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# 数学代写|应用数学代考Applied Mathematics代写|MATH311 Higher Derivatives

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## 数学代写|应用数学代考Applied Mathematics代写|Higher Derivatives

So far, we have considered Lagrangians of the form $L=L\left(s, x, x^{\prime}\right)$. An obvious generalization is to include higher derivatives in the Lagrangian. For instance, let us consider the second order problem
$$F: A \rightarrow \mathbb{R}, F(x)=\int_a^b L\left(s, x, x^{\prime}, x^{\prime \prime}\right) \mathrm{d} s,$$
where
$$A=\left{x \in C^4([a, b]) \mid x(a)=A_1, x^{\prime}(a)=A_2, x(b)=B_1, x^{\prime}(b)=B_2\right}$$
Moreover, the function $L:[a, b] \times \mathbb{R}^3 \rightarrow \mathbb{R}$ is assumed to be twice continuously differentiable in each of its arguments. Even though we have now included the second derivative $x^{\prime \prime}$ in the Lagrangian, we can proceed quite similar to Chapter 7.3. Again, our goal is to reformulate the necessary condition $\delta_x F(h)=0$ for all $h \in \operatorname{adm}(F, x)$ provided by Theorem $7.2 .7$ in a more practical manner:
Let $x$ be a local extremum and $h \in C^4([a, b])$ such that
$$h(a)=h^{\prime}(a)=h(b)=h^{\prime}(b)=0 .$$
Then, $x+t h \in A$ and
$$F(x+t h)=\int_a^b L\left(s, x+t h, x^{\prime}+t h^{\prime}, x^{\prime \prime}+t h^{\prime \prime}\right) \mathrm{d} s .$$

## 数学代写|应用数学代考Applied Mathematics代写|Several Functions

Another possible generalization is to consider Lagrangians which depend not on a single function $x$ and its derivatives but on several functions $x_1, \ldots, x_n$ and their derivatives. To illustrate this, let $n=2$ and let us focus on functionals of the form
$$F\left(x_1, x_2\right)=\int_a^b L\left(s, x_1, x_2, x_1^{\prime}, x_2^{\prime}\right) \mathrm{d} s,$$

where $x_1, x_2 \in C^2([a, b])$ satisfying the boundary conditions
\begin{aligned} & x_1(a)=A_1, \quad x_1(b)=B_1, \ & x_2(a)=A_2, \quad x_2(b)=B_2 . \end{aligned}
Once more, we can derive some kind of Euler-Lagrange equation which solutions are the extremals of $F$ by following similar arguments as presented in Chapter $7.2$ and $7.4$ :
Let $\left(x_1, x_2\right)$ be a local extremum of $F$ and let $h_1, h_2 \in C^2([a, b])$ such that
$$h_1(a)=h_2(b)=h_1(a)=h_2(b)=0$$

# 应用数学代考

## 数学代写|应用数学代考APPLIED MATHEMATICS代 写|HIGHER DERIVATIVES

$$F: A \rightarrow \mathbb{R}, F(x)=\int_a^b L\left(s, x, x^{\prime}, x^{\prime \prime}\right) \mathrm{d} s$$

$$h(a)=h^{\prime}(a)=h(b)=h^{\prime}(b)=0 .$$

$$F(x+t h)=\int_a^b L\left(s, x+t h, x^{\prime}+t h^{\prime}, x^{\prime \prime}+t h^{\prime \prime}\right) \mathrm{d} s .$$

## 数学代写|应用数学代考APPLIED MATHEMATICS代 写|SEVERAL FUNCTIONS

$$F\left(x_1, x_2\right)=\int_a^b L\left(s, x_1, x_2, x_1^{\prime}, x_2^{\prime}\right) \mathrm{d} s,$$

$$x_1(a)=A_1, \quad x_1(b)=B_1, \quad x_2(a)=A_2, \quad x_2(b)=B_2 .$$

$$h_1(a)=h_2(b)=h_1(a)=h_2(b)=0$$

## Matlab代写

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