All Questions
Tagged with classical-mechanics vector-fields
41
questions
11
votes
2
answers
1k
views
Does Hamilton Mechanics give a general phase-space conserving flux?
Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltonian theory like the flux of an ideal fluid, which doesn't change ...
- 175
9
votes
4
answers
2k
views
Time evolution operator in classical mechanics?
Hamilton's equation can be written in terms of Poisson brackets, as follows:
$$\dot{q} = \{q,H\}$$
$$\dot{p} = \{p,H\}$$
where $H$ is the Hamiltonian of the system. Now, wikipedia says that the ...
- 1,131
8
votes
1
answer
702
views
Can any symplectomorphism (1 Definition of canonical transformation) be represented by the flow of a vectorfield?
For this question I will use the definition that a canonical transformation is a map $T(q,p)$ from the phase space onto itself, which leaves the symplectic 2-form invariant (which is the definition of ...
- 6,763
6
votes
3
answers
2k
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Paths in phase space can never intersect, but why can't they merge?
Page 272 of No-Nonsense Classical Mechanics sketches why paths in phase space can never intersect:
Problem: It seems to me this reasoning only implies that paths can never "strictly" ...
- 337
6
votes
1
answer
5k
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Hamiltonian flow?
I was wondering what the Hamiltonian flow actually is?
Here is my idea, I just wanted to know if I am correct about this.
So let $(x(t),p(t))' = X_{H}(x(t),p(t))$ are the Hamilton's equations and $...
- 97
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
3
votes
1
answer
71
views
Two first integrals of an hamiltonian field $X_{H}$ are independent $\det \left[ \frac{\partial F_{i}}{\partial p_{k}} \right] \neq 0$
I want to understand how it is established if two first integrals of an hamiltonian field $X_{H}$ are independent.
One hypothesis is:
Considering two first integrals $F(q^i,p_k)$
$$\det \left[ \...
- 585
3
votes
2
answers
154
views
Vector function of vectors expansion
I am reading Landau's Mechanics. In the solution to the problem 4 on page 138, section 42, it is stated that an arbitrary vector function $\vec f(\vec r,\vec p)$ may be written as $\vec f=\vec r\phi_1+...
- 613
3
votes
1
answer
82
views
Do all Joukowski aerofoils violate no-penetration condition at trailing edge?
In our fluids course we calculated the velocity distribution around a completely symmetric Joukowski aerofoil (as shown below) and used the Kutta condition to ensure that the velocity was not infinite ...
- 2,604
3
votes
0
answers
85
views
About the equation $\frac {d^2} {dt^2}\vec x(t) = \nabla \times \vec F(x(t))$. Motion in a curl vector field
I was wondering if there is a physical interpretation of ODEs of the form
$$\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
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
2
votes
2
answers
110
views
In a gas of particles, how is the displacement vector related to the number density?
Suppose I have a gas of particles that is initially uniformly distributed so that the number density is $n_0$ (number of particles per unit volume), and then I displace the particles by the vector ...
- 852
2
votes
1
answer
192
views
Is there any special significance of force field in physics?
What is the formal definition of force field? Which is more fundamental force or field? Do field exist in nature (as force do i think as per section 12-1 of Feynman lecture volume 1, and page 8,9 of ...
- 181
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
87
views
How to compute the vector field from a potential in the complex plane?
I am watching this Youtube video and I have the following dumb question around 1:18:00: How do you draw the vector field for a given potential in the complex plane? He gives the potential $V(x) = x^4-...
- 123