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\title{Noether's Theorem for Functionals Depending on Higher-Order Derivatives}
\author{Vanessa E. McHale}
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\begin{abstract}
Presents proofs of some theorems involved in the calculus of variations with fields depending on higher order derivatives. We follow Gelfand and Fomin very closely.
\end{abstract}
\section{The Euler-Lagrange Equations in general}
Suppose we have a functional of the form $J[u]=\int F(x_i; u_j;\frac{\partial u_j}{\partial x_i};\frac{\partial^2u_j}{\partial x_j x_k};\cdots)dx_1\cdots dx_n$. We wish to find a sufficient condition that it is stationary.
\section{Calculation of $\delta u_{x_i x_j}$}
We get that $$(\bar{\delta u})_{x_ix_j}+\displaystyle\sum_{j,k=1}^nu_{x_i x_j x_k}\delta x_k$$
\section{General Expression for the variation of a functional}
\section{Conserved flows}
Let us suppose we are given a functional $J[u]$ which is invariant under a transformation of the form
$$x_i^*=\Phi_i(x,u,\partial_iu,\partial_i\partial_ju;\epsilon)~x_i+\epsilon\phi_i(x,u,\partial_i u)$$
$$u_j^*=\Psi_i(x,u,\partial_iu,\partial_i\partial_ju;\epsilon)~x_i+\epsilon\psi_j(x,u,\partial_i u)$$
that is, $\int F^*(u*)dx^*=\int F(u)dc$.
Then
$$\sum_{i=1}^n\frac{\partial}{\partial x_i}M=0$$
whenever $u_j$ are chosen to be extremal.
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