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The description of J. B. Barbour arXiv:0903.3489, 2009 is considered to be timeless but the term "change" appears in that text.

Essentially, a configuration space is considered: One calculates the sum of the variable "action" along each of the many trajectories linking two points in that space. One sum is absolutely the smallest. The "Variational Principle of Jacobi" says that only this is the real case, i.e. only the points lying on that trajectory obey the well-known laws of classical mechanics.

I am looking for a timeless description of the classical world. However, it is not clear to me whether this can be conceived without assuming that the components which are put together to define a point in configuration space have to be taken at one and the same instant.

I see several possibilities: (1) The variational minimum automatically yields those components of a point in conf. space that are valid at ONE instant. I would not understand HOW but that is not the problem of my present question.

(2) I could deliberately select the x and y coordinates of one object at time t1 and as the z coordinate I introduce a z-coordinate of another object, and in addition not taken at time t1 but another time t2. If I understand correctly the definition of a conf. space, only the coordinates of all objects count.

Having defined a point in conf. space with this contribution, there are 2 possibilities: (2a) the the selected collection of coordinates of all objects is indeed reached by the system at SOME (unknown) time. Then, (1) applies. No problem.

(2b) the system never reaches the selected point in conf. space because the system may not go to ALL places within that space.

My questions are: - is (1) the only relevant statement? Then, case (2b) should somehow be impossible, perhaps because the calculation of the "action" fails, or the search for the variational minimum goes awry.

or

  • Barbour's proposal is not correct, insofar as one has to know beforehand what "simultaneity" is (this would be a temporal notion).
Jürgen Krüger
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2 Answers2

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It seem that a good part of the 'timeless' formulations in Ref. 1 is directly or indirectly related to the well-known fact that (for autonomous systems) one may perform a Legendre transform $$\Delta t~:=~t_f-t_i\quad\longleftrightarrow\quad E $$ to a dual energy-variable $E$ instead of time. See my Phys.SE answers here & here for details.

References:

  1. J. Barbour, The Nature of Time, arXiv:0903.3489.
Qmechanic
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I think the issue here lies with the concept of configuration space. The language used in texts might sometimes be confusing especially when describing a configuration space for only a few individual particles. However, in the configuration space every single point represents an entirely separate state for the entire system as a whole. Even if you have a billion particles, the whole current state of the system is just represented by a single point in configuration space.

So as you trace out a trajectory in configuration space you're basically defining an order in which the system evolves. If you draw a line from point P1, to P2, to P3, then you're basically saying that the entire system evolves from state 1 to state 2 to state 3. Note that despite the fact that you're only talking about 3 points in configuration space they are indeed 3 separate full descriptions of all the billions of particles in your system.

Although we can look at it as an evolution between these states to understand the role time plays, note that there is no actual explicit reference to time. So you can think of it as "timeless" and just work out what trajectories exist in the configuration space.

ison
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