As an example let us get the equations in cylindrical coordinates
x = r cos ϕ , y = r sin ϕ , z = z , {\displaystyle x=r\cos \phi ,\,\,\,\,\,\,y=r\sin \phi ,\,\,\,\,\,\,z=z,}
T = m 2 [ r ˙ 2 + r 2 ϕ ˙ 2 + z ˙ 2 ] . {\displaystyle T={\frac {m}{2}}\left[{\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}+{\dot {z}}^{2}\right].}
∂ T r ˙ = m r ˙ , {\displaystyle {\frac {\partial T}{\dot {r}}}=m{\dot {r}},}
T ∂ r = m r ϕ ˙ 2 , {\displaystyle {\frac {T}{\partial r}}=mr{\dot {\phi }}^{2},}
∂ T ∂ ϕ ˙ = m r 2 ϕ ˙ , {\displaystyle {\frac {\partial T}{\partial {\dot {\phi }}}}=mr^{2}{\dot {\phi }},} ∂ T ∂ z ˙ = m z ˙ . {\displaystyle {\frac {\partial T}{\partial {\dot {z}}}}=m{\dot {z}}.}
δ r W = m [ r ¨ − r ϕ ˙ 2 ] δ r = R δ r , {\displaystyle \delta _{r}W=m\left[{\ddot {r}}-r{\dot {\phi }}^{2}\right]\delta r=R\delta r,}
δ ϕ W = m d d t ( r 2 ϕ ˙ ) δ ϕ = Φ r δ ϕ , {\displaystyle \delta _{\phi }W=m{\frac {d}{dt}}\left(r^{2}{\dot {\phi }}\right)\delta \phi =\Phi r\delta \phi ,}
δ z W = m z ¨ δ z = Z δ z ; {\displaystyle \delta _{z}W=m{\ddot {z}}\delta z=Z\delta z;}
or
m [ d 2 r d t 2 − r ( d ϕ d t ) 2 ] = R , {\displaystyle m\left[{\frac {d^{2}r}{dt^{2}}}-r\left({\frac {d\phi }{dt}}\right)^{2}\right]=R,}
m r d d t ( r 2 d ϕ d t ) = Φ , {\displaystyle {\frac {m}{r}}{\frac {d}{dt}}\left(r^{2}{\frac {d\phi }{dt}}\right)=\Phi ,} m d 2 z d t 2 = Z . {\displaystyle m{\frac {d^{2}z}{dt^{2}}}=Z.}
![{\displaystyle T={\frac {m}{2}}\left[{\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}+{\dot {z}}^{2}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23951332fd13fe20db447023205bc8bd1fbb3eaa)
![{\displaystyle \delta _{r}W=m\left[{\ddot {r}}-r{\dot {\phi }}^{2}\right]\delta r=R\delta r,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b66988410f7f3b2cd157b1576bd55fc67e733f4)
![{\displaystyle m\left[{\frac {d^{2}r}{dt^{2}}}-r\left({\frac {d\phi }{dt}}\right)^{2}\right]=R,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36d6002d298227f96918394cdee4a306d9f2b418)
