Noeter Theorem in mechanics

Noeter theorem (Теорема Нётер) relates the laws of conservation of a Lagrangian system with parametric symmetries of the Lagrangian. In this article, the application to the Classical mechanics is considered.

Statement of the Theorem
Consider a system described with lagrangian \(L=L(q(t),\dot q(t),t)\); dot means the derivative with respect to time \(t\); \(q\) are (generalized) coordinates.

Assume that the action


 * \( S=\int_{a}^{b} L(q,\dot q, t) \mathrm d t \)

remains invariant at the transformation


 * \( t \mapsto \tilde t = t + \varepsilon T(t) + O(\varepsilon^2)\)


 * \( q(t) \mapsto \tilde q(\tilde t) + \varepsilon Q(q(t),t) + O(\varepsilon^2)\)

where \(\varepsilon\) is infinitesimal parameter of the transform \(T\) is generator of transformation of time, and \(Q\) is generator of transformation of coordinates.

\(T\) is assumed to be [[scalar] function of scalar argument.

The generalized coordinates \(q\) may have any origin, and return values from some set of allowed coordinates; but the smoothness of function(s) \(q\) is assumed. and the generator \(Q\) should have values from the same set mentioned. In particular, \(q\) nay denote the set of vectors of cartesian coordinates of the particles involved.

The Lagrangian function \(L\) is assumed to remain the same.

Then, the Lagrange-Euler equation(s)


 * \( \displaystyle

\left( \frac{\partial L}    {\partial \dot q} \right)^{\bullet} = \frac{\partial L}{\partial q}\)

leads to evolution that preserves quantity


 * \(\displaystyle C = \left( \frac{ \partial L}{\partial \dot q} - L \right) T - \frac{\partial L}{\partial q} Q\)

id est, \(\mathrm d C / \mathrm d t = 0\)

Various symmetries of the system provide various laws of conservation.

3-dimensional space
For an isolated system in a uniform isotropic three–dimensional space, the 10 fundamental symmetries give the 10 basic laws of conservation.

The translation with respect to time give the conservation of energy.

The [translation]]s with respect to 3 coordinates give the conservation of 3 momenta.

The boosts along the 3 coordinates give the law of movement of the center of mass; each of the 3 coordinates it is linear function of time. (Which agree with the First Law of Newton.)

The rotations around 3 coordinates give the conservation of 3 angular momenta.

For the simle lagrangian of \(N\) elementary bodies in a 3-dimensional space,
 * \( \displaystyle

L= \sum_{k=0}^{N-1} M_k {\dot {\vec {q_k}}}^2 /2 - U(q)\)

where \(M_k\) are masses of these bodies, and \(U\) is the potential of interaction.

Then, the energy is expressed as


 * \(\displaystyle H = \sum_{k=0}^{N-1} ~M_k~ {\dot {\vec{q_k}}}^2 /2 + U(q) \)

The momentum (vector) is expressed as


 * \(\displaystyle \vec{P} = \sum_{k=0}^{N-1} ~M_k~ \dot{ \vec{q_k}}\)

The momentum (Displacian) is expressed as


 * \(\displaystyle \vec{D} = \sum_{k=0}^{N-1} ~M_k~ \vec{q_k} - \vec{P}~ t\)

and the angular momentum is expressed as


 * \(\displaystyle \vec{\Omega} = \sum_{k=0}^{N-1} ~M_k~ \vec{q_k} \times \dot {\vec {q_k}}\)

Together, the scalar \(H\) and three 3-vectors \(\vec P\), \(\vec D\), \(\vec \Omega\) give 10 laws of conservation. It is interesting, that these 10 quantities remain constants not only in a mechanical system, but also in the field theory.

For simple systems, the conservation laws determine the temporal evolution, allowing to avoid numerical solution of the Lagrange-Euler equations.

Violation of laws of conservation
Consideration of models that models that do not preserve some of the 10 fundamental laws of conservation above may be honest, if the cause violating the conservation, is explicitly declared. This cause may be an external field, or some special support that blocks some of degrees of freedom of a system.

If the cause that break the symmetry (and allow the violation of the conservation law) is hidden, then the consideration is qualified as error or fraud.

Errors and fraud
Historically, any device that pretends to violate the conservation of energy, is called perpetual motion.

Any device that pretends to violate the conservation of momentum, is called inertioid. In century 21, the most known inertioid is gratitsapa. In Russia, there is special institute, Khrunichev center, that is dedicated to engineering of inertioids for the application in the aerospace technology. Due to the principle of relativity of motion, violation of conservation of momentum leads to violation of conservation of energy too ; in this sense, gravitsqpa can be considered also as a perpetual motion machine. Indeed, it is qualified so in the popular publications .

Any device that pretends to violate the conservation of momentum (displacing) can be called varipend, after the invention by Sergei Butov.

In addition, one could expect also the appearance of the claims for violation of conservation of angular momentum; such a device could be called, for example, a twistogen (word invented by the Editor of TORI), but yet the databases do not find such a claim in the internet.

Usually, the frauders are pretty tricky hiding the source of the external force in the accessories of their device. For varipend, the variation of masses of the particles involved is assumed. For Gravitsapa, the high voltage discharge, evaporation of teflon and movement of a liquid by a "special trajectory, similar to tornado" are involved.

Models with limited range of applicability
According to Axiom 1, the range of applicability of any physical concept is limited. However, some models may have narrower range of application, than other, and for some specific cases, the models that do not support some of the laws of conservation may have sense.

The 10 conservation laws mentioned above are entangled with fundamental physical concepts, and not always it is possible to break some of them, keeping the others. The most known cases are the following:

(1) Movement of a charged particle in a stationary external field (foe example, electric or/and magnetic). Such a movement preserves the energy. If the field have certain symmetry, some other quantities may be preserved. For example, the movement of a chanted particle in the centrally–symmetric electric field presedtes not only energy, but also the 3 components of the angular momentum

(2) Movement of a wagon at the idealized railway; such a movement preserves the energy and the momentum.

(3) Movement of an experimentalist sitting a the Zhukovsky bench that preserves the vertical component of the angular momentum.

Extrapolation of properties of the case (3) above to the case (2) may lead to confusions and errors; one of example of such error is varipend by Sergei Butov; his device pretends to violate the conservation of momentum.

Various applications
The Noeter theorem applies not only to classical mechanics, but also to the field theory, and, in rapricular, to the paraxial optics and the Navier-Stokes equation.

In Optics, the conservation of quasi-energy prohibits the quantum annihilation of the optical solution; the conservation momentum in the medium with Kerr nonlinearity prohibits the self–bending of a beam by the gradient of intensity in its transversal profiles, and so on.

In many cases, the basic laws of conservation allows to reveal, that some result is just wrong, without to drill the deduction. In such a way, the Noeter theorem and the resulting conservations are important tools in scientific research in Physics and related area.

Keywords
Noeter theorem, Symmetry, Laws of Newton