Abstract
Thus far we have focusses on examples allowing an analytical solution of the equation(s) of motion. However, already our first mechanics problem, the mathematical pendulum, required the assumption that the amplitude is small, in order for us to arrive at a reasonably simple differential equation. In fact, for most problems a numerical solution is attempted first and sometimes is the only feasible option. In the following we study a numerical method, which is useful for solving problems involving few variables, e.g. the displacement of a one-dimensional oscillator. Subsequently we discuss a technique, the Molecular dynamics simulation technique, which can be used to solve large numbers of coupled equations of motion.
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Notes
- 1.
A system is a large box containing a (uniform) mass distribution. Large means that boundary effects can be neglected. Systems are distinguished according to whether they are isolated systems (no interaction with the outside world whatsoever), closed systems (energy exchange across the system’s boundaries is possible) or open systems (energy and mass exchange is possible).
- 2.
However, they may be closed (periodic motion).
- 3.
The existence of the right hand side is the subject of Birkhoff’s theorem (cf. [6]).
- 4.
Gibbs, Josiah Willard, American mathematician and physical chemist, *New Haven (Connecticut) 11.2.1839, †New Haven 28.4.1903; he is one of the fathers of modern statistical thermodynamics.
- 5.
Boltzmann, Ludwig, Austrian physicist, *Vienna 20.2.1844, †Duino (today Duino–Aurisina, near Trieste) 5.9.1906; he laid the foundation of statistical mechanics. There is a nice quote in Kerson Huang’s book on this topic which states: [His] H-theorem opened the door to an understanding of the macroscopic world on the basis of molecular dynamics.
- 6.
E.g. K. Huang (1963) Statistical Mechanics. Chap. 3.
- 7.
How this can be rationalized despite of the deterministic nature of the equations of motion is discussed in the next section.
- 8.
The irrelevant positions in the argument of f are omitted.
- 9.
Maxwell, James Clerk, Britisch physicist, *Edinburgh 13.6.1831, †Cambridge 5.11.1879; he was one of the most influential contributors to development of modern physics.
- 10.
Not to be confused with time.
- 11.
The insulated systems are ‘different’ only with respect to (most of) their initial conditions. However, they do share the same energy (hyper)surface.
- 12.
Liouville, Joseph, French mathematician, *Saint-Omer (Départment Pas-de-Calais) 24.3.1809, †Paris 8.9.1882.
- 13.
If a quantity, which here happens to the number of phase space points, satisfies the continuity equation, it means that a change of the amount of this quantity in a certain region of space is entirely due to the quantity flowing in or out of the region. In other words, there is not ‘production’ or ‘destruction’ of this quantity inside this region. An example for processes causing ‘production’ or ‘destruction’ are chemical reactions.
- 14.
In the case of gravitation the potential energy of a mass density, \(\rho (\vec {r}\,)\), is given by \(U= -(G/2) \int _V d^3 r d^3 r^{\prime } \rho (\vec {r}\,) | \vec {r} -\vec {r}\,^\prime |^{-1} \rho (\vec {r}\,^\prime )\). The volume V encloses the mass distribution. Thus \(U = g G M^2 V^{-1/3}\), where for a uniform mass distribution g is a geometry dependent factor.
- 15.
Poincaré, Jules Henri, French physicist and mathematician, *Nancy 29.4.1854, †Paris 17.7.1912.
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Hentschke, R. (2017). Many-Particle Mechanics. In: Classical Mechanics. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-48710-6_9
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