All Questions
50
questions
3
votes
0
answers
830
views
Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian
I want to show that the gauge transformation
$$L(q,\dot{q},t)\mapsto L^\prime(q,\dot{q},t):=L(q,\dot{q},t)+\frac{d}{dt}f(q, t)$$
corresponds to a canonical transformation of the Hamiltonian $H(p, q, ...
- 1,187
3
votes
0
answers
79
views
Hamiltonian definitions in the presence of boundary term [duplicate]
Consider a Lagrangian of the form
\begin{equation}
L(q,\dot{q})=L_1(q,\dot{q})+\frac{d L_2(q,\dot{q})}{dt}
\end{equation}
I understand that $\dot{L_2}$ does not modify the equations of motion, ...
- 459
2
votes
1
answer
2k
views
Why Lagrangian is unchanged under rotation and translation?
In Landau Mechanics, he derived the conservation of momentum assuming that $\delta L = 0$ under infinitesimal translation $\epsilon$. However, one just need the change of Lagrangian to be a total ...
- 121
6
votes
1
answer
1k
views
Momentum as derivative of on-shell action
In Landau & Lifshitz' book, I got stuck into this claim that the momentum is the derivative of the action as a function of coordinates i.e.
$$
\begin{equation}p_i = \frac{\partial S}{\partial x_i}\...
- 430
0
votes
1
answer
3k
views
What is the difference between generalized momentum and ordinary momentum?
I'm studying about motion equation of charge in electromagnetic field.
Lagrangian of charge in E.M field is
$L=-mc^2\sqrt{1-v^2/c^2}+\frac{e}{c}\mathbf{A}\cdot \boldsymbol{v}-e\phi$ .
Thus ...
- 305
1
vote
1
answer
2k
views
Do dimensions of the product $q_k p_k$ always equal to that of angular momentum?
I know that generalised coordinates and their conjugate momentum may or may not have the same dimensions as to that of length and linear momentum, but in one book I saw it was mentioned that their ...
- 1,043
0
votes
1
answer
128
views
Spherical momentums in terms of cartesian momentums and coordinates [closed]
I want to prove the equations for spherical momentas, in terms of Cartesian momentas and Cartesian coordinates. If $p_r=m\dot r$, $p_\theta=mr^2\dot\theta^2$, $p_\phi=mr^2\dot\phi\sin^2\theta$, prove ...
- 175
11
votes
2
answers
1k
views
Simple explanation of why momentum is a covector?
Can you give a simple, intuitive explanation (imagine you're talking to a schoolkid) of why mathematically speaking momentum is covector? And why you should not associate mass (scalar) times velocity (...
- 121
1
vote
2
answers
7k
views
Conjugate momentum in Cartesian coordinates
The conjugate Hamiltonian can be defined from the Lagrangian as,
$$ p_i ~=~ \frac{\partial L}{\partial \dot{q}^i}$$
Typically the momenta components are given in spherical polars $(r, \theta, \phi)$....
- 1,751
11
votes
3
answers
10k
views
What is the difference between kinetic momentum $p=mv$ and canonical momentum?
What is the difference, if any, between kinetic momentum $p=mv$ and canonical momentum? Why is canonical momentum important (specifically to classical field theory)?
- 530
1
vote
0
answers
108
views
Classical action [closed]
Any idea how to solve this problem?
In classical mechanics, the action $S$ is defined as
$$S[q(t)] = \int_{t_0}^t L(q(t'), \dot q(t'), t')\; dt'$$
where $L$ is the Lagrangian function (also ...
- 11
1
vote
1
answer
1k
views
Lorentz force with Lagrangian
I want to prove that
$$
\vec{F}=d\vec{p}/dt=q\vec{E}+(q/c) \cdot v\times \vec{B}
$$
in CGS system, using
$$
L=-mc^{2}/\gamma-q\phi+(q/c)\cdot \vec{v}\cdot \vec{A} \hspace{10mm} \tag 1
$$
and
$$
\...
- 297
2
votes
1
answer
442
views
What is the function type of the generalized momentum?
Let
$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$
denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action
$...
- 8,523
0
votes
1
answer
86
views
Does mass equal angular momentum?
At the wikipedia pages for angular momentum ($L$) and moment of inertia ($I$) we find the equations:
$$L=I \omega$$
$$I=m r^2$$
where $m$ is mass and $r$ is the distance between said mass and ...
- 1,517
7
votes
1
answer
382
views
Why isn't $F = \frac{\partial \mathcal{L}}{\partial q}$?
If momentum is, $$p = \frac{\partial \mathcal{L}}{\partial \dot{q}}$$ and force is,
$$ F = \frac{dp}{dt}$$
and by Euler-Langrange equations,
$$ \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{...
- 779