Hey, I have a theorem I cannot prove.(adsbygoogle = window.adsbygoogle || []).push({});

We have a function [itex]x^*[/itex] that maximizes or minimizes the integral:

[tex]\int^{t_1}_{t_0} F(t,x(t),\dot{x}(t))dt[/tex]

Our end point conditions are:

[tex]x(t_0) = x_0, x(t_1) \geq x_1[/tex]

I am told that [itex]x^*[/itex] has to satisfy the Euler equation. That I can fully understand since [itex]x^*(t_1)[/itex] can be equal to [itex]x_1[/itex]. However, then it gives me the transversality condition:

[tex]\left(\frac{\partial F}{\partial \dot{x}}\right)_{t=t_1} \leq 0 \text{ ( = 0 if $x^*(t_1) > x_1$)}[/tex]

I can understand the statement in the parentheses. However, I do not understand why [itex]\left(\frac{\partial F}{\partial \dot{x}}\right)_{t=t_1}[/itex] must be less than or equal to zero if [itex]x^*(t_1) = x_1[/itex]. Why can it not be more than zero?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# I Calculus of variations question

Tags:

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads for Calculus variations question |
---|

I Euler’s approach to variational calculus |

A Maximization problem using Euler Lagrange |

A Maximization Problem |

A Derivation of Euler Lagrange, variations |

I Calculus of variations |

**Physics Forums | Science Articles, Homework Help, Discussion**