Consider the functional \(S [ x , y , z ] = \int _ { t _ { 1 } } ^ { t _ { 2 } } d t L ( x , y , z , x ^ { \prime } , y ^ { \prime } , z ^ { \prime } )\) , where
\[ L = { \frac { m } { 2 } } ( x ^ { \prime 2 } + y ^ { \prime 2 } + z ^ { \prime 2 } ) - U ( x , y , z ) ,\tag{14} \]
and U is \(\mathrm { ~ a ~ } C ^ { 1 }\) function of \(x , y , z\) . Write the system of equations on \(x ( t ) , y ( t ) , z ( t )\) whose solution is an extremum of \(S [ x , y , z ]\) in a class of \(C ^ { 2 } ( [ t _ { 1 } , t _ { 2 } ] )\) functions with fixed boundary conditions.