All Questions
50
questions
-3
votes
2
answers
81
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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
82
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 ...
- 49
0
votes
1
answer
69
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 ...
- 7,398
1
vote
1
answer
66
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 ...
- 349
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:
$$p_k\equiv\dfrac{\partial\mathcal T}{\partial \dot q_k}\text{ and }p_k\...
- 296
1
vote
1
answer
54
views
Sufficient condition for conservation of conjugate momentum
Is the following statement true?
If $\frac{\partial \dot{q}}{\partial q}=0$, then the conjugate momentum $p_q$ is conserved.
We know that conjugate momentum of $q$ is conserved if $\frac{\partial L}{\...
- 41
0
votes
1
answer
80
views
Lagrangian and Hamiltonian Mechanics: Conjugate Momentum
I am a physics undergraduate student currently taking a classical mechanics course, and I am not able to understand what conjugate/canonical momentum is (physically). It is sometimes equal to the ...
1
vote
1
answer
51
views
Lagrange momentum for position change
After the tremendous help from @hft on my previous question, after thinking, new question popped up.
I want to compare how things behave when we do: $\frac{\partial S}{\partial t_2}$ and $\frac{\...
- 525
1
vote
2
answers
103
views
Momentum $p = \nabla S$
My book mentions the following equation:
$$p = \nabla S\tag{1.2}$$ where $S$ is the action integral, nabla operator is gradient, $p$ is momentum.
After discussing it with @hft, on here, it turns out ...
- 525
2
votes
0
answers
57
views
What are the extra terms in the generalized momentum regarding the Lagrangian formalism?
In the lectures, we have defined the generalized momentum in the Lagrangian to be:
$$p_i=\frac{\partial L}{\partial\dot q_i}.$$
But with this definition, if we do not make any assumptions about the ...
- 193
1
vote
1
answer
135
views
In a simple case of a particle in a uniform gravitational field, do we have translation invariance or not?
Consider a system where a particle is placed in a uniform gravitational field $\vec{F} = -mg\,\vec{e}_{z}$. The dynamics of this are clearly invariant under translations. When we take $z\rightarrow z+...
- 8,640
4
votes
3
answers
236
views
How to show the velocity of free motion is constant in Galileo's relativity principle?
Picture below is from Landau & Lifshitz's Mechanics. How to get the red line from green line?
- 351
0
votes
1
answer
299
views
Generalized vs conjugate momenta
For a given Lagrangian $L$, the $i$th generalized momentum is defined as
$$p_i = \frac{\partial L}{\partial \dot{q_i}}$$
where $\dot{q_i}$ is the time derivative of the $i$th generalized coordinate (i....
- 3,350
3
votes
2
answers
452
views
Help with geometric view of conjugate momenta and Legendre transformation
I'm familiar with the ''coordinate view'' of Lagrangian and Hamiltonian mechanics where if $\pmb{q}=(q^1,\dots, q^n)\in\mathbb{R}^n$ are any $n$ generalized coordinates and $L(\pmb{q},\dot{\pmb{q}})$ ...
- 477
2
votes
1
answer
99
views
What do you think about this particularization of the Euler-Lagrange equation that resembles Newton's 2nd Law?
For:
$$\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}}{\...
- 1,775
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
0
votes
2
answers
530
views
Why is derivative of Lagrangian with respect to generalized position and velocity equal to this?
I'm currently studying Lagrangian mechanics, and in the process, I've met the following equations in a couple of proofs.
$$
\frac{\partial \mathcal{L}}{\partial q_i} = \dot p_i
$$
$$
\frac{\partial \...
user334560
0
votes
0
answers
95
views
Conjugate momentum vs translation generator with non-standard kinetic term
I am reading this paper and for equation (2.5) (associated with the Lagrangian in eq 2.1) there is the claim that for a Lagrangian $L(\varphi,A,\dot{\varphi},\dot{A})$ containing an extra non-standard ...
3
votes
5
answers
370
views
Axiomatising classical mechanics to arrive at the principle of stationary action - what are the fundamental definitions of momentum, etc.?
$\newcommand{\d}{\mathrm{d}}\newcommand{\l}{\mathcal{L}}$Throughout all my study of physics, it has never been clear what is a definition, what is an axiom, what is a law and what is a proof in ...
- 221
3
votes
0
answers
121
views
Intuitive explanation on why velocity = 0 for a inverted pendulum on a wheel system
I believe I have solved below problem. I am not looking for help on problem-solving per se. I am just looking for an intuitive explanation.
Problem statement: wheel mass = $m_1$, even mass rod BC mass ...
- 131
1
vote
1
answer
108
views
Geometric meaning of conjugate momentum
Let's say I have a free particle moving in an $n$-dimensional Manifold $M$. There is a tangent space $TM$ associated with all possible infinitesimal motions of a particle at each point in this ...
- 137
2
votes
2
answers
156
views
How to derive the fact that $p\sim d/dx$ and $H\sim d/dt$ from classical mechanics?
I am trying to understand Noether's conserved quantities to shifts in time and or position. I have seen the derivation of the operators for Schrodinger's equation but not for classical mechanics.
Is ...
- 381
0
votes
0
answers
124
views
Lagrangian and Friction
How does lagrangian mechanics explain loss of momentum conservation in presence of friction?
My try is this:
The lagrange equation would then include a generalized force term $Q_i$:
$$\frac{d}{dt}\...
- 132
2
votes
2
answers
161
views
Conjugate momentum notation
I was reading Peter Mann's Lagrangian & Hamiltonian Dynamics, and I found this equation (page 115):
$$p_i := \frac{\partial L}{\partial \dot{q}^i}$$
where L is the Lagrangian. I understand this is ...
- 41
0
votes
1
answer
320
views
Gauge ivariance and canonical versus kinetic momenta for a charged particle in an EM field
I all, I am struggling to grasp the notion of gauge invariant when talking about an object like the canonical momenta $\frac{\partial L}{\partial \dot{q}_i}$ or kinetic momenta $m\dot{q}_i$.
I am very ...
- 1,031
3
votes
1
answer
151
views
Reducing the degrees of freedom of a Lagrangian in a spherical potential by using integrals of motion [duplicate]
I'm sure I've made a silly mistake here, so I would be very grateful if someone could help me clear it up! Here is my reasoning:
The Lagrangian in a spherical potential is
$$
\mathcal{L}=\frac{m\...
- 2,157
1
vote
2
answers
202
views
Does the conservation of $\frac{\partial L}{\partial\dot{q}_i}$ necessarily require $q_i$ to be cyclic?
If a generalized coordinate $q_i$ is cyclic, the conjugate momentum $p_i=\frac{\partial L}{\partial\dot{q}_i}$ is conserved.
Is the converse also true? To state more explicitly, if a conjugate ...
- 11.8k
0
votes
1
answer
328
views
Difference between kinematic momentum and conjugated momentum in purely mechanical setup
I don't know much about physics, but I wanted to understand what was the difference between the "kinematic momentum" and the conjugated momentum. As I understand it, kinematic momentum is mass times ...
- 111
1
vote
2
answers
352
views
Ambiguity in d'Alembert's principle
It seems to me that many different momenta $\dot{\bf p}_j $ can satisfy d'Alembert's principle:
$$\tag{1} \sum_{j=1}^N ( {\bf F}_j^{(a)} - \dot{\bf p}_j ) \cdot \delta {\bf r}_j~=~0 $$
in a ...
- 121
2
votes
2
answers
2k
views
Defining generalized momentum in terms of kinetic energy versus a Lagrangian
Reputable authors (e.g., Bergmann, Wells, Susskind) define generalized momentum using the Lagrangian $L$ as $$p_{i}\equiv\frac{\partial L}{\partial\dot{q}^{i}}.\tag{1}$$
Joos and Freeman define ...
- 2,015
3
votes
0
answers
830
views
Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian
I want to show that the gauge transformation
$$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
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
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
0
votes
1
answer
3k
views
Generalized momentum conjugate and potential $U(q, \dot q)$
On Goldstein's "Classical Mechanics" (first ed.), I have read that
if $q_j$ is a cyclic coordinate, its generalized momentum conjugate $p_j$ is costant.
He obtained that starting from Lagrange's ...
- 1,133
0
votes
1
answer
191
views
Non-relativistic Kepler orbits
Consider the Newtonian gravitational potential at a distance of Sun:
$$\varphi \left ( r \right )~=~-\frac{GM}{r}.$$
I write the classical Lagrangian in spherical coordinates for a planet with mass $...
- 307
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