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Minimize the functional from classical mechanics: \int _{{t_{1}}}^{{t_{2}}}({\mathrm  {Kinetic\,Energy}}-{\mathrm  {Potential\,Energy}})\,

Potential Energy U=cx^{2}\,

Kinetic Energy T={\frac  {1}{2}}m{\dot  {x}}^{2}\,

The functional to minimize is \int _{{t_{1}}}^{{t_{2}}}\left({\frac  {1}{2}}m{\dot  {x}}^{2}-cx^{2}\right)\,dt\,.

The Euler-Lagrange equation is {\frac  {\partial L}{\partial x}}={\frac  {d}{dt}}{\frac  {\partial L}{\partial {\dot  {x}}}}\,.

Since there is no t\, in the integrand, the E-L eqs reduce to

L-{\dot  {x}}{\frac  {\partial L}{\partial {\dot  {x}}}}=c_{1}\,.

c_{1}={\frac  {1}{2}}m{\dot  {x}}^{2}-cx^{2}-{\dot  {x}}(m{\dot  {x}})\,

c_{1}={\frac  {-1}{2}}m{\dot  {x}}^{2}-cx^{2}\,

c_{1}\leq 0\, and c_{1}=-A^{2}\,

A^{2}={\frac  {1}{2}}m{\dot  {x}}^{2}+cx^{2}\,

{\dot  {x}}={\sqrt  {{\frac  {2}{m}}}}{\sqrt  {A^{2}-cx(t)^{2}}}\,

{\frac  {dx}{{\sqrt  {c}}{\sqrt  {{\frac  {A^{2}}{c}}-x(t)^{2}}}}}={\sqrt  {{\frac  {2}{m}}}}dt\,

{\frac  {1}{{\sqrt  {c}}}}\sin ^{{-1}}\left({\frac  {x(t){\sqrt  {c}}}{A}}\right)={\sqrt  {{\frac  {2}{m}}}}t+c_{2}\,

x(t)={\frac  {A}{{\sqrt  {c}}}}\sin \left({\sqrt  {c}}\left[{\sqrt  {{\frac  {2}{m}}}}t+c_{2}\right]\right)\,

Calculus of Variations

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