I Calculus of variations question

1. Aug 16, 2017

Avatrin

Hey, I have a theorem I cannot prove.

We have a function $x^*$ that maximizes or minimizes the integral:
$$\int^{t_1}_{t_0} F(t,x(t),\dot{x}(t))dt$$

Our end point conditions are:
$$x(t_0) = x_0, x(t_1) \geq x_1$$

I am told that $x^*$ has to satisfy the Euler equation. That I can fully understand since $x^*(t_1)$ can be equal to $x_1$. However, then it gives me the transversality condition:
$$\left(\frac{\partial F}{\partial \dot{x}}\right)_{t=t_1} \leq 0 \text{ ( = 0 if x^*(t_1) > x_1)}$$

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

2. Aug 16, 2017

Orodruin

Staff Emeritus
The original variation of the functional (call the functional $\mathcal F$) is given by
$$\delta \mathcal F = \int_{t_0}^{t_1} \left( \frac{\partial F}{\partial x} \delta x + \frac{\partial F}{\partial \dot x} \delta \dot x\right) dt.$$
Integration by parts of the second term now leads to
$$\delta \mathcal F = \int_{t_0}^{t_1} \left( \frac{\partial F}{\partial x} - \frac{d}{dt} \frac{\partial F}{\partial \dot x} \right) \delta x \, dt + \left[\frac{\partial F}{\partial \dot x} \delta x\right]_{t=t_0}^{t_1}.$$
Since $\delta x$ is arbitrary, the Euler-Lagrange equation has to hold if $x$ is a stationary function of the functional. This, together with $\delta x(t_0) = 0$ leads to
$$\delta \mathcal F = \left. \frac{\partial F}{\partial \dot x}\right|_{t=t_1} \delta x(t_1).$$
It follows that to have a local stationary function, you must have
$$\left. \frac{\partial F}{\partial \dot x}\right|_{t=t_1} = 0.$$
However, there is also the option of having a function at the boundary $x(t_1) = x_1$. If you are at the boundary, then only variations $\delta x(t_1) \geq 0$ are allowed and therefore $\delta \mathcal F$ can only be positive if
$$\left. \frac{\partial F}{\partial \dot x}\right|_{t=t_1} > 0$$
implying that the value of $\mathcal F$ would increase with the allowed variation. Your condition
$$\left. \frac{\partial F}{\partial \dot x}\right|_{t=t_1} \leq 0$$
is therefore equivalent to requiring that $\delta \mathcal F \leq 0$, i.e., that $x^*$ maximises the functional (at least locally). For a minimisation, you would get the opposite inequality.