![{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {x}}}}\right)-{\frac {\partial L}{\partial x}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70763a6b5a61c85034a96b1cea82b87af6d0f188)
where ![{\displaystyle {\dot {x}}={\frac {dx}{dt}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87b167b70bcade5c55b0225ee0892acdd6526397)
The equation above is called Lagrange Equation.
Let the kinetic energy of the point mass be
and the potential energy be
.
is called Lagrangian. Then the kinetic energy is expressed by
![{\displaystyle T={\frac {1}{2}}m{\dot {x}}^{2}+{\frac {1}{2}}m{\dot {y}}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40d6d9f002eb0ab2ffb7b53b43efd9cce625394c)
![{\displaystyle ={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8bd7e236f13bef06eb1229223977d1ce7117c05)
Thus
![{\displaystyle T=T({\dot {x}},{\dot {y}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/651faffbd59621290c12b1a4f73e9df1713806c1)
![{\displaystyle U=U(x,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d404235b4f3fc26c56d871de62d65550c8b4d945)
Hence the Lagrangian
is
![{\displaystyle L=T-U}](https://wikimedia.org/api/rest_v1/media/math/render/svg/513800de5fe628518ce5e1247a7fc613ca95e79f)
![{\displaystyle =T({\dot {x}},{\dot {y}})-U(x,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/847a3d32f780734de7e70ef07c1c4d9baffe7945)
![{\displaystyle ={\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2})-U(x,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3df109c0ff5a460722842190840d4b27c542ffbb)
Therefore
relies on only
and
.
relies on only
and
. Thus
![{\displaystyle {\frac {\partial L}{\partial {\dot {x}}}}={\frac {\partial T}{\partial {\dot {x}}}}=m{\dot {x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/757bfda7a4bff0ca5b0a89fa05128413cd84de17)
![{\displaystyle {\frac {\partial L}{\partial {\dot {y}}}}={\frac {\partial T}{\partial {\dot {y}}}}=m{\dot {y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91eb7b6549360e216eef09ec48a67bdda5f47b86)
In the same way, we have
![{\displaystyle {\frac {\partial L}{\partial x}}=-{\frac {\partial U}{\partial x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4fec146af33320d6b7a10ecacec3129ace666789)
![{\displaystyle {\frac {\partial L}{\partial y}}=-{\frac {\partial U}{\partial y}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/219b432cc3163b7602cbc8e7eff7b42046ed0a8a)