Noeter Theorem
Noeter Theorem defines the conservation laws, that corresponds to the continuous symmetries of the action. In this article, the most of subscripts are omitted, assuming that the colleagues can recover them by themselves.
Background
The physical system is assumed to be determined through its Lagrangian, which is function of coordinates $x$, field $u$, and its derivative $u_\bullet$:
- $(1) ~ ~ ~ ~ ~ ~ ~ \mathcal L =\mathcal L(x,u,u_\bullet)$
Field and its derivatives are functions of coordinates; $u=u(x)$ and $u_\bullet=u_\bullet(x)$.
In general, $x$ may be a vector in some $N$dimensional space; and the field $u$ may also have several components, for example, $M$ components; $u_\bullet$ is assumed to be a matrix with the corresponding number of raws and columns, id set, $M \times N$ matrix.
Action is expressed as integral of Lagrangian:
- $(2) ~ ~ ~ ~ ~ ~ ~ A=\int \mathcal L \mathrm{d} x $
It is supposed that field $u$ realizes the stationary action, that leads to the Lagrange-Euler equation
- $\displaystyle ~ ~ ~ ~ ~ ~ ~
\frac{\partial L}{\partial u} = \left(\frac{\partial L}{\partial u_\bullet}\right)_\bullet$
Assume, the transformation of the coordinates $x$ and filed $u$ is parametrized with parameter $\varepsilon$ in such a way that the transformed coordinates $y$ are related to the initial coordinates $x$ with as follows:
- $(3) ~ ~ ~ ~ ~ ~ ~ y=x+\delta x(x)=x+\mathcal{X}(x) \varepsilon + \mathcal O(\varepsilon)^2$
The new field should be expressed through the initial field with relation
- $(4) ~ ~ ~ ~ ~ ~ ~ v=u+\delta u(x)=u+\Psi(x) \varepsilon + \mathcal O(\varepsilon)^2$
and its derivative
- $(5) ~ ~ ~ ~ ~ ~ ~ v_\bullet=u_\bullet+\delta u_\bullet (x)$
where $\varepsilon$ is parameter of transformation, and $\varepsilon$ is assumed to be small.
The invariance of action with respect to transform (3)-(5) means that
- $(8)\displaystyle ~ ~ ~ ~ ~ ~ ~ \int_{\Omega_\varepsilon} \mathcal L(y,v(y), v_\bullet(y)) ~\mathrm{d}^N(y)= \int_{\Omega} \mathcal L(x,u(x), u_\bullet(x)) ~\mathrm{d}^N(x)$
it is assumed that the $\Omega_\varepsilon$ is transform of $\Omega$.
Statement of the Theorem
The invariance (8) leads to the conservation of the Noeterian current
- $(9)\displaystyle ~ ~ ~ ~ ~ ~ ~ \Theta= \Theta(x,u,u_\bullet)= - \mathcal L \mathcal X
- \frac{ \partial \mathcal L}{\partial u_\bullet} \Psi + \frac{ \partial \mathcal L}{\partial u} u_\bullet \mathcal X$
In such a way, $\Theta$ is vector of length $N$. The law of conservation has the form
- $(10) ~ ~ ~ ~ ~ ~ ~ \Theta_\bullet = 0$
where the summation with respect to repeating subscripts is assumed.
Proof of the Noeter theorem
Using the infinitesimal generators $\chi = \varepsilon \mathcal{X}(x)$ and $\psi = \varepsilon \Psi(x)$, in the first order with respect to $\varepsilon$ , the transform of the coordinates and the field can be expressed as follows:
- $(11) \displaystyle ~ ~ ~ ~ ~ ~ ~ y=x+\chi~; ~ ~ x=y-\chi$
- $(12) \displaystyle ~ ~ ~ ~ ~ ~ ~ v(y)=u(x)+\psi = u(y\!-\!\chi) + \psi$
- $(13) \displaystyle ~ ~ ~ ~ ~ ~ ~ v_\bullet(y)=
\frac{\partial V(y)}{\partial y}= u_\bullet(y-\chi)(I-\chi_\bullet)+\psi_\bullet= u_\bullet(x)-u_\bullet \chi + \psi_\bullet $
where $I$ is diagonal identity operator. Then variation $\delta A$ of action $A$ can be expressed as follows:
- $(21) \displaystyle ~ ~ ~ ~ ~ ~ ~
\delta A = \int \mathcal{L}(y,v(y),v_\bullet(y)) \mathrm{d}^N y - \int \mathcal{L}(x,u(x),u_\bullet(x)) \mathrm{d}^N x=$
- $ \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ =
\int \mathcal{L}(y,v(y),v_\bullet(y)) (I+\chi_\bullet)\mathrm{d}^N x- \int \mathcal{L}(x,u(x),u_\bullet(x) \mathrm{d}^N x=$
- $ ~ ~ \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ =
\int \Big( \mathcal{L} \chi_\bullet+ \mathcal{L}\!\big(x\!+\!\chi,v(y),v_\bullet(y)\big) - \mathcal{L}(x,u(x),u_\bullet(x)\big) \Big) \mathrm{d}^N x$
- $ ~ ~ \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ = \int \Big(
\mathcal{L} \chi_\bullet+ \mathcal{L}\!\big(x\!+\!\chi,u(x)+\psi, u_\bullet(x)-u_\bullet \chi +\psi_\bullet \big) - \mathcal{L}(x,u(x),u_\bullet(x)\big) \Big) \mathrm{d}^N x=$
- $ ~ ~\displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ =
\int \Big( \mathcal{L} \chi_\bullet + \mathcal{L}_x \chi + \mathcal{L}_u u_\bullet \chi + \mathcal{L}_{u_\bullet} u_{\bullet\bullet} \chi$
- $ \phantom{(21 12345678901)}~ ~ \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~
- \mathcal{L}_u u_\bullet \chi - \mathcal{L}_{u_\bullet} u_{\bullet\bullet} \chi +\mathcal L _u \psi +\mathcal L _{u_\bullet} \psi_\bullet -\mathcal L_{u_\bullet} \chi \Big) \mathrm{d}^N x$
The first line of the integrand in the last expression already can be combined into the divergence, giving
- $ (22) \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ \delta A=
\int \Big( (\mathcal L \chi)_\bullet+ \mathcal L_u \cdot (\psi - u_\bullet \chi) + \mathcal L_{u_\bullet} \cdot (-u_{\bullet\bullet} \chi) + \mathcal L_u \psi_\bullet -\mathcal L_{u_\bullet} u_\bullet \chi \Big) \mathrm{d}^N x$
or, in slightly different form,
- $ (23) \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ \delta A=
\int \Big( (\mathcal L \chi)_\bullet+ \mathcal L_u \cdot (\psi - u_\bullet \chi) + \mathcal L_{u_\bullet} \cdot \big( \psi -u_{\bullet} \chi \big )_\bullet \Big) \mathrm{d}^N x$
Using the equation of Lagrange-Euler, $\mathcal L_u$ can be replaced to $\big(\mathcal L_{u_\bullet})_\bullet$ ; this gives
- $ (24) \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ \delta A=
\int \Big( (\mathcal L \chi)_\bullet+ \big(\mathcal L_{u_\bullet})_\bullet\cdot (\psi - u_\bullet \chi) + \mathcal L_{u_\bullet} \cdot \big( \psi -u_{\bullet} \chi \big)_\bullet\Big) \mathrm{d}^N x$
and the integrand can be represented as the divergence:
- $ (25) \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ \delta A=
\int \Big( \mathcal L \chi + \mathcal L_{u_\bullet} \cdot \big( \psi -u_{\bullet} \chi \big)\Big )_\bullet ~ \mathrm{d}^N x$
The integrand is the conservating current, corresponding to the symmetry of the action declared at the beginning. (End of proof)
The indices in (25) are traced in the special article Proof:Noeter theorem; with subscripts, the law of conservation can be written a follows:
- $ (26) ~ ~ ~ ~ ~ ~ \Big(
\mathcal L \mathcal X_k + ( \Psi_{\alpha} -u_{\alpha,j} \mathcal X_j ) \mathcal L_{u_{\alpha,k}} \Big) _{,k} =0$
Use of the Theorem
The integration of the current $\Theta$ with respect to spatial coordinates leads to the scalar quantity, that determines the time derivative. Let the time coordinate has number zero. Then, at the integration over a domain where the field vanish at the boundary, the integral of the 0th component of the current conserves:
- $(11) \displaystyle ~ ~ ~ ~ ~ ~ ~ \int_\Omega \Theta_{0} \mathrm d^{N-1} x=\mathrm {constant}$
For the case of conservation of electric charge, the Noeterian current has the same sense as the usual electric current. Often, there are a lot of transforms, that preserve the action. Each of them, by the Theorem above, gives the law of conservation. In particular, the translational symmetry of the space-time allows the four-parametric group of transform; each parameter gives the law of conservation; together they form the tensor of energy–momentum. The spatial integral of the zeroth components gives the vector of energy–momentum.
Conservation of Energy-momentum is one of the most fundamental principles of physics. Intents to negate this principle are described in the article Gravitsapa. Huge budget is spent for the development of the "propulsors without expulsion of the workint nedium" (движители без выброса рабочего тела); the Noeter theorem indicates that such activity is fraud.