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In the classical calculus of variations it is know that a necessary condition to a minimum of a functional $$\int_0^T L(x,\dot{x},t)dt$$ is the Legendre condition $$L_{\dot x \dot x} \geq 0.$$ If the inequality is strong, then the theory is further developed into the Jacobi's theory of conjugate points.

I am interested in the case when the inequality is not strict, but $L_{\dot x\dot x} = 0$ in an isolated point. Are there any known second order minimality conditions in that case?

Thank you.

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