v → = r θ ˙ e → θ + r cos θ φ ˙ e → φ {\displaystyle {\vec {v}}=r{\dot {\theta }}{\vec {e}}_{\theta }+r\cos \theta {\dot {\varphi }}{\vec {e}}_{\varphi }}
v 2 = r 2 θ ˙ 2 + r 2 cos 2 θ φ ˙ 2 {\displaystyle v^{2}=r^{2}{\dot {\theta }}^{2}+r^{2}\cos ^{2}\theta {\dot {\varphi }}^{2}}
T = ∫ 0 ℓ v 2 2 m l d r = m l ∫ 0 ℓ r 2 θ ˙ 2 + r 2 cos 2 θ φ ˙ 2 2 d r = m l ⋅ ( ℓ 3 θ ˙ 2 6 + ℓ 3 cos 2 θ φ ˙ 2 6 ) = 1 6 m ℓ 2 θ ˙ 2 + 1 6 m ℓ 2 cos 2 θ φ ˙ 2 {\displaystyle T=\int _{0}^{\ell }{\frac {v^{2}}{2}}{\frac {m}{l}}dr={\frac {m}{l}}\int _{0}^{\ell }{\frac {r^{2}{\dot {\theta }}^{2}+r^{2}\cos ^{2}\theta {\dot {\varphi }}^{2}}{2}}dr={\frac {m}{l}}\cdot \left({\frac {\ell ^{3}{\dot {\theta }}^{2}}{6}}+{\frac {\ell ^{3}\cos ^{2}\theta {\dot {\varphi }}^{2}}{6}}\right)={\frac {1}{6}}m\ell ^{2}{\dot {\theta }}^{2}+{\frac {1}{6}}m\ell ^{2}\cos ^{2}\theta {\dot {\varphi }}^{2}}
p θ = ∂ T ∂ θ ˙ = 1 3 m ℓ 2 θ ˙ θ ˙ = 3 p θ m ℓ 2 {\displaystyle p_{\theta }={\frac {\partial T}{\partial {\dot {\theta }}}}={\frac {1}{3}}m\ell ^{2}{\dot {\theta }}\quad {\dot {\theta }}={\frac {3p_{\theta }}{m\ell ^{2}}}}
p φ = ∂ T ∂ φ ˙ = 1 3 m ℓ 2 cos 2 θ φ ˙ φ ˙ = 3 p φ m ℓ 2 cos 2 θ {\displaystyle p_{\varphi }={\frac {\partial T}{\partial {\dot {\varphi }}}}={\frac {1}{3}}m\ell ^{2}\cos ^{2}\theta {\dot {\varphi }}\quad {\dot {\varphi }}={\frac {3p_{\varphi }}{m\ell ^{2}\cos ^{2}\theta }}}
H ( p θ , p φ , θ , φ ) = 3 p θ 2 2 m ℓ 2 + 3 p φ 2 2 m ℓ 2 cos 2 θ + m g ℓ 2 sin θ {\displaystyle H\left(p_{\theta },p_{\varphi },\theta ,\varphi \right)={\frac {3p_{\theta }^{2}}{2m\ell ^{2}}}+{\frac {3p_{\varphi }^{2}}{2m\ell ^{2}\cos ^{2}\theta }}+mg{\frac {\ell }{2}}\sin \theta }
∂ H ∂ φ = 0 ∂ H ∂ t = 0 {\displaystyle {\frac {\partial H}{\partial \varphi }}=0\quad {\frac {\partial H}{\partial t}}=0}
W = − α 0 t + α 1 φ + S ( θ ) {\displaystyle W=-\alpha _{0}t+\alpha _{1}\varphi +S(\theta )}
3 2 m ℓ 2 ( ∂ S ∂ θ ) 2 + 3 α 1 2 2 m ℓ 2 cos 2 θ + m g ℓ 2 sin θ = α 0 {\displaystyle {\frac {3}{2m\ell ^{2}}}\left({\frac {\partial S}{\partial \theta }}\right)^{2}+{\frac {3\alpha _{1}^{2}}{2m\ell ^{2}\cos ^{2}\theta }}+mg{\frac {\ell }{2}}\sin \theta =\alpha _{0}}
S = ∫ 0 θ 2 m ℓ 2 3 ( α 0 − 3 α 1 2 2 m ℓ 2 cos 2 ξ − m g ℓ 2 sin ξ ) d ξ = ∫ 0 θ 2 m ℓ 2 3 ( α 0 − m g ℓ 2 sin ξ ) − α 1 2 cos 2 ξ d ξ {\displaystyle S=\int _{0}^{\theta }{\sqrt {{\frac {2m\ell ^{2}}{3}}\left(\alpha _{0}-{\frac {3\alpha _{1}^{2}}{2m\ell ^{2}\cos ^{2}\xi }}-mg{\frac {\ell }{2}}\sin \xi \right)}}d\xi =\int _{0}^{\theta }{\sqrt {{\frac {2m\ell ^{2}}{3}}\left(\alpha _{0}-mg{\frac {\ell }{2}}\sin \xi \right)-{\frac {\alpha _{1}^{2}}{\cos ^{2}\xi }}}}d\xi }
∂ W ∂ α 0 = β 0 ∂ W ∂ α 1 = β 1 {\displaystyle {\frac {\partial W}{\partial \alpha _{0}}}=\beta _{0}\quad {\frac {\partial W}{\partial \alpha _{1}}}=\beta _{1}}
β 0 = − t + ∫ 0 θ m ℓ 2 3 2 m ℓ 2 3 ( α 0 − m g ℓ 2 sin ξ ) − α 1 2 cos 2 ξ d ξ {\displaystyle \beta _{0}=-t+\int _{0}^{\theta }{\frac {\frac {m\ell ^{2}}{3}}{\sqrt {{\frac {2m\ell ^{2}}{3}}\left(\alpha _{0}-mg{\frac {\ell }{2}}\sin \xi \right)-{\frac {\alpha _{1}^{2}}{\cos ^{2}\xi }}}}}d\xi }
β 1 = − φ + ∫ 0 θ − α 1 cos 2 ξ 2 m ℓ 2 3 ( α 0 − m g ℓ 2 sin ξ ) − α 1 2 cos 2 ξ d ξ {\displaystyle \beta _{1}=-\varphi +\int _{0}^{\theta }{\frac {-{\frac {\alpha _{1}}{\cos ^{2}\xi }}}{\sqrt {{\frac {2m\ell ^{2}}{3}}\left(\alpha _{0}-mg{\frac {\ell }{2}}\sin \xi \right)-{\frac {\alpha _{1}^{2}}{\cos ^{2}\xi }}}}}d\xi }
