If $- \infty < \alpha < \beta < \infty $ and $f \in C^{ 3} \left( [ \alpha , \beta ] \times { \bf R}^{ 2} , { \bf R} \right) $ is bounded, while $y \in C^{ 2} \left( [ \alpha , \beta ] , { \bf R} \right) $ solves the typical one-dimensional problem of the calculus of variations to minimize the function $$F \left( y \right) = \int_{ \alpha }^{ \beta }f \left( x, y(x), y'(x) \right) dx,$$ then for any ${ \phi } \in C^{ 2} \left( [ \alpha , \beta ] , { \bf R} \right) $ for which ${ \phi }^{ (k)}( \alpha ) = { \phi }^{ (k)}( \beta ) = 0$ for every $k \in \{ 0, 1, 2 \} $, we prove that $\int_{ \alpha }^{ \beta } \left( \frac{ { \partial }^{ 2}f }{ \partial y^{ 2} } { \phi }^{ 2} - \frac{ { \partial }^{ 3}f }{ \partial y^{ 2} \partial y' } 2 { \phi }^{ 3} \right) dx$ $\geq \int_{ \alpha }^{ \beta } \left( \frac{ { \partial }^{ 2}f }{ \partial y \partial y' } 2 \phi \phi ' + \frac{ { \partial }^{ 3}f }{ \partial y { \partial y'}^{ 2} } 2 { \phi }^{ 2} \phi ' + \frac{ { \partial }^{ 2}f }{ { \partial y'}^{ 2} } \phi \phi " + \frac{ { \partial }^{ 3}f }{ \partial y { \partial y'}^{ 2} } \phi ' { \phi }^{ 2} + \frac{ { \partial }^{ 3}f }{ { \partial y'}^{ 3} } \phi { \phi '}^{ 2} \right) dx$, so either the above are variational inequalities of motion or the Lagrangian of motion is not $C^{ 3}$.
Categories: math.OC