Classical mechanics: Jacobi variational principle

[Jurgen Kruger]

An isolated mechanical system can be represented by a point in a high-dimensional configuration space. This point evolves along a line. The variational principle of Jacobi says that, among many imagined trajectories between two points, only the SHORTEST is real and is associated with situations that fulfil classical mechanical laws.

A description in these terms is said to be timeless because "time" does not appear in the formulae. However, on the other hand, the term "change" appears in the pertinent texts. Can someone explain me how a "change" can be understood without a concept of (temporal !) simultaneity, i.e., without any reference to "time"? Naïvely I assume that the individual contributions to the ensemble of coordinates making up the point in configuration space have to be taken at the same instant in time.

I am looking for a truly timeless description of classical mechanics. Is this really the case for the principle of Jacobi?

 

[Greg Bernhardt]

Welcome to PF!

 

[dextercioby]

An isolated mechanical system can be represented by a point in a high-dimensional configuration space. This point evolves along a line. The variational principle of Jacobi says that, among many imagined trajectories between two points, only the SHORTEST is real and is associated with situations that fulfil classical mechanical laws.

A description in these terms is said to be timeless because "time" does not appear in the formulae. However, on the other hand, the term "change" appears in the pertinent texts. Can someone explain me how a "change" can be understood without a concept of (temporal !) simultaneity, i.e., without any reference to "time"? Naïvely I assume that the individual contributions to the ensemble of coordinates making up the point in configuration space have to be taken at the same instant in time.

I am looking for a truly timeless description of classical mechanics. Is this really the case for the principle of Jacobi?

Can you tell us which text you have read to come to this description? Do you know what a phase space is and how it is linked with the notion of time?

 

[zwierz]

Ok let's clarify that; suppose we have a Lagrangian system $L=T-V(q),\quad T=\frac{1}{2}g_{ij}(q)\dot q^i\dot q^j$ here $q=(q^1,\ldots,q^m)$ are the local coordinates on a configuration manifold. (The form $\frac{1}{2}g_{ij}(q)$ is positive definite)
1) If $q(t)$ is a motion of the system with energy constant $h$ that is $T+V=h$ then the curve $q(t)$ is a geodesic of the Jacobi metric:
$$\tilde g_{ij}=g_{ij}(q)(h-V).\qquad (*)$$
2) If a curve $q(\xi)$ is a geodesic of the metric (*) then we can reparameterize this curve $\xi\to t$ such that $q(\xi(t))$ becomes a motion of the system on the energy level $h$.
This reparametresation is constructed as follows
$$\frac{1}{2}g_{ij}(q(\xi))\frac{dq^i}{d\xi} \frac{dq^j}{d\xi} \dot \xi^2+V(q(\xi))=h.$$

 

[zwierz]

ts, only the SHORTEST is

broadly speaking that is wrong. A geodesic gives shortest distance when end points are sufficiently close to each other

 

[Jurgen Kruger]

broadly speaking that is wrong. A geodesic gives shortest distance when end points are sufficiently close to each other

Unfortunalety I cannot understand the contribution of Zwierz. Can you express that in words?

Thank you for the reactions so far. However, my description was not clear insofar as one must say, starting from the configuration space: One can calculate the action sum along each of the many trajectories linking two points. 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.

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

I am indeed 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 that are put together to define a point in configuration space have to be taken at one and the same instant. (This would be a temporal notion). I hope that I am not right, but can someone explain me, why?

 

[zwierz]

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.

Your assertion is wrong. Indeed, take two close to each other points on two dimensional sphere. The sphere is endowed with the standard metric inherited from $\mathbb{R}^3$. You will have two geodesics connecting these points: maximal geodesic and the minimal one. Both correspond to motions of the mass point on the sphere when active forces are not applied.

 

[zwierz]

J. B. Barbour arXiv:0903.3489, 2009

looks like waste

 

[Jurgen Kruger]

Zwierz: "waste" means that there are errors in Barbours article. Can you specify them? Are there other ways of expressing fundamental equations of classical mechanics without referring to "time"?