Momentum

In Physical, the momentum is the physical size associated with the Speed and the Masse of an object. It forms part, with the energy, of the values which preserve at the time of the interactions between elements of the system. This law, initially empirical, was explained by the Théorème of Noether and is related to the symmetry of the equations of physics by translation in space.

Momentum and impulse are often confused because of their coincidence in the majority of the cases. Nevertheless in theory these two sizes are distinct.

In traditional mechanics

In Mechanical traditional, the momentum of a material point of mass $m\; \backslash ,$ animated a speed $\backslash \; vec\; \{v\}$, is defined like speed and bulk product:

$\backslash \; vec\; \{p\}\; =m\; \backslash \; vec\; \{v\}$

It is thus, like speed, a vectorial size .

The unit IF of the momentum is the kg $\backslash \; times$ m $\backslash \; times$ S ^-1.

Variation of momentum

A variation of consecutive momentum to the action of a force is calculated as being the Intégrale of the force throughout action of the force. It is calculated:

$\backslash \; mathbf\; \{I\}\; =\; \backslash \; int\; \backslash \; mathbf\; \{F\}\; \backslash ,\; dt$

What, by using the definition of the force, gives:

$\backslash \; mathbf\; \{I\}\; =\; \backslash \; int\; \backslash \; frac\; \{D\; \backslash \; mathbf\; \{p\}\}\; \{dt\}\; \backslash ,\; dt\; =\; \backslash \; int\; D\; \backslash \; mathbf\; \{p\}\; =\; \backslash \; Delta\; \backslash \; mathbf\; \{p\}$

The use, derived from Anglo-Saxon name pulse , is to call this size impulse . Nevertheless, in any rigor, in French impulse indicates the linear Moment, size of the Lagrangian Mécanique. When the duration of action of the force is very short, size I the preceding one is called mechanical Percussion, because of its importance in the theory of the shocks.

Theorem of the center of inertia for a system

In traditional mechanics, the application of the laws of Newton makes it possible to show the theorem of the center of inertia which seems the generalization of the second law of Newton for a non-linear system (solid or together of material points, together of solids):

If $M\; \backslash ,$ indicates the total mass of the system and $G\; \backslash ,$ its Center of inertia, then, the momentum of the system is:

$\backslash \; vec\; \{P\}\; =M\; \backslash \; vec\; \{V\_\; \{G\}\}$

$\backslash \; vec\; \{V\_\; \{G\}\}$ thus indicating the speed of the center of inertia of the system and $M$ total mass of the system.

The theorem is stated then as follows: the variation of the momentum of the system is equal to the sum of the external forces being exerted on the system:

$=\; \backslash \; sum\; \backslash \; vec\; \{F\_\; \{ext.\}\}$

This relation is fundamental: it is it which makes it possible to study the movement of a solid without needing to know the interatomic bonding strengths. It is used to study the fall of an apple that the movement of the Moon around the Earth.

An important particular case: if one imagines the shock of two objects (or particles) for which the external forces (with the system made up of these 2 objects) is null (or negligible), then the total momentum is preserved: it is the same one after the shock as before the shock, and this in spite of the interactions which took place during the shock. It is besides the study of the shocks which led Descartes to think that a certain quantity of the movement was necessarily preserved.

In Lagrangian mechanics

In Mechanical Lagrangian, if one notes $L\; (X,\; \backslash \; dowry\; \{X\})\; \backslash ,$ the Lagrangien of the system with $x\; \backslash ,$ a coordinate of position of the system and $\backslash \; dowry\; \{X\}\; \backslash ,$ its derivative compared to time, one obtains the component of the momentum following the direction X by:

$p\_x\; =\; \backslash \; frac\; \{\backslash \; partial\; \{L\}\}\; \{\backslash \; partial\; \{\backslash \; dowry\; \{X\}\}\}$

This relation, which actually defines the moment combined of the position (or impulse), is not however general. In the case in particular of a particle in charge moving in an electromagnetic field the quantity of movement is defined by:

$p\_x\; =\; \backslash \; frac\; \{\backslash \; partial\; \{L\}\}\; \{\backslash \; partial\; \{\backslash \; dowry\; \{X\}\}\}\; -\; Q\; A\_x$ where Q is the electric Charge of the particle and has is the Potentiel vector.

In relativistic mechanics

The momentum is a preserved size at the time of transformations of translation. If not, that would imply a modification without cause of the position of the Center of gravity of a system of two elastic bodies which are struck.

Also, when Albert Einstein formulated his theory of the restricted Relativité, it adapted the definition of the momentum so that this one is also preserved at the time of relativistic transformations. The size thus obtained is called a 4-moment , it is a vector quantity with four dimensions which combines the traditional momentum and the energy:

where

$\backslash \; mathbf\; \{E\}\; =\; \backslash \; gamma\; m\; c^2$ is total energy

$\backslash \; mathbf\; \{p\}\; =\; \backslash \; gamma\; m\; \backslash \; mathbf\; \{v\}$ is the relativistic momentum

$\backslash \; gamma\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; sqrt\; \{1\; -\; \backslash \; frac\; \{v^2\}\; \{c^2\}\}\}$ is a factor called relativistic gamma

$c\; \backslash ,$ is the Speed of light

The “length” of this vector is the size which remains invariant during translation:

$\backslash \; mathbf\; \{p\}\; \backslash \; cdot\; \backslash \; mathbf\; \{p\}\; -\; (E/c)^2$

The objects of null mass, such as the Photon S, have also a 4-moment when the pseudo-standard of p is null. One has in this cas :

$\backslash \; mathbf\; \{E\}\; ^2-p^2\; c^2\; =\; m^2\; c^4=0$

In quantum mechanics

In Mechanical quantum, the momentum is defined as a Opérateur acting on the Fonction of wave. The Principe of uncertainty of Heisenberg imposes a limit on the precision with which the momentum and the position of a simple observable system can be simultaneously known.

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