Momentum#Conservation of momentum

The momentum conservation law is one of the most important conservation laws of physics, and states that the total momentum is constant in a closed system. " Closed system " means that the system has no interaction with its surroundings.

The conservation of momentum applies both in classical mechanics and in the special theory of relativity and quantum mechanics. It applies regardless of the conservation of energy and is about in the description of collision processes is of fundamental importance, where the law states that the total momentum of all collision partners before and after the collision must be equal. Momentum conservation applies, if received in the collision, the kinetic energy is ( elastic collision ), and if this is not the case ( inelastic collision).

Conservation of momentum in Newtonian mechanics

The momentum conservation law follows directly from the second and third Newtonian axiom. In accordance to Newton's second axiom is the change in the pulse by the time equal to the force acting on a body external force, that is:

If there are no forces from the outside, it must be according to the third Newtonian axiom type for each forcing an equal but oppositely acting force ( " action = reaction = "); the vector sum of these forces is therefore zero. Since this is true for all the forces, is the vector sum of the forces occurring in the system and thus also the change of the total momentum zero. Thus, the following applies:

Which is a constant. When the pulse is only dependent on the speed, this means that the center of gravity moves at a constant speed.

The conservation of momentum is equivalent with the statement that moves the center of gravity of a system without external force at a constant speed and direction ( this is a generalization of the first Newtonian axiom, which was originally formulated only for individual body ).

Momentum conservation in the Lagrangian formalism

In the Lagrangian formalism, the momentum conservation for a free particle follows directly from the equations of motion. The Lagrangian for a particle in a potential is generally

The equations of motion are:

Does not depend on (ie: Due to the potential of no force acts on the particle, one speaks of a free particle ), as follows:

This is exactly equivalent to conservation of momentum of Newtonian mechanics.

In the Lagrangian formalism, a corresponding derivation is also possible for the conservation of angular momentum, if one uses generalized coordinates.

Momentum conservation as a consequence of the homogeneity and isotropy of space

Conservation of momentum

Under the homogeneity of space is defined as a shift-invariance; i.e., a process at the point A does not occur differently when it occurs instead at any other point B. There is no physical difference between the points A and B in the sense that the space B for other properties than possessed A. From this property, the following conservation of momentum in the following manner:

It was a generalized coordinate which describes a shift and the Lagrange function must remain invariant in accordance with the homogeneity of the area under this shift. Then is a cyclic coordinate and the corresponding generalized momentum is conserved.

The vector is thus shifted in any direction, is then obtained by Taylor expansion:

The expression:

Is a vector indicating the displacement direction. The corresponding to the cyclic coordinate generalized impulse is then

It was assumed in the first calculation step that the potential V does not depend on the generalized velocity. Now use that:

Applies. Finally, in order follows:

Accordingly, the projection of the total momentum is obtained in the direction of the displacement. If is a unit vector, the generalized impulse with this projection is identical. This is not the case, it differs from it by a constant factor.

Note: The Noether theorem

The above- derived conservation laws are actually special cases of a more general formulation, which was commissioned by Emmy Noether. By Noether 's theorem is generally defined circumstances under which it is a size of a physical system is a conserved quantity and what it looks like.

Momentum conservation in the crystal lattice

A special case is an ideal crystal lattice, in which the translation ( shift ) by a lattice vector a symmetry operation is, so again leads to indistinguishable from the original lattice arrangement; other shifting of a grid, not the grid points coincide with the original grid points. In this case, the conservation of momentum applies, with the restriction that for a pulse multiplied by Planck's quantum of action lattice vector of the reciprocal lattice can be added:

So it can not be transferred in any pulse level of the crystal lattice, but only in discrete steps as determined by the reciprocal lattice. When the pulse for the smallest such a step is too small, such as visible light in the interior of a crystal, again applies the momentum in the free space. Therefore, visible light is not diffracted in crystals, however, can ray radiation, having a higher momentum, being diffracted. Taking into account the conservation of momentum of the reciprocal lattice vector in this case is equivalent to the Bragg equation.

Momentum conservation in flowing fluids

In a flow space are the incoming and outgoing pulse currents with the external, acting on these flow space forces always in equilibrium ( balanced force balance ). Hence, in each coordinate direction:

The forces include this impulse forces, pressure forces, wall forces, inertia forces and frictional forces.

• Classical Mechanics
• Physical basic concept