Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

New features!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

123 submissions , 104 unreviewed
3,547 questions , 1,198 unanswered
4,549 answers , 19,356 comments
1,470 users with positive rep
410 active unimported users
More ...

Lagrangian formalism and Contact Bundles

+ 6 like - 0 dislike
46 views

In his Applied Differential Geometry book, William Burke says the following after telling that the action should be the integral of a function $L$:

A line integral makes geometric sense only if it's integrand is a 1-form. Is $Ldt$ a 1-form? Well, that's the wrong question. The correct question is: On what space is $Ldt$ a 1-form? It is not a 1-form on configuration space, the space of positions, because it can have a non-linear dependence on velocities. A 1-form must be a linear operator on the tangent vectors. The correct space for $L dt$ is the line-element contact bundle of the configuration space.

Now, why intuitively the correct setting for lagrangian mechanics is on the contact bundle? I understand the contact bundle as pairs $(p,[v])$ where $p$ is a point in configuration space and $[v]$ is an equivalence class of vectors, explicitly $v\sim kv$.

Thinking not on all that arguments for selecting the space on which $Ldt$ is a $1$-form, physically, how can we intuit that the contact bundle is useful for that? I mean, is there some observation in classical mechanics that guides us in building the theory on that space?

This post imported from StackExchange Physics at 2014-04-01 12:39 (UCT), posted by SE-user user1620696
asked Feb 21, 2014 in Theoretical Physics by user1620696 (160 points) [ no revision ]

1 Answer

+ 1 like - 0 dislike

The special feature of contact geometry is the contact 1-form $\lambda$, which satisfies $\lambda\wedge d\lambda\ne0$ (let's restrict to 3-dimensions). In our Lagrangian mechanics example, $\lambda = dq-vdt$. You want this to pull-back to zero on the ``permissible'' curves in phase space -- these curves represent the motions of your system.

For a more detailed but tangential explanation, see:
http://mathoverflow.net/questions/72498/what-is-the-role-of-contact-geometry-in-the-hamiltonian-mechanics

This post imported from StackExchange Physics at 2014-04-01 12:39 (UCT), posted by SE-user Chris Gerig
answered Feb 22, 2014 by Chris Gerig (540 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$y$\varnothing$icsOverflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
To avoid this verification in future, please log in or register.




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...