Point mechanics

Two- body problem of two attractive mass points (blue)

The point mechanics is a branch of mechanics , in which, apart from the nature and shape of the body, the movements of interest and on their in center of mass concentrated mass are reduced. This idealization simplifies the mathematical treatment and had its first successes in celestial mechanics , where Isaac Newton succeeded in mathematically representing the Kepler orbits around the sun , see picture. Because the dimensions of the celestial bodies in the solar system are small compared to their distance and the influence of their own rotations is negligible.

The concept of the mass point is successful in many areas and applications and it can also be used to deal with aspects of relativity and quantum mechanics . The presentation remains clear and also accessible to beginners.

History

Point mechanics goes back to the founder of theoretical physics Isaac Newton. He succeeded in deriving the law of gravitation from Kepler's laws and generalizing the area law to any central forces . This success led to efforts to attribute all natural phenomena to forces that act between two point-like particles and depend only on their distance, a conception that dominated physics well into the 19th century.

Point dynamics has found a new application in providing mechanical analogies of physical phenomena with which James Clerk Maxwell, for example , came to his fundamental equations of electromagnetism . The equations of motion can be transferred from the mechanical system to the analog, whereby the Lagrangian equations gained a universal meaning for the whole of physics.

Mechanically similar behavior of systems was already noticed by Aristotle , Galileo Galilei and Isaac Newton , but only MJ Bertrand pronounced the principle of mechanical similarity based on point mechanics with full rigor.

Areas of application and limitations

The idealization of a body as a mass point is effective when a characteristic dimension of the body is small compared to the accuracy with which length measurements are made, and errors in the order of magnitude of the ratio of the distances set to zero to the other distances can be accepted. The location of the mass point is generally the center of mass or - which is often the same - the center of gravity of the body it represents.

Point mechanics cannot deal with effects that go back to the expansion of real bodies. Self-rotation with precession and nutation or deformations can be mapped more appropriately with other disciplines such as rigid body or continuum mechanics , where mathematics is, however, much more complicated, not least because the rigid body has six and the deformable one has an infinite number of degrees of freedom . The meaning of the theorem of twist as an independent principle cannot be represented in point mechanics, see theorem of twist in point mechanics , because in point mechanics the conservation of angular momentum is a consequence of the conservation of momentum and of the fact that mass points can only absorb central forces.

Since the establishment of theoretical physics by Isaac Newton in 1687, elementary point mechanics has been presented practically unchanged in textbooks, even though it is logically unsatisfactory in some respects. These deficiencies in Newtonian mechanics have been remedied by relativity and quantum mechanics , but there are a considerable number of statements that are the same in classical and modern physics. These statements all concern fundamental physical quantities which describe processes through their exchange. Newtonian, Einsteinian and quantum mechanics have in common that colliding bodies or particles exchange momentum and energy. Only the kinematics , i.e. the geometric representation of movement through space, is fundamentally different in modern physics and classical mechanics .

Equations of motion and integrals of motion

The movement of a mass point is described by its three-dimensional position vector as a function of the time parameter, which is treated like an additional coordinate. While the mass point moves through space, time derivatives can be used to determine its speed and acceleration . Conversely, in physical laws such as Newton's law of gravitation or Coulomb's law , the acceleration of the mass point is given and the path-time law then results from double time integration. A family of trajectories characterize the type of movement associated with the acceleration curve . The Kepler orbits are the type of movement of the planets in the gravitational field of the solar system.

Values ​​are important that are constant over time for a body typically moving on a railway. These constants are called integrals of the motion or first integrals of a type of motion and have fundamental representatives in the conservation laws . For example, the total energy of an isolated system of mass points is an integral of the motion due to the law of conservation of energy . From the first integrals, important conclusions can often be drawn about the course of the movement, see for example the following section # Newton's law of gravitation . The integrals are a function of the location and the speed but are constant along the trajectory, which is why the function value is already fixed with the initial conditions.

Newton's law of gravitation

See also : Newton's law of gravitation

Two attractive mass points m ₁ and m ₂

Newton's law of gravitation states that two interacting mass points are accelerated towards each other, which is noticeable as a force of attraction, see picture. In a modern perspective, the gravitational field is an acceleration field and not a force field. Newton had derived his law of gravity from Kepler's laws and Johann I Bernoulli was able to show in 1710 that a central force reciprocal to the square of the distance always leads to a Kepler movement on a conic section , as in Newton's law .

The type of motion of Newton's theory of gravity are accordingly Kepler orbits , as celestial bodies follow. The motion integral in the form of the specific orbital energy determines the type of motion, which is elliptical, hyperbolic or parabolic, depending on whether the integral is negative, positive or zero. The two-body problem can be solved analytically, which is generally not (analytically) possible with the three -body problem and N-body problem with N> 3.

In the case of gravitationally interacting mass points, there are seven integrals of motion that go back to the conservation laws of momentum , angular momentum and energy . The gravitational field generated by them also participates in the exchange of momentum and energy between the bodies , which in contrast to the mass points cannot be localized but is spread over the entire space. The Newtonian gravitational field can absorb energy, but not momentum, which it therefore immediately releases again to other mass points (principle Actio and Reactio ).

The constant total energy stored in the two-body system of the two-body problem can be split into the center of gravity or external energy, which is the constant kinetic energy of the center of gravity, and the therefore also constant internal energy. The center of gravity energy can be reduced to zero through the transition to the center of gravity system , in which the total momentum disappears, while the internal energy is invariant to Galileo transformations . In the absence of total momentum, the total energy is minimal and equals the internal energy. With this decomposition, the solution of the two-body problem is simplified. The splitting into path-transformable external energy and a minimum value of the energy, the internal or rest energy , is also important in Einstein's mechanics .

dynamics

Dynamics describes the movement of a body in such a way that the body absorbs or transfers certain exchangeable physical quantities from other bodies or systems in the course of movement . In particle mechanics momentum, angular momentum and energy are interchangeable and they are conservation laws ( conservation of momentum , angular momentum conservation , energy conservation ). They can therefore neither be created nor destroyed, but only accepted, given away or kept. The gravitational field takes part in the exchange and is able to transfer momentum and energy from one body to another.

Momentum balance

Two mass points can transmit momentum through forces which, according to the Actio and Reactio principle, always exert opposite and opposite pairs to one another, such as a collision . The modeled real bodies are more or less deformed, in which the mutual force effects reveal themselves. The momentum is transmitted according to Newton's second law, force equals mass times acceleration , where the product of mass and acceleration is the rate of change of momentum. The increase or decrease in momentum is the result of the impulse exerted , through which the exerting system loses momentum to the same extent as it transmits through the impulse. The gravitational field can also cause a change in momentum due to the force of attraction , but since it cannot absorb any momentum itself, it must momentarily give it to other mass points (principle of action and reaction). Jean-Baptiste le Rond d'Alembert saw the product of mass and acceleration as the “ inertial force ” and the gravitational force as an “ external force ”. A distinction must be made between this and the reaction force due to geometrical bonds on rails or surfaces, a force that only arises in the course of movement.

Angular momentum balance

The angular momentum , like the momentum, is a conserved quantity and can only be changed by applying a torque . Angular momentum is withdrawn from the system that exerts the torque to the same extent as it is transmitted by the torque.

In a system of mass points, the angular momentum can be broken down into the orbital angular momentum of the center of mass and an intrinsic angular momentum around the center of mass. Both angular momentum, orbital and intrinsic angular momentum, are conserved quantities, but only the intrinsic angular momentum is invariant to Galileo transformations of the reference system. Two non-rotating observers, moving uniformly against each other, always perceive matching intrinsic angular impulses of the bodies they are observing.

Energy balance

The energy of a particle can only be changed by applying a force on him work done along a path. The system that exerts the force loses the same amount of energy as is transmitted through the work. The speed of energy transfer is the work per unit of time or the power . Relative to a fixed reference point, the force produces a torque that does work in the rotation, so that the mass point is given rotational energy .

The gravitational field is able to absorb, store or give off energy from mass points.

Mass conservation

With the momentum, the mass must also fulfill a conservation law in Newtonian dynamics.

Because the total momentum of a closed system of mass points is, unlike their masses, dependent on the speed of the reference system used. In the center of gravity system , the total momentum is constantly zero, even if the mass points exchange momentum with one another. An observer who moves uniformly in relation to this perceives a total impulse which is also constant and proportional to his relative speed and the total mass. Because this can be determined at every relative speed, the total mass must be the same at all times. This also applies if a speed dependency of the individual masses is permitted.

In Newtonian mechanics, heavy mass and inert mass are necessarily proportional to each other.

Because the two-body problem is in the pulse of the movement integrally only the heavy mass one ( "gravitational charge") and its current speed. In a collision with the same speed, the same bodies would show an impulse proportional to their inertial mass , which in total is also a conservation quantity. In the momentum equations, the velocities with the heavy or inert masses appear as coefficients. If both equations were linearly independent, the then inevitably constant velocities could be calculated from them and thus only straight movements on the Kepler orbits would be allowed, which obviously does not apply. The linear dependence of the equations leads to the proportionality of the inert and heavy masses in Newtonian mechanics.

Aspects of Modern Physics

Basics of mechanics

Newton's and Einstein's mechanics can be derived from three laws:

1. Momentum and energy are conserved quantities.
2. Velocity and momentum are parallel and their proportionality factor, the mass , can depend on the velocity: ${\displaystyle {\vec {v}}}$ ${\displaystyle {\vec {p}}}$ ${\displaystyle m}$ ${\displaystyle {\vec {p}}=m(v){\vec {v}}}$
3. Fundamental equation of dynamics: The change of the energy d E corresponding to the change of the pulse , when the transport speed of the energy is equal in speed . ${\displaystyle \mathrm {d} {\vec {p}}}$ ${\displaystyle \mathrm {d} E={\vec {v}}\cdot \mathrm {d} {\vec {p}}}$

Mechanics are defined by the energy-momentum relation . ${\displaystyle E=E({\vec {p}})}$

Newtonian mechanics

In Newton's mechanics , the mass m is constant and from which the energy-momentum relation is derived ${\displaystyle {\vec {p}}=m{\vec {v}}}$ ${\displaystyle \mathrm {d} E={\tfrac {1}{m}}{\vec {p}}\cdot \mathrm {d} {\vec {p}}={\tfrac {1}{m}}\mathrm {d} (p^{2}/2)}$

${\displaystyle E({\vec {p}})={\frac {p^{2}}{2m}}+E_{0}={\frac {m}{2}}v^{2}+E_{0}}$

derives. The constant is the internal energy at . ${\displaystyle E_{0}}$ ${\displaystyle {\vec {p}}={\vec {0}}}$

Einstein's mechanics

See also : Special Theory of Relativity

In Einstein 's mechanics with the speed of light c and from the speed-momentum relation (2) and the fundamental equation (3) we get: ${\displaystyle E=c^{2}m(v)}$

${\displaystyle \mathrm {d} E={\vec {v}}\cdot \mathrm {d} {\vec {p}}={\frac {c^{2}}{E}}{\vec {p}}\cdot \mathrm {d} {\vec {p}}={\frac {c^{2}}{E}}\mathrm {d} (p^{2}/2)}$

or where the rest energy is. Insertion of momentum and energy as a function of mass results ${\displaystyle E^{2}=(cp)^{2}+E_{0}^{2}}$ ${\displaystyle E_{0}}$

${\displaystyle E={\frac {E_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$ and ${\displaystyle m(v)={\frac {\frac {E_{0}}{c^{2}}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$

The relationship between speed and momentum (2) leads to the relativistic momentum , which then makes it clear that velocities no longer add up as easily as in Newtonian mechanics. The relativistic addition theorem for speeds is derived from the consideration of a decaying particle .

In the theory of relativity , every exchange is limited by the speed of light . For example, while the gravitational field transfers momentum from one mass point to the other, the law of conservation of momentum may appear to be violated, because the part of the momentum to be transferred is absorbed in the field, which is known as retardation, see also Retarded potential in electromagnetism.

The law of conservation of energy in Einstein's mechanics breaks down in the limiting case of small velocities into a law of conservation for internal energy and one for mass. In general, Einstein's mechanics contain Newton's as a limiting case for small energies and impulses.

Quantum mechanics

See also : Kinematics (particle processes)

Feynman diagram for the decay of a neutron  n into proton  p , electron  e ^- and electron antineutrino  mediated via a W boson W ^- . ${\displaystyle {\overline {\nu }}_{e}}$ 

In quantum mechanics, one can also work with mass points if, in addition to momentum, energy and / or mass, they are also assigned a natural angular momentum in the form of a spin and other quantities such as charge . In quantum mechanics, these physical quantities can only assume discrete values ​​and are also subject to conservation laws. From the conservation of angular momentum, for example, be seen that when β ^- decay of the neutron into a proton and an electron ℏ each having the same spin / 2, while the charge conservation is already ensured by the three particles, a fourth particles with spin ℏ / 2 arise must, see picture.

See also

• Applications: ballistics , mathematical pendulum , Foucault's pendulum
• Trajectories: trajectory parabola , brachistochrones , tractrix
• Statistical Mechanics

Web links

• Wagner, Paul: Introduction to Physics I, II Mechanics: Dynamics of mass points. Technical Information Library , 2015, accessed February 26, 2020 . 

Footnotes

1. ↑ Falk (1966), Hamel (1912), pp. 66 ff. And 77–166., See literature, or point mechanics. Spektrumverlag, 1998, accessed February 26, 2020 . 
2. ↑ ^{a b} Wilderich Tuschmann, Peter Hawig: Sofia Kowalewskaja . A life dedicated to math and emancipation. Birkhäuser Verlag, Basel 1993, ISBN 978-3-0348-5721-5 , p. 119 f ., doi : 10.1007 / 978-3-0348-5720-8 ( limited preview in Google book search [accessed on May 25, 2017]).
3. ↑ ^{a b} Falk (1966), pp. V ff.
4. ↑ Stäckel (1908), pp. 494 + 498.
5. ↑ Stäckel (1908), p. 464.
6. ↑ Stäckel (1908), p. 449.
7. ↑ Stäckel (1908), p. 451 ff.
8. ↑ Stäckel (1908), p. 478, see JM Bertrand: Note about the similarity in mechanics . In: École polytechnique (ed.): Journal de l'École polytechnique . Tome XIX, No. 32 . Bachelier, Paris 1848, p. 189–197 (French, bnf.fr [accessed February 28, 2020] Original title: Note sur la similitude en méchanique .).
9. ↑ Falk (1966), p. 1 f.
10. ↑ Hamel (1912), p. 66 f.
11. ↑ Falk (1966), p. 10.
12. ↑ Falk (1966), p. 18.
13. ↑ Falk (1966), p. 22.
14. ↑ Stäckel (1908), p. 494.
15. ↑ Falk (1966), p. 23 ff.
16. ↑ Falk (1966), p. 75.
17. ↑ ^{a b} Falk (1966), p. 73.
18. ↑ ^{a b} Falk (1966), p. 76.
19. ↑ Falk (1966), pp. 70 f.
20. ↑ Falk (1966), p. 102.
21. ↑ Falk (1966), p. 80.
22. ↑ Falk (1966), p. 66.
23. ↑ Falk (1966), p. 67.
24. ↑ Falk (1966), p. 79.
25. ↑ Falk (1966), p. 77.
26. ↑ Falk (1966), p. 81 f.
27. ↑ Falk (1966), p. 84.
28. ↑ Falk (1966), p. 87.
29. ↑ Falk (1966), p. 97.
30. ↑ Falk (1966), p. 114.
31. ↑ Falk (1966), p. 105 f.

literature

• Gottfried Falk : Theoretical physics based on general dynamics . Elementary point mechanics. 1st volume. Springer-Verlag, Berlin, Heidelberg 1966, DNB  456597212 , doi : 10.1007 / 978-3-642-94958-6 .
• Georg Hamel : Elementary Mechanics . A textbook. BG Teubner, Leipzig and Berlin 1912, p. 66 f . ( archive.org [accessed February 26, 2020]).
• Paul Stäckel , edited by Felix Klein and Conr. Müller: Encyclopedia of the Mathematical Sciences with inclusion of its applications . Mechanics. Ed .: Academies of Sciences in Göttingen , Leipzig, Munich and Vienna. Fourth volume, 1st part volume, Art. 6.1: Point dynamics. BG Teubner Verlag , 1908, ISBN 978-3-663-16021-2 , p. 449 ff ., doi : 10.1007 / 978-3-663-16021-2 ( wikisource.org [accessed January 24, 2020]).