All Questions
14
questions
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.
$$
\begin{equation}p_i = \frac{\partial S}{\partial x_i}\...
- 430
11
votes
3
answers
10k
views
What is the difference between kinetic momentum $p=mv$ and canonical momentum?
What is the difference, if any, between kinetic momentum $p=mv$ and canonical momentum? Why is canonical momentum important (specifically to classical field theory)?
- 530
11
votes
2
answers
1k
views
Simple explanation of why momentum is a covector?
Can you give a simple, intuitive explanation (imagine you're talking to a schoolkid) of why mathematically speaking momentum is covector? And why you should not associate mass (scalar) times velocity (...
- 121
9
votes
2
answers
3k
views
How does the canonical momentum $p_i\equiv\frac{\partial L}{\partial\dot q_i}$ transform under a coordinates change $\mathbf q\to\mathbf Q$?
The canonical momentum is defined as
$$p_{i} = \frac {\partial L}{\partial \dot{q_{i}}}, $$
where $L$ is the Lagrangian.
So actually how does $p_{i}$ transform in one coordinate system $\textbf{q}$ to ...
- 91
3
votes
2
answers
452
views
Help with geometric view of conjugate momenta and Legendre transformation
I'm familiar with the ''coordinate view'' of Lagrangian and Hamiltonian mechanics where if $\pmb{q}=(q^1,\dots, q^n)\in\mathbb{R}^n$ are any $n$ generalized coordinates and $L(\pmb{q},\dot{\pmb{q}})$ ...
- 477
3
votes
0
answers
830
views
Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian
I want to show that the gauge transformation
$$L(q,\dot{q},t)\mapsto L^\prime(q,\dot{q},t):=L(q,\dot{q},t)+\frac{d}{dt}f(q, t)$$
corresponds to a canonical transformation of the Hamiltonian $H(p, q, ...
- 1,187
3
votes
0
answers
79
views
Hamiltonian definitions in the presence of boundary term [duplicate]
Consider a Lagrangian of the form
\begin{equation}
L(q,\dot{q})=L_1(q,\dot{q})+\frac{d L_2(q,\dot{q})}{dt}
\end{equation}
I understand that $\dot{L_2}$ does not modify the equations of motion, ...
- 459
2
votes
1
answer
442
views
What is the function type of the generalized momentum?
Let
$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$
denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action
$...
- 8,523
1
vote
1
answer
135
views
In a simple case of a particle in a uniform gravitational field, do we have translation invariance or not?
Consider a system where a particle is placed in a uniform gravitational field $\vec{F} = -mg\,\vec{e}_{z}$. The dynamics of this are clearly invariant under translations. When we take $z\rightarrow z+...
- 8,640
8
votes
5
answers
716
views
Why can't we obtain a Hamiltonian from the Lagrangian by only substituting?
This question may sound a bit dumb. Why can't we obtain the Hamiltonian of a system simply by finding $\dot{q}$ in terms of $p$ and then evaluating the Lagrangian with $\dot{q} = \dot{q}(p)$? Wouldn't ...
- 590
3
votes
1
answer
664
views
Hamilton-Jacobi formalism and on-shell actions
My question is essentially how to extract the canonical momentum out of an on-shell action.
The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which ...
- 649
1
vote
2
answers
103
views
Momentum $p = \nabla S$
My book mentions the following equation:
$$p = \nabla S\tag{1.2}$$ where $S$ is the action integral, nabla operator is gradient, $p$ is momentum.
After discussing it with @hft, on here, it turns out ...
- 525
1
vote
2
answers
202
views
Does the conservation of $\frac{\partial L}{\partial\dot{q}_i}$ necessarily require $q_i$ to be cyclic?
If a generalized coordinate $q_i$ is cyclic, the conjugate momentum $p_i=\frac{\partial L}{\partial\dot{q}_i}$ is conserved.
Is the converse also true? To state more explicitly, if a conjugate ...
- 11.8k
0
votes
1
answer
87
views
Help with understanding virtual displacement in Lagrangian
I know that these screen shots are not nice but I have a simple question buried in a lot of information
My question
Why can't we just repeat what they did with equation (7.132) to equation (7.140) ...
- 283