Difference between revisions of "Noeter Theorem"
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| + | #redirect[[Noeter Theorem]] |
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| − | '''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. |
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| − | ==Background== |
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| − | The physical system is assumed to be determined through its [[Lagrangian]], which is function of coordinates $x$, field $u$, and its derivative $u_\bullet$: |
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| − | : $(1) ~ ~ ~ ~ ~ ~ ~ \mathcal L =\mathcal L(x,u,u_\bullet)$ |
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| − | Field and its derivatives are functions of coordinates; |
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| − | $u=u(x)$ and |
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| − | $u_\bullet=u_\bullet(x)$. |
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| − | 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. |
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| − | Action is expressed as integral of Lagrangian: |
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| − | : $(2) ~ ~ ~ ~ ~ ~ ~ A=\int \mathcal L \mathrm{d} x $ |
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| − | It is supposed that field $u$ realizes the [[stationary action]], that leads to the [[Lagrange-Euler equation]] |
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| − | : $\displaystyle ~ ~ ~ ~ ~ ~ ~ |
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| − | \frac{\partial L}{\partial u} = |
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| − | \left(\frac{\partial L}{\partial u_\bullet}\right)_\bullet$ |
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| − | |||
| − | Assume, the transformation of the coordinates $x$ and filed $u$ |
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| − | is parametrized with parameter $\varepsilon$ in such a way that the transformed coordinates $y$ are related to the initial coordinates $x$ with as follows: |
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| − | : $(3) ~ ~ ~ ~ ~ ~ ~ y=x+\delta x(x)=x+\mathcal{X}(x) \varepsilon + \mathcal O(\varepsilon)^2$ |
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| − | The new field should be expressed through the initial field with relation |
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| − | : $(4) ~ ~ ~ ~ ~ ~ ~ v=u+\delta u(x)=u+\Psi(x) \varepsilon + \mathcal O(\varepsilon)^2$ |
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| − | and its derivative |
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| − | : $(5) ~ ~ ~ ~ ~ ~ ~ v_\bullet=u_\bullet+\delta u_\bullet (x)$ |
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| − | where $\varepsilon$ is parameter of transformation, and $\varepsilon$ is assumed to be small. |
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| − | The invariance of action with respect to transform (3)-(5) means that |
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| − | : $(8)\displaystyle ~ ~ ~ ~ ~ ~ ~ \int_{\Omega_\varepsilon} \mathcal L(y,v(y), v_\bullet(y)) ~\mathrm{d}^N(y)= \int_{\Omega} \mathcal L(x,u(x), u_\bullet(x)) ~\mathrm{d}^N(x)$ |
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| − | it is assumed that the $\Omega_\varepsilon$ is transform of $\Omega$. |
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| − | ==Statement of the Theorem== |
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| − | The invariance (8) leads to the conservation of the Noeterian current |
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| − | : $(9)\displaystyle ~ ~ ~ ~ ~ ~ ~ \Theta= \Theta(x,u,u_\bullet)= - \mathcal L \mathcal X |
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| − | - \frac{ \partial \mathcal L}{\partial u_\bullet} \Psi |
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| − | + \frac{ \partial \mathcal L}{\partial u} u_\bullet \mathcal X$ |
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| − | |||
| − | In such a way, $\Theta$ is vector of length $N$. The law of conservation has the form |
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| − | : $(10) ~ ~ ~ ~ ~ ~ ~ \Theta_\bullet = 0$ |
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| − | where the summation with respect to repeating subscripts is assumed. |
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| − | |||
| − | ==Proof of the Noeter theorem== |
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| − | Using the infinitesimal generators |
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| − | $\chi = \varepsilon \mathcal{X}(x)$ and |
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| − | $\psi = \varepsilon \Psi(x)$, in the first order with respect to $\varepsilon$ , the transform of the coordinates and the field can be expressed as follows: |
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| − | : $(11) \displaystyle ~ ~ ~ ~ ~ ~ ~ y=x+\chi~; ~ ~ x=y-\chi$ |
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| − | : $(12) \displaystyle ~ ~ ~ ~ ~ ~ ~ v(y)=u(x)+\psi = u(y\!-\!\chi) + \psi$ |
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| − | : $(13) \displaystyle ~ ~ ~ ~ ~ ~ ~ v_\bullet(y)= |
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| − | \frac{\partial V(y)}{\partial y}= |
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| − | u_\bullet(y-\chi)(I-\chi_\bullet)+\psi_\bullet= |
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| − | u_\bullet(x)-u_\bullet \chi + \psi_\bullet |
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| − | $ |
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| − | |||
| − | where $I$ is diagonal identity operator. |
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| − | Then variation $\delta A$ of action $A$ can be expressed as follows: |
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| − | : $(21) \displaystyle ~ ~ ~ ~ ~ ~ ~ |
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| − | \delta A = |
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| − | \int \mathcal{L}(y,v(y),v_\bullet(y)) \mathrm{d}^N y - |
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| − | \int \mathcal{L}(x,u(x),u_\bullet(x)) \mathrm{d}^N x=$ |
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| − | : $ \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ = |
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| − | \int \mathcal{L}(y,v(y),v_\bullet(y)) (I+\chi_\bullet)\mathrm{d}^N x- |
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| − | \int \mathcal{L}(x,u(x),u_\bullet(x) \mathrm{d}^N x=$ |
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| − | |||
| − | : $ ~ ~ \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ = |
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| − | \int \Big( |
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| − | \mathcal{L} \chi_\bullet+ |
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| − | \mathcal{L}\!\big(x\!+\!\chi,v(y),v_\bullet(y)\big) |
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| − | - |
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| − | \mathcal{L}(x,u(x),u_\bullet(x)\big) \Big) \mathrm{d}^N x$ |
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| − | |||
| − | : $ ~ ~ \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ = \int \Big( |
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| − | \mathcal{L} \chi_\bullet+ |
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| − | \mathcal{L}\!\big(x\!+\!\chi,u(x)+\psi, u_\bullet(x)-u_\bullet \chi +\psi_\bullet \big) |
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| − | - \mathcal{L}(x,u(x),u_\bullet(x)\big) \Big) \mathrm{d}^N x=$ |
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| − | |||
| − | : $ ~ ~\displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ = |
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| − | \int \Big( |
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| − | \mathcal{L} \chi_\bullet |
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| − | + \mathcal{L}_x \chi |
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| − | + \mathcal{L}_u u_\bullet \chi |
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| − | + \mathcal{L}_{u_\bullet} u_{\bullet\bullet} \chi$ |
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| − | |||
| − | : $ \phantom{(21 12345678901)}~ ~ \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ |
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| − | - \mathcal{L}_u u_\bullet \chi |
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| − | - \mathcal{L}_{u_\bullet} u_{\bullet\bullet} \chi |
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| − | +\mathcal L _u \psi |
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| − | +\mathcal L _{u_\bullet} \psi_\bullet -\mathcal L_{u_\bullet} \chi \Big) \mathrm{d}^N x$ |
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| − | The first line of the integrand in the last expression already can be combined into the divergence, giving |
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| − | : $ (22) \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ \delta A= |
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| − | \int \Big( |
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| − | (\mathcal L \chi)_\bullet+ |
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| − | \mathcal L_u \cdot (\psi - u_\bullet \chi) + |
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| − | \mathcal L_{u_\bullet} \cdot (-u_{\bullet\bullet} \chi) + \mathcal L_u \psi_\bullet -\mathcal L_{u_\bullet} u_\bullet \chi |
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| − | \Big) \mathrm{d}^N x$ |
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| − | |||
| − | or, in slightly different form, |
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| − | : $ (23) \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ \delta A= |
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| − | \int \Big( |
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| − | (\mathcal L \chi)_\bullet+ |
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| − | \mathcal L_u \cdot (\psi - u_\bullet \chi) + |
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| − | \mathcal L_{u_\bullet} \cdot \big( \psi -u_{\bullet} \chi \big )_\bullet |
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| − | \Big) \mathrm{d}^N x$ |
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| − | Using the equation of Lagrange-Euler, $\mathcal L_u$ can be replaced to $\big(\mathcal L_{u_\bullet})_\bullet$ ; |
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| − | this gives |
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| − | : $ (24) \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ \delta A= |
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| − | \int \Big( |
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| − | (\mathcal L \chi)_\bullet+ |
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| − | \big(\mathcal L_{u_\bullet})_\bullet\cdot (\psi - u_\bullet \chi) + |
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| − | \mathcal L_{u_\bullet} \cdot \big( \psi -u_{\bullet} \chi \big)_\bullet\Big) \mathrm{d}^N x$ |
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| − | and the integrand can be represented as the divergence: |
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| − | : $ (25) \displaystyle ~ ~ ~ ~ ~ ~ ~ ~ ~ \delta A= |
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| − | \int \Big( |
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| − | \mathcal L \chi + |
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| − | \mathcal L_{u_\bullet} \cdot \big( \psi -u_{\bullet} \chi \big)\Big )_\bullet ~ \mathrm{d}^N x$ |
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| − | The integrand is the conservating current, corresponding to the symmetry of the action declared at the beginning. |
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| − | (End of proof) |
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| − | The indices in (25) are traced in the special article [[Proof:Noeter theorem]]; with subscripts, the law of conservation |
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| − | can be written a follows: |
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| − | : $ (26) ~ ~ ~ ~ ~ ~ \Big( |
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| − | \mathcal L \mathcal X_k + |
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| − | ( \Psi_{\alpha} -u_{\alpha,j} \mathcal X_j ) \mathcal L_{u_{\alpha,k}} \Big) _{,k} =0$ |
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| − | ==Use of the Theorem== |
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| − | 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: |
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| − | : $(11) \displaystyle ~ ~ ~ ~ ~ ~ ~ \int_\Omega \Theta_{0} \mathrm d^{N-1} x=\mathrm {constant}$ |
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| − | 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. |
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| − | 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]]. |
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| − | ==References== |
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| − | http://en.wikipedia.org/wiki/Noether's_theorem |
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| − | [[Category:Conservation laws]] |
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| − | [[Category:Articles in English]] |
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Latest revision as of 07:03, 1 December 2018
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