͱબͿ͜ͱͰɺS(α, α, 0) = 0 ͕੒Γཱͪ·͢ɻϋ ϛϧτϯɾϠίϏํఔࣜ (9) ͷղʹ΋ಉ༷ͷఆ਺ͷࣗ༝౓͕͋ΔͷͰɺ͜ͷૢ࡞͸ɺS(α, q, t) ͕ (9) ͷղͰ͋Δͱݴ͏લఏʹ΋ໃ६͠·ͤΜɻ Ҏ্ͷٞ࿦ʹΑΓɺϋϛϧτϯɾϠίϏํఔࣜͷղ S(α, q, t)ʢ΋͘͠͸ɺӡಈํఔࣜʹै͏࣌ ؒൃల (α, β) → (q(t), p(t)) Λಋ͘ਖ਼४ม׵ʹର͢ΔୈҰछͷ฼ؔ਺ −S(α, q, t)ʣ͸ɺڥք৚݅ q(0) = α, q(t) = q Λຬ࣮ͨ͢ܦ࿏ q(α, q, t′) ʹԊͬͨ࡞༻ੵ෼ W(α, q, t) ʹҰக͢Δ͜ͱ͕Θ͔Γ ·ͨ͠ɻ 6 ࡞༻ੵ෼Λग़ൃ఺ͱͨٞ͠࿦ ͜͜Ͱ͸ɺલઅͷ݁Ռͷٯ͕੒Γཱͭ͜ͱΛࣔ͠·͢ɻͭ·Γɺ(15) Ͱఆٛ͞ΕΔ࡞༻ੵ෼Λ͋ ΒͨΊͯ W(α, q, t) ͱஔ͘ͱɺ−W(α, q, t) ͸ɺӡಈํఔࣜʹै͏࣌ؒൃల (α, β) → (q(t), p(t)) Λ ಋ͘ਖ਼४ม׵ʹର͢ΔୈҰछͷ฼ؔ਺Ͱ͋Γɺͦͯ͠ɺW(α, q, t) ͸ϋϛϧτϯɾϠίϏํఔࣜΛ ຬͨ͠·͢ɻ ·ͣɺલઅͰ͸ɺ࡞༻ੵ෼ʹؚ·ΕΔʮڥք৚݅ q(0) = α, q(t) = q Λຬ࣮ͨ͢ܦ࿏ʯΛ q(α, q, t′) ͱදه͍ͯ͠·͕ͨ͠ɺ͜͜Ͱ͸؆୯ʹ q(t′) ͱදه͠·͢ɻ͜ͷ໿ଋͷݩʹɺ࡞༻ੵ෼ W(α, q, t) ͸ɺ࣍Ͱ༩͑ΒΕ·͢ɻ W(α, q, t) = t 0 L(q(t′), ˙ q(t′), t′)dt′ ͦͯ͠ɺ͜ͷؔ਺ W(α, q, t) ͷ 3 ͭͷҾ਺ɺ͢ͳΘͪɺ࢝୺ͷ࠲ඪ αɺऴ୺ͷ࠲ඪ qɺ͓Αͼɺ ऴ୺ͷ࣌ࠁ t ͷ஋ΛͦΕͧΕ α → α + dαɺq → q + dqɺt → t + dt ͱඍখมԽͤͨ͞ͱͯ͠ɺ͜Ε ʹ൐͏࡞༻ੵ෼ͷඍখมԽ W(α, q, t) → W(α, q, t) + dW Λܭࢉ͠·͢ɻ୺఺Λมߋ͢Δͱ్தͷ 6 

ܦ࿏ q(t′) ΋มԽ͢ΔͷͰɺܦ࿏ͷඍখมԽΛ δq(t′) ͱهड़͢Δͱɺ͕࣍੒Γཱͪ·͢ɻ dW = t+dt 0 L (q(t′) + δq(t′), ˙ q(t′) + δ ˙ q(t′), t′) dt′ − t 0 L (q(t′), ˙ q(t′), t′) dt′ = L (q(t), ˙ q(t), t) dt + t 0 i ∂L ∂qi δqi (t′) + ∂L ∂ ˙ qi δ ˙ qi (t′) dt′ ͜͜Ͱɺ ୈೋ߲Λ෦෼ੵ෼ͯ͠ɺ ࣮ܦ࿏্Ͱ੒Γཱͭϥάϥϯδϡͷӡಈํఔࣜ d dt′ ∂L ∂ ˙ qi − ∂L ∂qi = 0 Λ༻͍·͢ɻ t 0 i ∂L ∂qi δqi (t′) + ∂L ∂ ˙ qi δ ˙ qi (t′) dt′ = t 0 i ∂L ∂qi − d dt′ ∂L ∂ ˙ qi δqi (t′) dt′ + i ∂L ∂ ˙ qi δqi (t′) t′=t t′=0 = i pi (t)δqi (t) − i pi (0)δqi (0) = i pi (t)δqi (t) − i βi dαi ͜͜Ͱ͸ɺҰൠԽӡಈྔͷఆٛ pi = ∂L ∂ ˙ qi Λ༻͍ͯɺ͞Βʹɺ࢝୺ʹ͓͚Δӡಈྔ pi (0) = βi ͱ ࢝୺ͷ࠲ඪมԽ δq(0) = dα Λ༻͍·ͨ͠ɻ·ͨɺऴ୺ͷ࠲ඪͷมԽ dq ͸ɺ࣌ࠁ t ʹ͓͚ΔมԽ δq(t) ʹՃ͑ͯɺऴ୺ͷ࣌ࠁͦͷ΋ͷͷมԽ dt ͷӨڹ΋͋ΔͷͰɺશମͱͯ͠ɺdq = δq(t) + ˙ q(t)dtɺ ͢ͳΘͪ δq(t) = dq − ˙ q(t)dt ͕੒Γཱͪ·͢ɻ͜ΕΛ୅ೖͯ͠੔ཧ͢Δͱɺ࣍ͷΑ͏ʹͳΓ·͢ɻ dW = L (q(t), ˙ q(t), t) − i pi (t) ˙ q(t) dt + i pi (t)dqi − i βi dαi = −H (q(t), p(t), t) dt + i pi (t)dqi − i βi dαi (16) ࠷ޙͷมܗͰ͸ɺϋϛϧτχΞϯͷఆٛ H(p, q, t) = i pi ˙ qi − L (q, ˙ q, t) Λ༻͍·ͨ͠ɻຊདྷͷ ʢؔ਺ͱͯ͠ͷʣϋϛϧτχΞϯ͸ɺ i pi ˙ qi − L (q, ˙ q, t) Λม਺ (q, p, t) Ͱॻ͖௚ͨ͠΋ͷͰ͕͢ɺ ͜͜Ͱ͸ɺಛఆͷ࣌ࠁ t ʹ͓͚Δ࣮ܦ࿏্ͷ஋ q(t), p(t) Λ༻͍Ε͹ɺগͳ͘ͱ΋਺஋ͱͯ͠ɺ H (q(t), p(t), t) = i pi (t) ˙ qi (t) − L (q(t), ˙ q(t), t) ͕੒Γཱͭ͜ͱΛ༻͍͍ͯ·͢ɻ Ұํɺ࡞༻ੵ෼ W ͸ಠཱม਺ (α, q, t) ͷؔ਺ͳͷͰɺ͜ΕΒͷม਺ͷඍখมԽʹ൐͏ W ࣗ਎ ͷมԽʹ͍ͭͯɺҰൠʹ͕࣍੒Γཱͪ·͢ɻ dW = i ∂W(α, q, t) ∂qi dqi + i ∂W(α, q, t) ∂αi dαi + ∂W(α, q, t) ∂t dt ͜ΕΒ 2 ͭͷ dW ͕೚ҙͷඍখมԽ dqi , dαi , dt ʹରͯ͠౳͘͠ͳΔ͜ͱ͔Βɺ࣍ͷؔ܎͕੒Γ 7 

ཱͪ·͢ɻ pi (t) = ∂W(α, q, t) ∂qi (17) βi = − ∂W(α, q, t) ∂αi (18) H(q(t), p(t), t) = − ∂W(α, q, t) ∂t (19) ͜͜Ͱɺq(t), p(t) ͸ಛఆͷ࣌ࠁ t ʹ͓͚Δ࣮ܦ࿏্ͷ஋Λද͢఺ʹ஫ҙ͠·͢ɻ͢Δͱɺ (17)(18) ͸ɺୈҰछͷ฼ؔ਺Λ −W(α, q, t) ͱͨ͠ͱ͖ͷਖ਼४ม׵ͷม׵ࣜʹͳ͓ͬͯΓɺॳظঢ় ଶ (α, β) ͔Β࣌ࠁ t ʹ͓͚Δ࣮ܦ࿏্ͷঢ়ଶ (q(t), p(t)) ΁ͷม׵ΛҾ͖ى͜͠·͢ɻ͜ΕͰɺ ʮ࡞ ༻ੵ෼ͷූ߸ҧ͍͕࣌ؒൃలͷਖ਼४ม׵ͷ฼ؔ਺ʹͳΔʯ͜ͱ͕ࣔ͞Ε·ͨ͠ɻ ͞Βʹɺ(19) ʹ (17) Λ୅ೖ͢Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ H q, ∂W(α, q, t) ∂q , t + ∂W(α, q, t) ∂t = 0 ͜Ε͸ɺϋϛϧτϯɾϠίϏํఔࣜʹଞͳΓ·ͤΜɻ͜ΕͰɺ࡞༻ੵ෼ W(α, q, t) ͕ϋϛϧτ ϯɾϠίϏํఔࣜͷղͰ͋Δ͜ͱ͕ࣔ͞Ε·ͨ͠ɻ ͜͜·Ͱͷٞ࿦ʹΑΓɺ࣍ͷ 3 ͭ͸͢΂ͯಉ஋ʹͳΔ͜ͱ͕ࣔ͞Ε·ͨ͠ɻ 1. S(α, q, t) ͸ɺϋϛϧτϯɾϠίϏํఔࣜͷղͰ͋Δ 2. −S(α, q, t) ͸ɺӡಈํఔࣜʹै͏࣌ؒൃల (α, β) → (q(t), p(t)) Λಋ͘ਖ਼४ม׵ʹର͢ΔୈҰछ ͷ฼ؔ਺Ͱ͋Δ 3. S(α, q, t) ͸ɺڥք৚݅ q(0) = α, q(t) = q Λຬ࣮ͨ͢ܦ࿏ʹԊͬͨ࡞༻ੵ෼ W(α, q, t) Ͱ͋Δ 7 ྺ࢙తܦҢ ຊߘͰ͸ɺ ʮӡಈํఔࣜʹै͏࣌ؒൃలΛಋ͘ਖ਼४ม׵ͷ฼ؔ਺ʯͱͯؔ͠਺ S(α, q, t) Λಋೖ͠ ্ͨͰɺ͜Ε͕ϋϛϧτϯɾϠίϏํఔࣜͷղͱͯ͠ಘΒΕΔ͜ͱɺͦͯ͠ɺ࣮ܦ࿏ʹԊͬͨ࡞༻ ੵ෼ʹҰக͢Δ͜ͱΛࣔ͠·ͨ͠ɻैͬͯɺS(α, q, t) ͕ܾ·Ε͹ɺ(5)(6) ʹࣔͨ͠ਖ਼४ม׵ͷม׵ ͔ࣜΒɺ୅਺తૢ࡞Ͱӡಈํఔࣜͷղ q = q(α, β, t), p = p(α, β, t) ͕ಘΒΕΔ͜ͱ͕ࣗ໌ʹཧղͰ ͖·͢ɻ Ұํɺؔ਺ S(α, q, t) ʹ૬౰͢Δ֓೦Λ͸͡Ίʹఏএͨ͠ϋϛϧτϯ͸ɺ࣮ܦ࿏ʹԊͬͨ࡞༻ੵ ෼ͱͯ͜͠ΕΛఆٛ͠·ͨ͠ɻϋϛϧτϯ͸ɺ͜ͷࡍɺ࢝୺ͷ࠲ඪ q0 ͱ࣌ࠁ t0 ɺ͓Αͼɺऴ୺ͷ ࠲ඪ q ͱ࣌ࠁ t Λࢦఆͯ͠ɺ͜ΕΒΛ݁Ϳ࣮ܦ࿏ʹԊͬͨ࡞༻ੵ෼ SH (q0 , t0 , q, t) = t t0 L (q(t′), ˙ q(t′), t′) dt′ Λར༻͠·ͨ͠ɻ͜ΕΛʮϋϛϧτϯͷओؔ਺ʯͱݺͼ·͢ɻ 8 

͜ͷ৔߹ɺ(16) Λಋ͍ͨ࣌ͱಉ༷ͷܭࢉʹΑΓɺ྆୺఺ͷඍখมԽʹ൐͏࡞༻ੵ෼ͷඍখม Խ͸ɺ dSH = −H (q(t), p(t), t) dt + H (q(t0 ), p(t0 ), t) dt0 + i pi (t)dqi − i pi (t0 )dαi ͱͳΓ·͢ɻ͜ΕΛҰൠతͳදࣜ dSH = i ∂SH (q0 , t0 , q, t) ∂qi dqi + i ∂SH (q0 , t0 , q, t) ∂q0i dq0i + ∂SH (q0 , t0 , q, t) ∂t dt + ∂SH (q0 , t0 , q, t) ∂t0 dt0 ͱൺֱ͢Δͱɺ࣍ͷ 4 ͭͷؔ܎͕ࣜಘΒΕ·͢ɻ ∂SH (q0 , t0 , q, t) ∂qi = pi (t), ∂SH (q0 , t0 , q, t) ∂q0i = −pi (t0 ) (20) ∂SH (q0 , t0 , q, t) ∂t = −H(q, p, t), ∂SH (q0 , t0 , q, t) ∂t0 = H(q0 , p(t0 ), t0 ) (21) (20) ͷؔ܎Λ (21) ͷӈลʹ୅ೖ͢Δͱɺ͕࣍ಘΒΕ·͢ɻ ∂SH (q0 , t0 , q, t) ∂t = −H q, ∂SH (q0 , t0 , q, t) ∂qi , t (22) ∂SH (q0 , t0 , q, t) ∂t0 = H q0 , − ∂SH (q0 , t0 , q, t) ∂q0i , t0 (23) ؔ਺ SH (q0 , t0 , q, t) ʹ͸ɺਖ਼४ม׵ͷ฼ؔ਺ͱ͍͏ҙຯ߹͍͸͋Γ·ͤΜ͕ɺ࣮ܦ࿏ʹԊͬͨ࡞ ༻ੵ෼͔ΒಘΒΕΔؔ਺ͱͯ͠ɺӡಈํఔࣜͷղɺ͢ͳΘͪɺ࣮ܦ࿏Λಋͨ͘Ίͷ৘ใؚ͕·Εͯ ͍Δ΋ͷͱظ଴͞Ε·͢ɻ࣮ࡍɺ(22)(23) ͷ 2 ͭͷภඍ෼ํఔࣜΛຬͨ͢ղ SH (q0 , t0 , q, t) ͕ಘΒ Εͨͱ͢Ε͹ɺ(20) Λ 2N ݸͷม਺ (q, p) ʹର͢Δ 2N ຊͷ࿈ཱํఔࣜͱΈͳͯ͠ղ͘͜ͱͰɺ (q, p) Λ (t0 , q0 , p(t0 ), t) Ͱॻ͖Լͤ·͢ɻ͜Ε͸ɺॳظ৚݅ q(t0 ) = q0 , p(t0 ) ʹରԠ͢Δɺ࣌ࠁ t ͷ ࣮ܦ࿏্ͷ࠲ඪͱӡಈྔΛද͓ͯ͠Γɺॳظ৚݅ q(t0 ) = q0 , p(t0 ) ʹର͢Δӡಈํఔࣜͷղͱߟ͑ ΒΕ·͢ɻ ͔͠͠ͳ͕ΒɺҰൠʹɺ2 ͭͷภඍ෼ํఔࣜΛಉ࣌ʹղ͘࡞ۀ͸ࠔ೉Ͱ͋Γɺ͜ͷख๏ʹ͸࣮༻ ੑͷ؍఺Ͱ՝୊͕͋Γ·ͨ͠ɻϠίϏ͸ɺϋϛϧτϯͷ͜ͷΞΠσΞΛվྑͯ͠ɺ୯Ұͷภඍ෼ํ ఔࣜʢ͢ͳΘͪɺϋϛϧτϯɾϠίϏํఔࣜʣΛղ͘͜ͱͰӡಈํఔࣜͷղ͕ಘΒΕΔ͜ͱΛࣔ͠ ·ͨ͠ɻ͜Ε͕ɺຊߘͰઆ໌ͨ͠ख๏ʹͳΓ·͢ɻ ͳ͓ɺຊߘͷઆ໌Ͱ͸ɺϋϛϧτϯɾϠίϏํఔࣜͷղ S(α, q, t) ͕ʮӡಈํఔࣜʹै͏࣌ؒൃ లΛಋ͘ਖ਼४ม׵ͷ฼ؔ਺ʯʹͳΔ͜ͱΛূ໌͢Δ͜ͱͰɺਖ਼४ม׵ͷม׵ࣜ (5)(6) Λ௨ͯ͠ӡ ಈํఔࣜͷղ͕ಘΒΕΔ͜ͱΛࣔ͠·ͨ͠ɻҰํɺਖ਼४ม׵ͷ฼ؔ਺ͱ͍͏ߟ͑ํΛ༻͍ͣʹɺϋ ϛϧτϯɾϠίϏํఔࣜͷղ S(α, q, t) ͔Βม׵ࣜ (5)(6) Λ௨ͯ͠ಘΒΕΔ q(t), p(t) ͕ϋϛϧτ ϯͷӡಈํఔࣜΛຬͨ͢͜ͱΛ௚઀ܭࢉͰࣔ͢͜ͱ΋ՄೳͰ͢ɻ͜ΕΛҎԼͰࣔ͠·͢ɻ ࠓɺؔ਺ S(α, q, t) ͕ϋϛϧτϯɾϠίϏํఔࣜ (9) Λຬ͓ͨͯ͠Γɺม׵ࣜ (5)(6) Λղ͍ͯಘ ΒΕͨ݁Ռ͕ q(α, β, t), p(α, β, t) Ͱ͋Δͱ͠·͢ɻ͜ͷ݁ՌΛ؆୯ʹ q(t), p(t) ͱදهͯ͠ɺ͜Ε 9 

Β͕ϋϛϧτϯͷӡಈํఔࣜΛຬͨ͢͜ͱΛࣔ͠·͢ɻͳ͓ɺม׵ࣜ (5)(6) ͕ਖ਼४ม׵ʹ൐͏ม ׵ͷ৔߹ɺ͜Ε͕ਖ਼ଇͳม׵ʢՄٯͳม׵ʣͰ͋Δͱ͍͏લఏ͔Βɺม׵ͷϠίϏߦྻ ∂2S(α, q, t) ∂αi ∂qj ͸ٯߦྻΛ࣋ͪ·͢ɻࠓ͸ਖ਼४ม׵ͱ͍͏લఏ͸͋Γ·ͤΜ͕ɺ෺ཧతͳཁ੥ͱͯ͠ɺٯߦྻ͕ଘ ࡏ͢Δղ S(α, q, t) ͕ಘΒΕͨ΋ͷͱԾఆ͠·͢ɻҰൠʹɺ͜ͷ৚݅Λຬͨ͢ϋϛϧτϯɾϠίϏ ํఔࣜͷղΛʮ׬શղʯͱݺͼ·͢ɻ ·ͣɺϋϛϧτϯɾϠίϏํఔࣜΛ࠶ܝ͢Δͱɺ࣍ʹͳΓ·͢ɻ ∂S(α, q, t) ∂t + H q, ∂S(α, q, t) ∂q , t = 0 (24) ͦͯ͠ɺಘΒΕͨ q(t), p(t) Λ࠶౓ (5)(6) ʹ୅ೖ͢Δͱɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ pi (t) = ∂S(α, q, t) ∂qi q=q(t) (25) βi = − ∂S(α, q, t) ∂αi q=q(t) (26) ͜͜Ͱɺ(26) ͷ྆ลΛ࣌ؒ t Ͱશඍ෼͠·͢ɻβi ͸࣌ؒʹґଘ͠ͳ͍ఆ਺ͳͷͰɺࠨล͸ 0 ʹ ͳΓ·͢ɻҰํɺӈล͸ɺ࣍ͷΑ͏ʹܭࢉ͞Ε·͢ɻ − d dt ∂S(α, q, t) ∂αi q=q(t) = − j ∂2S(α, q, t) ∂αi ∂qj (t) q=q(t) ˙ qj (t) − ∂2S(α, q, t) ∂αi ∂t q=q(t) (27) ͜ΕΑΓɺ࣍ͷؔ܎͕ಘΒΕ·͢ɻ ∂2S(α, q, t) ∂αi ∂t q=q(t) = − j ∂2S(α, q, t) ∂αi ∂qj q=q(t) ˙ qj (t) (28) ࣍ʹɺ(24) ͷ྆ลΛ αi Ͱภඍ෼͢Δͱɺ͕࣍ಘΒΕ·͢ɻ ∂2S(α, q, t) ∂αi ∂t + j ∂H(q, p, t) ∂pj p= ∂S(α,q,t) ∂q × ∂2S(α, q, t) ∂αi ∂qj = 0 ͜Εʹɺq = q(t) Λ୅ೖ͢Δͱɺ(25) Λ༻͍ͯɺ͕࣍ಘΒΕ·͢ɻ ∂2S(α, q, t) ∂αi ∂t q=q(t) + j ∂H(q, p, t) ∂pj p=p(t) ∂2S(α, q, t) ∂αi ∂qj q=q(t) = 0 (29) (28)(29) ͔Β ∂2S(α, q, t) ∂αi ∂t q=q(t) Λফڈ͢Δͱɺ͕࣍ಘΒΕ·͢ɻ j ∂2S(α, q, t) ∂αi ∂qj q=q(t) ˙ qj (t) − ∂H(q, p, t) ∂pj p=p(t) = 0 ্ࣜʹ͸ม׵ͷϠίϏߦྻ ∂2S(α, q, t) ∂αi ∂qj ؚ͕·Ε͓ͯΓɺ͜Ε͕ٯߦྻΛ࣋ͭ͜ͱ͔Βɺ ˙ qj (t) − ∂H(q, p, t) ∂pj p=p(t) = 0 (30) 10 

͕੒Γཱͪ·͢ɻ͜Ε͸ɺҰൠԽ࠲ඪ q(t) ʹର͢ΔϋϛϧτϯͷӡಈํఔࣜʹҰக͍ͯ͠·͢ɻ ଓ͍ͯɺ(25) ͷ྆ลΛ࣌ؒ t Ͱશඍ෼͠·͢ɻ ˙ pi (t) = j ∂2S(α, q, t) ∂qi ∂qj q=q(t) ˙ qj (t) + ∂2S(α, q, t) ∂qi ∂t q=q(t) (31) ಉ༷ʹɺ(24) ͷ྆ลΛ qi Ͱภඍ෼͢Δͱ͕࣍ಘΒΕ·͢ɻ ∂2S(α, q, t) ∂qi ∂t + ∂H(q, p, t) ∂qi p= ∂S(α,q,t) ∂q + j ∂H(q, p, t) ∂pj p= ∂S(α,q,t) ∂q × ∂2S(α, q, t) ∂qi ∂qj = 0 ͜Εʹɺq = q(t) Λ୅ೖ͢Δͱɺ(25) Λ༻͍ͯɺ͕࣍ಘΒΕ·͢ɻ ∂2S(α, q, t) ∂qi ∂t q=q(t) + ∂H(q, p, t) ∂qi p=p(t) + j ∂H(q, p, t) ∂pj p=p(t) ∂2S(α, q, t) ∂qi ∂qj q=q(t) = 0 ͜Εʹ (30) Λ୅ೖ͠·͢ɻ ∂2S(α, q, t) ∂qi ∂t q=q(t) + ∂H(q, p, t) ∂qi p=p(t) + j ˙ qj (t) ∂2S(α, q, t) ∂qi ∂qj q=q(t) = 0 ࠨลͷୈ 1 ߲ͱୈ 3 ߲ͷ࿨͸ɺ(31) ΑΓ ˙ pj (t) ʹҰக͓ͯ͠Γɺ͕࣍ಘΒΕ·͢ɻ ˙ pj (t) + ∂H(q, p, t) ∂qi p=p(t) = 0 ͜Ε͸ɺҰൠԽӡಈྔ p(t) ʹର͢ΔϋϛϧτϯͷӡಈํఔࣜʹҰக͍ͯ͠·͢ɻҎ্ʹΑΓɺ ϋϛϧτϯɾϠίϏํఔࣜͷղ S(α, q, t) ͔ΒಘΒΕΔ q(t), p(t) ͸ɺϋϛϧτϯͷӡಈํఔࣜΛຬ ͨ͢͜ͱ͕ࣔ͞Ε·ͨ͠ɻ 11