stylistic:tone
Kinematics

In spherical coordinates, the position of a point or particle (although better written as a triple (r, θ, φ) {\displaystyle (r,\theta,\varphi)}) can be written as r = r r ^. {\displaystyle \mathbf {r} =r\mathbf {\hat {r}}.} Its velocity is then v = d r d t = r ˙ r ^ + r θ ˙ θ ^ + r φ ˙ sin θ φ ^ {\displaystyle \mathbf {v} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}={\dot {r}}\mathbf {\hat {r}} +r\,{\dot {\theta }}\,{\hat {\boldsymbol {\theta }}}+r\,{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\varphi }}} } and its acceleration is a = d v d t = + (r ¨ − r θ ˙ 2 − r φ ˙ 2 sin 2 θ) r ^ + (r θ ¨ + 2 r ˙ θ ˙ − r φ ˙ 2 sin θ cos θ) θ ^ + (r φ ¨ sin θ + 2 r ˙ φ ˙ sin θ + 2 r θ ˙ φ ˙ cos θ) φ ^ {\displaystyle {\begin{aligned}\mathbf {a} ={}&{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}\\={}&{\hphantom {+}}\;\left({\ddot {r}}-r\,{\dot {\theta }}^{2}-r\,{\dot {\varphi }}^{2}\sin ^{2}\theta \right)\mathbf {\hat {r}} \\&{}+\left(r\,{\ddot {\theta }}+2{\dot {r}}\,{\dot {\theta }}-r\,{\dot {\varphi }}^{2}\sin \theta \cos \theta \right){\hat {\boldsymbol {\theta }}}\\&{}+\left(r{\ddot {\varphi }}\,\sin \theta +2{\dot {r}}\,{\dot {\varphi }}\,\sin \theta +2r\,{\dot {\theta }}\,{\dot {\varphi }}\,\cos \theta \right){\hat {\boldsymbol {\varphi }}}\end{aligned}}}

The angular momentum is L = r × p = r × m v = m r 2 (− φ ˙ sin θ θ ^ + θ ˙ φ ^) {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {r} \times m\mathbf {v} =mr^{2}\left(-{\dot {\varphi }}\sin \theta \,\mathbf {\hat {\boldsymbol {\theta }}} +{\dot {\theta }}\,{\hat {\boldsymbol {\varphi }}}\right)} Where m {\displaystyle m} is mass. In the case of a constant φ or else θ = π / 2, this reduces to vector calculus in polar coordinates.

The corresponding angular momentum operator then follows from the phase-space reformulation of the above, L = − i ℏ r × ∇ = i ℏ (θ ^ sin (θ) ∂ ∂ ϕ − ϕ ^ ∂ ∂ θ). {\displaystyle \mathbf {L} =-i\hbar ~\mathbf {r} \times \nabla =i\hbar \left({\frac {\hat {\boldsymbol {\theta }}}{\sin(\theta)}}{\frac {\partial }{\partial \phi }}-{\hat {\boldsymbol {\phi }}}{\frac {\partial }{\partial \theta }}\right).}

The torque is given as τ = d L d t = r × F = − m (2 r r ˙ φ ˙ sin θ + r 2 φ ¨ sin θ + 2 r 2 θ ˙ φ ˙ cos θ) θ ^ + m (r 2 θ ¨ + 2 r r ˙ θ ˙ − r 2 φ ˙ 2 sin θ cos θ) φ ^ {\displaystyle \mathbf {\tau } ={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}=\mathbf {r} \times \mathbf {F} =-m\left(2r{\dot {r}}{\dot {\varphi }}\sin \theta +r^{2}{\ddot {\varphi }}\sin {\theta }+2r^{2}{\dot {\theta }}{\dot {\varphi }}\cos {\theta }\right){\hat {\boldsymbol {\theta }}}+m\left(r^{2}{\ddot {\theta }}+2r{\dot {r}}{\dot {\theta }}-r^{2}{\dot {\varphi }}^{2}\sin \theta \cos \theta \right){\hat {\boldsymbol {\varphi }}}}

The kinetic energy is given as E k = 1 2 m {\displaystyle E_{k}={\frac {1}{2}}m\left}