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fixed broken links to eom.springer.de
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edit approved Jul 16, 2022 at 8:58
The Amplitwist
edit approved Jul 16, 2022 at 8:58
The Amplitwist
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I believe that a piecewise smooth extremum would have to satisfy the Weierstrass-Erdmann corner conditionsWeierstrass-Erdmann corner conditions. If these conditions ensure that the extremum is in fact $C^1$, then it solves the Euler-Lagrange equations everywhere. At this point one can appeal to the regularity of solutions of ODEs, which can easily get you $C^2$ or even higher smoothness, depending on the regularity of the functional form of the ODE itself.

For more generality, look up the sufficient conditions for a strong variational extremum (starting at Weak and Strong Extrema herehere).

I believe that a piecewise smooth extremum would have to satisfy the Weierstrass-Erdmann corner conditions. If these conditions ensure that the extremum is in fact $C^1$, then it solves the Euler-Lagrange equations everywhere. At this point one can appeal to the regularity of solutions of ODEs, which can easily get you $C^2$ or even higher smoothness, depending on the regularity of the functional form of the ODE itself.

For more generality, look up the sufficient conditions for a strong variational extremum (starting at Weak and Strong Extrema here).

I believe that a piecewise smooth extremum would have to satisfy the Weierstrass-Erdmann corner conditions. If these conditions ensure that the extremum is in fact $C^1$, then it solves the Euler-Lagrange equations everywhere. At this point one can appeal to the regularity of solutions of ODEs, which can easily get you $C^2$ or even higher smoothness, depending on the regularity of the functional form of the ODE itself.

For more generality, look up the sufficient conditions for a strong variational extremum (starting at Weak and Strong Extrema here).

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Igor Khavkine
Igor Khavkine
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I believe that a piecewise smooth extremum would have to satisfy the Weierstrass-Erdmann corner conditions. If these conditions ensure that the extremum is in fact $C^1$, then it solves the Euler-Lagrange equations everywhere. At this point one can appeal to the regularity of solutions of ODEs, which can easily get you $C^2$ or even higher smoothness, depending on the regularity of the functional form of the ODE itself.

For more generality, look up the sufficient conditions for a strong variational extremum (starting at Weak and Strong Extrema here).