Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with volume
Search options answers only
not deleted
user 35354
2
votes
Accepted
The value of the volume form on an orthonormal frame
When applying the two constructions to the volume form the former produces the annoying factors you noticed:
$$dx^1 \wedge \cdots \wedge dx^n\left(\frac{\partial}{\partial x^1}, \ldots, \frac{\partial} …
- 78.1k
3
votes
Accepted
Liouville Theorem analogue in generalized velocities?
.$$
Therefore, the canonical volume in terms of Lagrangian variables is
$$dp\wedge dq = q d\dot{q} \wedge dq\:.$$
Since the left-hand side is preserved by solutions of the equation of motion and $q=q(t … So the apparantly natural volume constructed out of Lagrangian variables is not constant in time on the motion of the system, differently from the canonical volume. …
- 78.1k
17
votes
Accepted
Is a canonical transformation equivalent to a transformation that preserves volume and orien...
they are not equivalent since (for time-independent transformations) canonical is equivalent to $$\sum_{k=1}^n dq^k\wedge dp_k = \sum_{k=1}^n dQ^k\wedge dP_k\tag{1}$$
whereas conservation of oriented volume …
- 78.1k
16
votes
Accepted
Liouville's theorem and conservation of phase space volume
If $Q_k\equiv 0$, even if $H$ explicitely depend on time, the solutions of Hamilton equations preserve, in time, the canonical volume:
$$dq^1 \wedge \cdots \wedge dq^n \wedge dp_1 \wedge \cdots \wedge … d p_n\:.$$
In the presence of dissipative forces which cannot be included in the Lagrangian, the term $Q_k$ show up and the volume above generally fails to be preserved. …
- 78.1k