All Questions
50
questions
-3
votes
2
answers
76
views
Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
In Lagrangian mechanics the momentum is defined as:
$$p=\frac{\partial \mathcal{L}}{\partial \dot q}$$
Also we can define it as:
$$p=\frac{\partial S}{\partial q}$$
where $S$ is Hamilton's principal ...
- 443
0
votes
2
answers
79
views
Generalized momentum
I am studying Hamiltonian Mechanics and I was questioning about some laws of conservation:
in an isolate system, the Lagrangian $\mathcal{L}=\mathcal{L}(q,\dot q)$ is a function of the generalized ...
- 59
0
votes
1
answer
68
views
Changing variables from $\dot{q}$ to $p$ in Lagrangian instead of Legendre Transformation
This question is motivated by a perceived incompleteness in the responses to this question, which asks why we can't just substitute $\dot{q}(p)$ into $L(q,\dot{q})$ to convert it to $L(q,p)$, which ...
- 7,388
1
vote
1
answer
59
views
Landau/Lifshitz action as a function of coordinates [duplicate]
In Landau/Lifshitz' "Mechanics", $\S43$, 3ed, the authors consider the action of a mechanical system as a function of its final time $t$ and its final position $q$. They consider paths ...
- 277
1
vote
1
answer
49
views
Definition of generalized momenta in analytical mechanics
I've seen mainly two definitions of generalized momenta, $p_k$, and I wasn't sure which one is always true/ the correct one:
$$p_k\equiv\dfrac{\partial\mathcal T}{\partial \dot q_k}\text{ and }p_k\...
- 296
1
vote
1
answer
54
views
Sufficient condition for conservation of conjugate momentum
Is the following statement true?
If $\frac{\partial \dot{q}}{\partial q}=0$, then the conjugate momentum $p_q$ is conserved.
We know that conjugate momentum of $q$ is conserved if $\frac{\partial L}{\...
- 41
0
votes
1
answer
80
views
Lagrangian and Hamiltonian Mechanics: Conjugate Momentum
I am a physics undergraduate student currently taking a classical mechanics course, and I am not able to understand what conjugate/canonical momentum is (physically). It is sometimes equal to the ...
1
vote
1
answer
51
views
Lagrange momentum for position change
After the tremendous help from @hft on my previous question, after thinking, new question popped up.
I want to compare how things behave when we do: $\frac{\partial S}{\partial t_2}$ and $\frac{\...
- 525
1
vote
2
answers
101
views
Momentum $p = \nabla S$
My book mentions the following equation:
$$p = \nabla S\tag{1.2}$$ where $S$ is the action integral, nabla operator is gradient, $p$ is momentum.
After discussing it with @hft, on here, it turns out ...
- 525
2
votes
0
answers
57
views
What are the extra terms in the generalized momentum regarding the Lagrangian formalism?
In the lectures, we have defined the generalized momentum in the Lagrangian to be:
$$p_i=\frac{\partial L}{\partial\dot q_i}.$$
But with this definition, if we do not make any assumptions about the ...
- 193
1
vote
1
answer
134
views
In a simple case of a particle in a uniform gravitational field, do we have translation invariance or not?
Consider a system where a particle is placed in a uniform gravitational field $\vec{F} = -mg\,\vec{e}_{z}$. The dynamics of this are clearly invariant under translations. When we take $z\rightarrow z+...
- 8,640
4
votes
3
answers
232
views
How to show the velocity of free motion is constant in Galileo's relativity principle?
Picture below is from Landau & Lifshitz's Mechanics. How to get the red line from green line?
- 351
0
votes
1
answer
293
views
Generalized vs conjugate momenta
For a given Lagrangian $L$, the $i$th generalized momentum is defined as
$$p_i = \frac{\partial L}{\partial \dot{q_i}}$$
where $\dot{q_i}$ is the time derivative of the $i$th generalized coordinate (i....
- 3,350
3
votes
2
answers
449
views
Help with geometric view of conjugate momenta and Legendre transformation
I'm familiar with the ''coordinate view'' of Lagrangian and Hamiltonian mechanics where if $\pmb{q}=(q^1,\dots, q^n)\in\mathbb{R}^n$ are any $n$ generalized coordinates and $L(\pmb{q},\dot{\pmb{q}})$ ...
- 477
2
votes
1
answer
98
views
What do you think about this particularization of the Euler-Lagrange equation that resembles Newton's 2nd Law?
For:
$$\mathcal{L}=\mathcal{L}(q_j,\dot{q_j},t)=T-V$$
the Euler-Lagrange equation is simply:
$$\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\mathcal{\dot{q_j}}} \right)-\frac{\partial \mathcal{L}}{\...
- 1,645