Prove that the Lagrangian Density $\mathscr{L}$, which generates a given set of Euler-Lagrange equations, is not unique.

Question

Prove that the Lagrangian Density $\mathscr{L}$, which generates a given set of Euler-Lagrange equations, is not unique.

Hint 1: Adding a divergence to $\mathscr{L}$ does not alter the Euler-Lagrange equation.

Attempt: Let $\mathscr{L´}=\mathscr{L}+\sum_{k}\frac{\partial f_k}{\partial x_k}$

Where $$\mathscr{L}=\mathscr{L}\big(x_k, \varphi_j, \frac{\partial \varphi_k}{\partial x_k}\big)$$ $$f_k=f_k(\varphi_j)$$ $j=1,...,m$ indexes the dependent field variables.

$k=1,...,n$ indexes the independent variables.

Hint 2: Then prove that: $$\frac{\delta \mathscr{L´}}{\delta \varphi_j}=\frac{\delta \mathscr{L}}{\delta \varphi_j}$$

Attempt: Now we define $$\frac{\delta \mathscr{L}}{\delta \varphi_j}=\frac{\partial \mathscr{L}}{\partial \varphi_j}-\sum_{l} \frac{\partial}{\partial x_l} \frac{\partial \mathscr{L}}{\partial (\partial \varphi_j/\partial x_l)}$$

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But i can´t keep following the hints to get the proof done:

Answer 1

Let's see that $$ \sum_k\frac{\partial f_k}{\partial x_k}=\sum_{k,i}\frac{\partial f_k}{\partial \varphi_i}\frac{\partial \varphi_i}{\partial x_k} $$ so that \begin{align} \frac{\partial\mathscr{L}'}{\partial\varphi_j}&=\frac{\partial\mathscr{L}}{\partial\varphi_j}+\frac{\partial}{\partial\varphi_j}\sum_k\frac{\partial f_k}{\partial x_k}=\\ &=\frac{\partial\mathscr{L}}{\partial\varphi_j}+\sum_{k,i}\frac{\partial^2 f_k}{\partial \varphi_j\partial \varphi_i}\frac{\partial \varphi_i}{\partial x_k}= \end{align} and \begin{align} \frac{\partial \mathscr{L}'}{\partial(\partial \varphi_j/\partial x_l)}&=\frac{\partial \mathscr{L}}{\partial(\partial \varphi_j/\partial x_l)}+\frac{\partial}{\partial(\partial \varphi_j/\partial x_l)}\sum_k\frac{\partial f_k}{\partial x_k}=\\ &=\frac{\partial \mathscr{L}}{\partial(\partial \varphi_j/\partial x_l)}+\frac{\partial f_l}{\partial \varphi_j}\\ \sum_l\frac{\partial}{\partial x_l}\frac{\partial \mathscr{L}'}{\partial(\partial \varphi_j/\partial x_l)}&=\sum_l\frac{\partial}{\partial x_l}\frac{\partial \mathscr{L}}{\partial(\partial \varphi_j/\partial x_l)}+\sum_l\frac{\partial}{\partial x_l}\frac{\partial f_l}{\partial \varphi_j}=\\ &=\sum_l\frac{\partial}{\partial x_l}\frac{\partial \mathscr{L}}{\partial(\partial \varphi_j/\partial x_l)}+\sum_{l,i}\frac{\partial^2 f_l}{\partial\varphi_i\partial \varphi_j}\frac{\partial\varphi_i}{\partial x_l} \end{align}