The Lagrangian of a particle moving in a plane under the influence of a central potential is given by \[L = \frac{1}{2}m\left( {{{\dot r}^2} + {r^2}{{\dot \theta }^2}} \right) - V\left( r \right).\] The generalized momenta corresponding to r and θ are given by
The eigen values of the matrix \[\left[ {\begin{array}{*{20}{c}}
{\cos \theta }&{ - \sin \theta } \\
{\sin \theta }&{\cos \theta }
\end{array}} \right]\] are
Sij and Aij represent a symmetric and an anti-symmetric real-valued tensor respectively in three-dimension. The number of independent components of Sij and Aij
The Fourier transform F(k) of a function f(x) is defined as $$F\left( k \right)\int_{ - \infty }^\infty {dxf\left( x \right)\exp \left( {ikx} \right).} $$ Then F(k) for f(x) = exp(-x2) is $$\left[ {{\text{Given: }}\int_{ - \infty }^\infty {\exp \left( { - {x^2}} \right)dx = \sqrt \pi } } \right]$$
