All Questions
6
questions
3
votes
0
answers
85
views
About the equation $\frac {d^2} {dt^2}\vec x(t) = \nabla \times \vec F(x(t))$. Motion in a curl vector field
I was wondering if there is a physical interpretation of ODEs of the form
$$\frac d{dt}\vec x(t)=\vec y(t)$$
$$ \frac d{dt} \vec y(t) = \nabla \times \vec F(x(t))$$
(or equivalently $\frac {d^2} {dt^2}...
- 208
1
vote
1
answer
318
views
On the Hamiltonian vector fields of classical Hamiltonian mechanics
Notation: I denote phase space as the symplectic manifold $(M,\omega)$, in which $\omega=\sum_i\mathrm dp_i\wedge\mathrm dq_i$ in canonical coordinates.
In definitions of Hamiltonian vector fields I ...
- 6,963
5
votes
1
answer
756
views
Liouville's volume theorem in differential forms language
I will cast my question mostly in words. I usually find Liouville's volume theorem cast in two forms:
Easy way (without differential forms language): Phase space volume remains preserved under ...
- 837
0
votes
1
answer
121
views
Clarification about some steps in the derivation of the Lie derivative (mechanics)
First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...
- 912
2
votes
2
answers
1k
views
What are the mathematical models for force, acceleration and velocity?
In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector ...
- 905
2
votes
1
answer
597
views
Lagrangian vector field expression
The Lagrangian vector field $X_L$ on the tangent bundle is given in page 4 of Marco Mazzucchelli's "critical Point Theory for Lagrangian systems" to be;
\begin{equation}
X_L=\sum^M_{j=1}\bigg(v^j\frac{...
- 2,441