Getting the Euler Equation from the Pontryagin Maximum Principle

In the calculus of variations, control variables are rates of change of state variables and are unrestricted in value. With the Pontryagin Maximum Principle, we use the state equations in the statement of the problem. The functional stays the same, but we must define the state equations. Thus,

$$J = \int_{x_{1}}^{x_{2}} f\{y(x),y^{'}(x);x\}dx$$

as before, and we define $y^{'}(x)=u$ as the place for the control variable to do its magic.

The Hamiltoninan is

$$H(y,u,\lambda,x)=f\{y(x),y^{'}(x);x\}+\lambda y^{'}(x)$$

Maximizing the Hamiltonian by choice of $y^{'}(x)$ requires

$$\frac{\partial H}{\partial y^{'}} = \frac{\partial f}{\partia... ...a = 0 \Rightarrow \lambda = -\frac{\partial f}{\partial y^{'}}$$

We can differentiate this result with respect to $x$ to get

$$\lambda^{'}=-\frac{d}{dx}\frac{\partial f}{\partial y^{'}}$$

By definition, $\lambda^{'} = -\frac{\partial H}{\partial y}$. We can substitute this into the above equation to get the Euler equation.

$$0 = \frac{\partial f}{\partial y}-\frac{d}{dx} \frac{\partial f}{\partial y^{'}}$$