Conservation of energy-momentum

Conservation of energy-momentum is one of the most basic principles of the modern physics.

While the space-time (configurational space]] is considered as uniform, the physical theories are supposed to remain invariant at the translations, rotations and boosts. In particular, the invariance with respect to translation in time and space leads to conservation of energy and momentum. Usually, the physical models (concepts) allow definition of such quantities as energy and momentum; and the conservation of these quantities simplifies consideration of physical system.

Classical mechanics
In Classical mechanics, momentum \(\vec P\) of a system appears as sum of momenta of the elementary bodies (particles) of the system,

(1) \(\displaystyle ~ ~ ~ \vec P= \sum_{n=1}^N \vec p_n = \sum_{n=1}^N m_n \vec v_n\)

where \(\vec p_n\) is momentum of the \(n\)th particle, \(m_n\) is its mass, and \(\vec v_n\) is its velocity, and \(N\) is number of particles. Velocities \(\vec v_n\) and momenta \(\vec p_n\) are supposed to be functions of time, but, for the isolated system, \(\vec P\) is supposed to remain constant.

Similarly, the energy \(E\) is expressed as (2) \(\displaystyle ~ ~ ~ E= \sum_{n=1}^N E_n = \sum_{n=1}^N m_n v_n^2/2\)

Gravitation
In careful consideration of gravity, the total energy of a system is just zero; so, the conservation of energy does not prohibit creation of Universe "from nothing".

In addition, the space–time is not flat, and the conservation laws cannot be written in the global form. The inhomogeneity of space can be detected at the precise measurement of frequency. Precision of order of 18 decimal digits is sufficient to detect (and measure) the altitude of the clock with respect to the sea level, that can be interpreted as distance from the "center of the Earth".

Optics
In certain optical models, the translational and rotational invariance takes place; that leads to conservation of certain quantities, similar to energy and momentum. The analogy of the law of conservation of energy-momentum applies to some nonlinear media. For example, such a conservation prohibits the bending of a ray with gradient of intensity by the nonlinearity of a uniform medium.

Frauds
Since the establishing of Fundamental Laws of Conservation, the numerous attempts to "brake" these laws are registered.

The most often are reports about the perpetial motion machines.

Less often are so-called inertioid]s that claim the violation of conservation of momentum. Inertioid is any device that generates a support-less [[force, id est, provides the inertial propulsion. In centiry 21, in Russia, the Khrunichev State Research and Production Space Center  develops the "reactive motors without exhaust of the working mass" . Development of the inertioids in that center causes doubts also in all other achievement reported at its cite.