Momentum conservation law

The law of conservation of momentum helps to understand the behavior of a shot put pendulum

The law of conservation of momentum is one of the most important conservation laws in physics and states that the total momentum of a mechanically closed system is constant. “Mechanically closed system” means that the system does not interact with its environment.

The conservation of momentum applies in classical mechanics as well as in special relativity and quantum mechanics . It applies regardless of the conservation of energy and is of fundamental importance, for example, when describing collision processes , where the proposition says that the total momentum of all collision partners is the same before and after the collision. The conservation of momentum applies both if the kinetic energy is retained during the collision ( elastic collision ) and if this is not the case (inelastic collision).

The law of conservation of momentum is a direct consequence of the homogeneity of space , i.e. the fact that the behavior of an object is only determined by the values ​​of the physical quantities at its location, but not by the location itself.

Conservation of momentum in Newtonian mechanics

The law of conservation of momentum follows directly from Newton's second and third axioms . According to Newton's second axiom, the change in the momentum of a body over time is equal to the external force acting on it . This law, also called the law of momentum , reads ${\ displaystyle {\ dot {\ vec {p}}}}$${\ displaystyle {\ vec {p}}}$${\ displaystyle {\ vec {F}}}$

${\ displaystyle {\ dot {\ vec {p}}} = {\ vec {F}}}$.

If there are no external forces, according to Newton's third axiom ( “actio = reactio” ) there must be an equally large but opposing force (the so-called counter force) for each force ; the vector sum of these two forces is therefore zero. Since this applies to all forces, the vector sum of all forces occurring in the system and thus also the change in the total momentum is zero. Thus applies

${\ displaystyle {\ vec {F}} = \ sum _ {i = 1} ^ {n} {\ vec {F}} _ {i} = \ sum _ {i = 1} ^ {n} {\ dot {\ vec {p}}} _ {i} = {\ dot {\ vec {p}}} = {\ vec {0}}}$,

therefore the total momentum is a constant vector. If the momentum only depends on the speed, this means that the center of mass moves at a constant speed. ${\ displaystyle {\ vec {p}}}$

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

Conservation of momentum in the Lagrange formalism

In the Lagrange formalism , the conservation of momentum for a free particle follows from the equations of motion. For the Lagrangian for a particle in a potential, the following applies in general ${\ displaystyle L}$${\ displaystyle V (q)}$

${\ displaystyle L = {\ frac {1} {2}} m {\ dot {q}} ^ {2} -V (q)}$

with a generalized coordinate and the particle mass . The equations of motion are ${\ displaystyle q}$${\ displaystyle m}$

${\ displaystyle {\ frac {\ partial L} {\ partial q}} - {\ frac {\ rm {d}} {{\ rm {d}} t}} {\ frac {\ partial L} {\ partial {\ dot {q}}}} = 0}$

and after substituting the above term for ${\ displaystyle L}$

${\ displaystyle - {\ frac {\ partial V} {\ partial q}} - {\ frac {\ rm {d}} {{\ rm {d}} t}} (m {\ dot {q}}) = 0}$.

If it does not depend on, then the partial derivative of the potential according to the generalized coordinate results in the value zero. It remains ${\ displaystyle V}$${\ displaystyle q}$

${\ displaystyle {\ frac {\ rm {d}} {{\ rm {d}} t}} (m {\ dot {q}}) = 0}$.

If one chooses for a position coordinate, then the conservation of momentum of Newtonian mechanics results. ${\ displaystyle q}$

Conservation of momentum as a result of the homogeneity of space

According to Noether's theorem, there is a conserved quantity for every continuous symmetry. The physical symmetry that corresponds to the conservation of momentum is the " homogeneity of space".

Homogeneity of the space means that the system under consideration is shift- invariant, i. that is, a process at point A will not be different if it takes place at any other point B instead . There is no physical difference between points A and B in the sense that the space at B would have different properties than at A.

Let L be the Lagrangian of a physical system, which thus has the effect . The Noether theorem now says: If the effect under a transformation ${\ textstyle S = \ int L \, {\ text {d}} t}$

{\ displaystyle {\ begin {alignedat} {2} q_ {i} & \ mapsto q '_ {i} && = q_ {i} + \ delta \ \ psi _ {i} (q, {\ dot {q} }, t) \\ t & \ mapsto t '&& = t + \ varepsilon \ \ varphi (q, {\ dot {q}}, t) \ end {alignedat}}}

remains invariant, then is

${\ displaystyle Q: = \ sum _ {i} {\ frac {\ partial L} {\ partial {\ dot {q}} _ {i}}} \ psi _ {i} + \ left (L- \ sum _ {i} {\ frac {\ partial L} {\ partial {\ dot {q}} _ {i}}} {\ dot {q}} _ {i} \ right) \ \ varphi}$

a conservation size. The directions in space or time and , in which small shifts or should be carried out, can vary spatially and temporally for a general transformation, which is why and are shown above . ${\ displaystyle \ psi _ {i}}$${\ displaystyle \ varphi}$${\ displaystyle \ delta}$${\ displaystyle \ varepsilon}$${\ displaystyle \ psi _ {i} (q, {\ dot {q}}, t)}$${\ displaystyle \ varphi (q, {\ dot {q}}, t)}$

From the homogeneity of the space it follows that anything can be added to the space coordinates without changing the Lagrangian. In the above general formulation of Noether's theorem, this corresponds to the special case . There are three spatial coordinates, in each of the three spatial directions , and we can shift the coordinates by something that is spatially and temporally constant , without the Lagrangian changing. With Noether theorem we get the three conserved quantities , which are the conjugate momenta to the three space coordinates: ${\ displaystyle \ varepsilon = 0}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle z}$${\ displaystyle \ delta}$${\ displaystyle k = x, y, z}$${\ textstyle Q_ {k} = {\ frac {\ partial L} {\ partial {\ dot {q}} _ {k}}}}$

${\ displaystyle Q_ {k} = {\ frac {\ partial L} {\ partial {\ dot {q}} _ {k}}} = p_ {k}}$

The conservation of these three quantities is precisely the law of conservation of momentum:

${\ displaystyle 0 = {\ frac {{\ text {d}} Q_ {k}} {{\ text {d}} t}} = {\ frac {\ text {d}} {{\ text {d} } t}} {\ frac {\ partial L} {\ partial {\ dot {q}} _ {k}}} = {\ frac {\ text {d}} {{\ text {d}} t}} p_ {k}}$

This applies to all three spatial directions . ${\ displaystyle k = x, y, z}$

Conservation of momentum in the crystal lattice

A special case is an ideal crystal lattice , in which the translation (shift) around a lattice vector is a symmetry operation, that is, leads to an arrangement that is indistinguishable from the original lattice; other shifts result in a grid whose grid points no longer coincide with the original grid points. In this case, conservation of momentum applies with the restriction that a lattice vector of the reciprocal lattice multiplied by Planck's constant can be added to the momentum : ${\ displaystyle \ hbar}$${\ displaystyle {\ vec {G}}}$

${\ displaystyle {\ vec {p}} _ {\ text {after}} = {\ vec {p}} _ {\ text {before}} + \ hbar {\ vec {G}}}$

Impulse cannot be transferred to the crystal lattice to any extent, but only in discrete steps that are determined by the reciprocal lattice. If the momentum for the smallest such step is too small, e.g. B. With visible light inside a crystal, the conservation of momentum applies again as in free space. Therefore, visible light in crystals is not diffracted, but X-rays , which have a higher momentum, can be diffracted. The conservation of momentum taking into account the reciprocal lattice vector is in this case equivalent to the Bragg equation .

Conservation of momentum in flowing fluids

In a flow space, the incoming and outgoing momentum currents are always in equilibrium with the external forces acting on this flow space (balanced force balance). Therefore the following applies for each coordinate direction:

${\ displaystyle \ rho Ac ^ {2} + \ sum {F} = 0}$

The forces include impulse forces, pressure forces, wall forces, inertia forces and frictional forces. The other variables in the equation are: density of the fluid , cross-sectional area through which the flow passes , flow velocity of the fluid${\ displaystyle F}$${\ displaystyle \ rho,}$${\ displaystyle A,}$${\ displaystyle c.}$

Individual evidence

1. C. Gerthsen, H. Vogel: Gerthsen Physik . Springer, 2013, ISBN 978-3-662-07464-0 ( limited preview in Google Book Search [accessed April 13, 2020]).
2. LD Landau, EM Lifshitz: Course of theoretical physics . 3rd ed. 1. Mechanics. Butterworth-Heinemann, 1976, ISBN 0-7506-2896-0 (English, online [PDF; 47.5 MB ; accessed on April 13, 2020] Russian: Курс теоретической физики Ландау и Лифшица, Механика . Translated by JB Sykes, JS Bell).
3. Thorsten Fließbach: Mechanics . 6th edition. Spektrum, Heidelberg / Berlin 2009, ISBN 978-3-8274-1433-5 ( limited preview in Google book search [accessed April 13, 2020]).