Conservation law

Definition. @olver86 section 4.3.
Consider a system of DEs ${\mathrm{\Delta }}_{\nu }(x,u^{(n)})=0$. A conservation law is an expression

which vanishes for all solutions $u=f(x)$ of the system. Here $P=(P_1(x,u^{(n)}),\dots ,P_p(x,u^{(n)}))$ is a $p$-tuple of smooth functions and $\text{Div}$ is the total divergence.
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Particular cases

System of ODEs

In the particular case of a system of ODEs, where we have only 1 independent variable and $u$ represents functions

a conservation law is of the form $D_{x}P=0$ for every solution of the system, that is, $P(x,u^{(n)})$ must be constant along solutions. Son in this case a conservation law is nothing but a first integral of the system.

In a dynamical context

Suppose that our system of DE arise in a dynamical context, and that one of the independent variables is the time $t$ and the other $x_1,\dots ,x_p$ are spatial variables. The conservation law is given by

The function $T$ is called the conserved density and $X=(X_1,\dots ,X_p)$ are called the associated flux. This is completely related to the note four-current . Indeed, an example of conservation law is the continuity equation.

This should be related to evolution equation, but I have to think about it more...

Characteristic of a conservation law

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