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[Remark: I admit that my first attempt on What makes a space a real space? was rather ill-posed and led to some confusion. Sorry for that, but please give me a second try. Part of the confusion arose from the ununsual term "real space". The far better term would have been "physical space" which I hope to be unambiguous.]

In the following let a "mechanical system" be a system of $n$ spatial objects moving in physical space.

Consider you are given a function $q:\mathbb{R} \rightarrow \mathcal{M}^n$ with $\mathcal{M}$ a mathematical space.

Question 1

When would you assume that $q$ describes the evolution of a mechanical system?

I would, if I found or was given a function $\mathcal{L}(x,\dot{x},t)$ such that the Lagrange equations hold for all times $t$ when inserting $q(t)$ for $x$ and $\dot{q}(t)$ for $\dot{x}$, and which I can consider physically plausible. In this case I would interpret $\mathcal{M}$ as physical space, maybe distorted by generalized coordinates.

I would not, if there is provably no such function. In this case $q$ describes the evolution of a general dynamical system, but not a mechanical one.

Question 2

Are there mathematically definable conditions on $\mathcal{L}$ for being physically plausible, or possible, let's say?

Besides, maybe, that $\mathcal{L}$ must be of the form $\mathcal{T} - \mathcal{V}$.

Question 3

Can anyone give an example of a non-mechanical system that behaves like a mechanical one (in the narrow sense above)?

One answer to this question comes immediately in mind: a system that is designed to behave like a mechanical one, e.g. a computer running a simulation. I'd like to exclude such cases if I could, and restrict the question to "natural" systems.

Question 4

What about possible worlds with a physical space like ours but in which the Lagrange equations do not hold?

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For future, please don't pack few questions in one, rather try to post one more general or just few separate. –  mbq Feb 10 '11 at 10:05
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1 Answer

The initial definition given here of a mechanical system;

"In the following let a "mechanical system" be a system of n spatial objects moving in physical space."

is much broader than the restriction to a 'standard' Lagrangian framework would allow. By 'standard' I mean a Lagrangian depending only on q and its first time derivative, q', as well as, possibly, time itself. If one generalises the Lagrangian framework to allow dependence on higher order time derivatives, with the corresponding change in the Lagrangian equations of motion, then one comes closer to allowing for the variety permitted in the initial definition.

But the rest of the question suggests that systems governed by a 'standard' Lagrangian are the real focus of interest here. The restriction of L to the form, T - V, queried in the 2nd question is devoid of content unless, as usual, one restricts the appearance of the time derivative, q', to T. Usually (but not always), such T's depend on q' via quadratic forms like q'Mq', where M is a matrix/tensor depending only on q.

As for examples of systems evolving in accordance with equations of motion that would follow from such a Lagrangian formalism, but not be mechanical systems as per the initial definition, there may well be examples in the mathematical models used by economists and some social scientists to describe evolution of market variables or population parameters.

If one allows the exponent, n, of the mathematical space, M, in the initial definition to be infinite (corresponding to an infinite number of degrees of freedom) then the evolution of fields, such as the electromagnetic field, would provide examples. The fields take there values at all the points of physical space, but those values evolve in field-space, not physical space, so they are not mechanical according to the initial definition.

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There are also many "cartoon" models used in mathematical physics which are not realistic but feature the most important characteristics of realistic systems, depending on what we want to study of course. I think of baker maps, cat maps, kicked tops, etc... –  Raskolnikov Nov 18 '10 at 17:17
    
What about generalized potentials which may depend on the velocities, too? –  Hans Stricker Nov 18 '10 at 17:49
    
Thanks for the elaborate answer! What would be really great was a specific example of a model from economy that looks like a Lagrangian system (with a "physical" Lagrangian). –  Hans Stricker Nov 18 '10 at 17:52
    
May I sum up your point like this: For each class of "allowed" Lagrangians we get another class of "mechanical systems". And thus, the whole approach becomes vacuous? Because for arbritrary Lagrangians every dynamical system is a "mechanical" system? –  Hans Stricker Nov 18 '10 at 18:04
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