Two examples for limit relations in classical physics

2 Two examples for limit relations in classical physics

As our first example we consider the relation between relativistic mechanics and NM. As is well-known, in relativistic mechanics a new fundamental constant, the speed of light $c$ , appears, which is absent (infinitely large) in NM. Otherwise, the mathematical structures of both theories are similar. The basic equations of relativistic mechanics differ from (2) only by factors $γ$ , which depend on $v ∕c$ and disappear (reduce to $1$ ) if $c$ becomes large,
$∘ ----------- v2 γ = 1 - ---, lim γ = 1. 2 c→ ∞ c$ (3)

The relation between relativistic mechanics and NM may be summarized as follows:

Both theories have the same mathematical structure: ordinary differential equations for trajectories. A new fundamental constant $c$ appears in relativistic mechanics.
The limit $1 ∕c → 0$ transforms the basic equations of relativistic mechanics in the basic equations of NM; the same is true for the solutions of relativistic mechanics and NM respectively.

We see that relativistic mechanics and NM provide a perfect realization of a limit relation (NM is the limit theory of relativistic mechanics) or a covering relation (relativistic mechanics is the covering theory of NM), respectively. The significant feature is the appearance of a new fundamental constant which allows for a transition between two different theories of the same mathematical type. We will refer to the type of limit relation encountered in this first example as ’standard limit’ relation.

Our second example concerns the relation between NM and a probabilistic version of NM, which can be constructed according to the following well-known recipe. We consider a phase space probability density $ρ (x, p, t )$ and assume that the total differential of $ρ$ vanishes,
$∂ ρ ∂ ρ ∂ ρ d ρ (x, p, t) = ------dx + -------dp + -----dt = 0. ∂ x k ∂ p k ∂ t k k$ (4)

This means that $ρ$ is assumed to be constant along arbitrary infinitesimal changes of $xk , pk , t$ . Next we postulate that the movement in phase space follows classical mechanics, i.e. we set $dx = (p ∕m )dt k k$ and $dp = - ( ∂ V ∕ ∂ x )dt k k$ . This leads to the partial differential equation (Liouville equation)
$∂--ρ- pk---∂-ρ--- ∂-V----∂--ρ-- + - = 0, ∂ t m ∂ xk ∂ xk ∂ pk$ (5)

which has to be solved by choosing initial values $ρ (x, p, 0)$ for the new dynamical variable $ρ (x, p, t )$ . The relation between the probabilistic version (of NM) and NM may be summarized as follows:

The probabilistic version and NM have a different mathematical structure; the probabilistic version is ruled by a partial differential equation, NM by an ordinary differential equation. No new constant appears in the probabilistic version.
The probabilistic version and NM belong to fundamentally different epistemological categories. NM is a deterministic theory. The probabilistic version is a probabilistic (indeterministic) theory; predictions about individual events cannot be made because the initial values for individual particles are unknown.

The absence of a new fundamental constant prevents a simple transition between the two theories as found in our first example. Nevertheless, a kind of limit relation can be established by means of appropriate (singular) initial values. A probability density which is sharply peaked at $t = 0$ retains its shape at later times. Inserting the Ansatz
$(3) (3 ) ρ (x, p, t) = δ (x - r (t ))δ (p - p (t )),$ (6)

into Eq. (5), it is easily shown that admissible particle trajectories $rk (t )$ , $pk (t )$ are just given by the solutions of Newton’s equations (2). Thus, NM can be considered as a limit theory of the probabilistic version in the sense that the manifold of solutions of a properly (with regard to singular initial values) generalized version of the probabilistic version leads to NM. This limit relation is ’weaker’ than the one encountered in our first example, because there is no mapping of individual solutions. By allowing for singular solutions we have essentially constructed the union of the deterministic theory NM and the original probabilistic version of NM; it is then no surprise that the generalized probabilistic version theory contains NM as a special case. Considered from a formal point of view, however, the (generalized) probabilistic version is a perfect covering theory since its manifold of solutions is larger than that of NM. We shall refer to the kind of limit relation found in this second example as a ’deterministic’ limit relation.

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