All Questions
Tagged with classical-mechanics vector-fields
41
questions
2
votes
1
answer
87
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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
1
vote
3
answers
106
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The conservative force [closed]
I read about the definition of the curl.
It's the measure of the rotation of the vector field around a specific point
I understand this, but I would like to know what does the "curl of the ...
- 11
0
votes
0
answers
65
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Hamiltonian flows and Poisson Brackets confusion
I have been using results from this paper in calculations. In sections 2.4 and 3.4 they perform a canonical transformation into new coordinates consisting of constants of motion.
My question is ...
- 298
9
votes
4
answers
2k
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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
-3
votes
3
answers
311
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Why is vector calculus so much more important in classical electrodynamics than in classical mechanics?
In this question "vector calculus" refers to the integration and differentiation of vector fields.
Why is vector calculus so much more important in classical electrodynamics than in ...
- 111
0
votes
1
answer
177
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Curl of a velocity [closed]
In classical mechanics, is the curl of $\vec{v}$ always zero? As $\nabla$ is in position space and not in velocity space ($\nabla_v$). What am I missing regarding $\nabla$ operator in different spaces?...
user292464
2
votes
1
answer
200
views
What is the vector field associated with potential energy?
The mere concept of a line integral is defined for a vector field, and I thus thought the following was a rigorous and general definition of potential energy:
Definition: Given a conservative force ...
- 379
0
votes
1
answer
26
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Calculating work done when the lower bound of integral is greater than the upper bound
In this video, Dr. Peter Dourmashkin explained friction as an example of a force by which the work done is not path independent. In $2$$:$$50$ min of the video, when we're coming back, he said, $d\...
1
vote
2
answers
129
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Time derivative of unit velocity vector?
Let's say I have some parametric curve describing the evolution of a particle $\mathbf{r}(t)$. The velocity is $\mathbf{v}(t) = d\mathbf{r}/dt$ of course. I am trying to understand what the expression ...
-1
votes
1
answer
37
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Conservation and potential with non-cartesian forces
I understand how to determine if a force is conservative from
\begin{equation}
\nabla\times \mathbf{F}=0 \implies \mathbf{F}\text{ is conservative}
\end{equation}
When $F$ is in cartesian coordinates.
...
- 19
3
votes
0
answers
85
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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
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
0
votes
1
answer
206
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Displacement vector for rotational motion
When I try to find out how to compute work for rotational motion. I found an equation from a book online with a figure and equation as follows:
$ \vec{s}\ =\vec{\theta}\ \times\vec{r} $
$Thus,$
$d\...
- 135
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
1
vote
1
answer
66
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Conditions on $\phi$ and $\boldsymbol{A}$ for when $\boldsymbol{B}$ is uniform
I'm reading "Classical Mechanics" (5ed) by Berkshire and Kibble, in the example for uniform magnetic field on pg.243 (Chapter 10 Lagrangian Mechanics) I came across this
A charged particle ...
- 1,047
1
vote
1
answer
318
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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
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
0
votes
0
answers
177
views
Calculate position vector of a particle in given force field
Suppose I have a force field in XY plane as
f(x,y) = yî - xĵ
Suppose a point object of 1kg is put at position (1,0) in this field at t=0 with zero initial velocity....
- 1
1
vote
2
answers
333
views
Analogues to Hamilton's equations in Infinitesimal Canonical Transformations
This is from chapter 4 of David Tong's notes on Classical Dynamics (Hamiltonian Formalism).
Let's say you make an infinitesimal canonical transformation (with $\alpha$ as the infinitesimal parameter) ...
- 773
0
votes
1
answer
214
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Assignment of energy functions to flows is "equivariant"?
I am trying to understand the 2012 blog post What is a symplectic manifold, really?
It says (with correction of a typo in the second point):
If $f: M \to \mathbb{R}$ is a smooth compactly ...
- 103
0
votes
2
answers
201
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Potential Minimum Confusion
Today my lecturer mentioned the notion of vector field and potential, he also said that if the vector field is a force field then there is a potential energy given by: $F(x)=-\dfrac{dU}{dx}$. (I have ...
- 157
1
vote
0
answers
65
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Quantum mechanics in phase space - what are coordinate components?
I'm trying to understand the answer provided by Qmechanic to this question: What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum ...
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votes
2
answers
1k
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What is gradient with respect to components of a position vector?
I am reading "Classical mechanics" by Goldstein, Poole and Safko, Third edition.
Kindly please refer to page no 10, last paragraph.
They write
the subscript $i$ on the del operator indicates that ...
- 1,009
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
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
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vote
0
answers
20
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Are the gradient field are the only fields which are only conservative? [duplicate]
I have found that gradient fields are always conservative. But for my knowledge I wanna ask "are the gradient fields are only fields which are conservative"? I mean is it necessary that a field which ...
- 1,149
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
0
votes
1
answer
1k
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Vector field of a simple pendulum [closed]
Classical Mechanics by John Taylor walks through an example of a skateboard on a frictionless half-pipe of radius $R=5.0$m. This is equivalent to a frictionless pendulum, I believe.
The example goes ...
- 753
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 $F(q^i,p_k)$
$$\det \left[ \...
- 585
1
vote
2
answers
2k
views
Does path independence of line integral imply that the given vector field is conservative (that is, it is negative gradient of some scalar potential)?
For a vector field, is path independence of line integral a necessary and sufficient condition for the field to be conservative or is it just a necessary condition? Please provide proof if possible.
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votes
1
answer
2k
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What is the curl of $k\hat{r}/r^n$?
I'm trying to find the curl of ${\bf F}(r) = k \hat{r}/r^n$. I think that this converts to:
$$
k\left(\frac{\hat{x}}{r} + \frac{\hat{y}}{r} + \frac{\hat{z}}{r}\right)\frac{1}{(x^2 + y^2 + z^2)^{n/2}}
...
- 13
2
votes
0
answers
568
views
Torque of a distributed load: Integrating with dF vs using resultant force
We have a rod that rotates around an fixed point A, which coincides with the end of the rod. The mass distribution along the rod is uniform. We know that the torque generated by the force field at ...
- 556
0
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1
answer
215
views
Solenoidal forces
As far as I know a solenoidal vector field is such one that
$$\vec\nabla\cdot \vec F=0.$$
However I saw a book on mechanics defining a solenoidal force as one for which the infinitesimal work ...
- 17.8k
0
votes
1
answer
121
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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
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votes
2
answers
105
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What is the meaning of this definition of potential energy?
The isolated system of particles is being observed. In the coursebook, $\vec F_\mu$ is by definition the vector sum of forces of all other particles acting on $\mu$-th particle. Usually, potential ...
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
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votes
2
answers
1k
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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 ...
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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
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 ...
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2
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110
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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
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2
answers
1k
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Simulation of physics of chains/ropes in force fields resources?
I'm thinking about a project to tackle, and I'd like to make a simulation that allows the user to define a rope or chain of length L, pin it at arbitrary points r1, r2.... etc. and draw the resulting ...
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