On a formal viewpoint on the D’Alembert–Lagrange Principle

[wrobel]

I would like to discuss the following article: https://web.ma.utexas.edu/mp_arc/c/20/20-87.pdf
This article is pure educational and it does not contain any research level results. My aim was to build the theory of the D’Alembert–Lagrange Principle in mathematically closed and modern form.
Thanks for any comment.

 

[dextercioby]

Is it traditional in Russian literature to call the tangent bundle of configuration space the "phase space"? I am asking, because in "Western" literature it is the cotangent bundle getting that name (V.I. Arnol'd calls this on page 68 of this book in English on mathematical mechanics, and he is Russian, too).

 

[wrobel]

mmmmmmm I get used to think that in Lagrangian context phase space is the tangent bundle because $L=L(x,\dot x)$ and in Hamiltonian context phase space is the cotangent bundle since $H=H(x,p)$

 

[weirdoguy]

Hm, I did my masters in geometrical foundations of field theory (Tulczyjew triples) and I read a lot of papers about foundations of mechanics as well and I've never seen anyone calling tangent bundle a "phase space".

 

[vanhees71]

True, but that's semantics ;-)). I've also never heard the tangent bundle called phase space. It's always the co-tangent bundle.

I often wonder whether one can avoid the Hamiltonian formalism entirely. The main purpose to learn classical analytical mechanics for physicists in the 21st century of course is quantum theory, and I often wonder, whether you can do the heuristics with the operator formalism avoiding phase space and Poisson bracketes and only using configuration space and Lagrange brackets.

Usually the Lagrange formulation in quantum theory is only used in the context of the path integral, where it is derived by integrating out the canonical momenta in the Hamiltonian version of the path integral, which is always necessarily the same starting point, because the naive Lagrange version only works for special cases, but that's another story, more suitable for another thread in the QM forum.

 

[wrobel]

I read a lot of papers about foundations of mechanics as well and I've never seen anyone calling tangent bundle a "phase space".

Nonholonomic Mechanics and Control (Interdisciplinary Applied Mathematics) (Anthony Bloch, et al)

 

[wrobel]

The main purpose to learn classical analytical mechanics for physicists in the 21st century of course is quantum theory

for physicists -- yes, but there is a big field called dynamical systems and there is a big field called control theory