All Questions
Tagged with vector-fieldsclassical-mechanics
44 questions
1vote
0answers
42views
How much does classical mechanics depend on the choice of symplectic form?
TlDr; a different choice of symplectic structure on a phase-space $\mathcal{M}$ affects the Hamiltonian mechanics insofar as it could affect what the canonical coordinates are, but is this the only ...
- 230
0votes
0answers
35views
Kinematical and dynamical symmetries and their relevance
T.i.l that kinematical symmetries are symmetries generated by a Killing vector field $ \pi_*(X_q) $, $$ {\cal L}_{\pi_*(X_q)}g=0,$$ which is given by the pushforward $ \pi_*(X_q) $ of $X_q$, where $$ \...
- 429
0votes
0answers
69views
Force in thermodynamic configuration space
Consider a thermodynamic system whose internal energy $U$ may not be conserved in general. It's a direct consequence of the First Principle that the variations in internal energy do not depend on the ...
- 2,164
2votes
1answer
136views
How to compute the vector field from a potential in the complex plane?
I am watching this Youtube video and I have the following dumb question around 1:18:00: How do you draw the vector field for a given potential in the complex plane? He gives the potential $V(x) = x^4-...
1vote
3answers
135views
The conservative force [closed]
I read about the definition of the curl. It's the measure of the rotation of the vector field around a specific point I understand this, but I would like to know what does the "curl of the ...
- 11
0votes
0answers
97views
Hamiltonian flows and Poisson Brackets confusion
I have been using results from this paper in calculations. In sections 2.4 and 3.4 they perform a canonical transformation into new coordinates consisting of constants of motion. My question is ...
9votes
4answers
2kviews
Time evolution operator in classical mechanics?
Hamilton's equation can be written in terms of Poisson brackets, as follows: $$\dot{q} = \{q,H\}$$ $$\dot{p} = \{p,H\}$$ where $H$ is the Hamiltonian of the system. Now, wikipedia says that the ...
- 1,283
-3votes
3answers
488views
Why is vector calculus so much more important in classical electrodynamics than in classical mechanics?
In this question "vector calculus" refers to the integration and differentiation of vector fields. Why is vector calculus so much more important in classical electrodynamics than in ...
- 111
0votes
1answer
219views
Curl of a velocity [closed]
In classical mechanics, is the curl of $\vec{v}$ always zero? As $\nabla$ is in position space and not in velocity space ($\nabla_v$). What am I missing regarding $\nabla$ operator in different spaces?...
user292464
2votes
1answer
228views
What is the vector field associated with potential energy?
The mere concept of a line integral is defined for a vector field, and I thus thought the following was a rigorous and general definition of potential energy: Definition: Given a conservative force ...
- 379
0votes
1answer
37views
Calculating work done when the lower bound of integral is greater than the upper bound
In this video, Dr. Peter Dourmashkin explained friction as an example of a force by which the work done is not path independent. In $2$$:$$50$ min of the video, when we're coming back, he said, $d\...
2votes
2answers
163views
Time derivative of unit velocity vector?
Let's say I have some parametric curve describing the evolution of a particle $\mathbf{r}(t)$. The velocity is $\mathbf{v}(t) = d\mathbf{r}/dt$ of course. I am trying to understand what the expression ...
-1votes
1answer
48views
Conservation and potential with non-cartesian forces
I understand how to determine if a force is conservative from \begin{equation} \nabla\times \mathbf{F}=0 \implies \mathbf{F}\text{ is conservative} \end{equation} When $F$ is in cartesian coordinates. ...
- 19
3votes
0answers
92views
About the equation $\frac {d^2} {dt^2}\vec x(t) = \nabla \times \vec F(x(t))$. Motion in a curl vector field
I was wondering if there is a physical interpretation of ODEs of the form $$\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
3votes
2answers
168views
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+...
- 627
