As an example let us get the equations in polar coordinates for motion in a plane
Here x = r cos ϕ , y = r sin ϕ {\displaystyle x=r\cos \phi ,\,\,\,\,\,\,\,\,\,y=r\sin \phi }
x ˙ 2 + y ˙ 2 = v 2 = r ˙ 2 + r 2 ϕ ˙ 2 {\displaystyle {\dot {x}}^{2}+{\dot {y}}^{2}=v^{2}={\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}}
and
T = m 2 [ r ˙ 2 + r 2 ϕ ˙ 2 ] {\displaystyle T={\frac {m}{2}}\left[{\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right]}
∂ T ∂ r ˙ = m r ˙ {\displaystyle {\frac {\partial T}{\partial {\dot {r}}}}=m{\dot {r}}}
∂ T ∂ r = m r ϕ ˙ 2 . {\displaystyle {\frac {\partial T}{\partial r}}=mr{\dot {\phi }}^{2}.} δ r W = m [ r ¨ − r ϕ ˙ 2 ] δ r = R δ r {\displaystyle \delta _{r}W=m[{\ddot {r}}-r{\dot {\phi }}^{2}]\delta r=R\delta r} if R {\displaystyle R} is the impressed force resolved along the radius vector. ∂ T ∂ ϕ ˙ = m r 2 ϕ ˙ , {\displaystyle {\frac {\partial T}{\partial {\dot {\phi }}}}=mr^{2}{\dot {\phi }},}
∂ T ∂ ϕ = 0. {\displaystyle {\frac {\partial T}{\partial \phi }}=0.}
δ ϕ W = m d d t ( r 2 ϕ ˙ ) δ ϕ = Φ r δ ϕ {\displaystyle \delta _{\phi }W=m{\frac {d}{dt}}(r^{2}{\dot {\phi }})\delta \phi =\Phi r\delta \phi }
if Φ {\displaystyle \Phi } is the impressed force resolved perpendicular to the radius vector.
In a more familiar form
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 .}
![{\displaystyle T={\frac {m}{2}}\left[{\dot {r}}^{2}+r^{2}{\dot {\phi }}^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34322bc09d0451ffefca9b24aeba1b61ccaaa66f)
![{\displaystyle \delta _{r}W=m[{\ddot {r}}-r{\dot {\phi }}^{2}]\delta r=R\delta r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4024bd152a0a2774dfd1bbf9564178a83945fd5f)
if 
is the impressed force resolved along the radius vector.
is the impressed force resolved perpendicular to the radius vector.
![{\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)
