All Questions
44
questions
0
votes
2
answers
82
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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
1
vote
1
answer
75
views
Analogy of Euler-Lagrange-equation and Continuity equation
It seems to me that there is a link between the continuity equation
$$\nabla\rho u + \frac{\partial \rho}{\partial t} = 0$$
and the Euler-Lagrange equation for Lagrangian mechanics
$$\nabla_q L - \...
- 19
3
votes
5
answers
939
views
What is the point of knowing symmetries, conservation quantities of a system?
I think this kind of question has been asked, but i couldn’t find it.
Well i have already know things like symmetries, conserved quantities and Noether’s theorem, as well as their role in particle ...
- 311
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
1
vote
0
answers
54
views
Doubt Regarding Noether's theorem for time-dependent systems
I'm having problems showing Noether's theorem when the lagrangian is time dependent. I'm trying to do it not using infinitesimal transformations from the beginning, but continuous transformations of a ...
1
vote
0
answers
26
views
Is the invariance of the Lagrangian under some transformation equivalent to the covariance of the motion equation? [duplicate]
Take the Lagrangian $L=\frac{1}{2}m{{\left( \frac{{\rm{d}}}{{\rm{d}}t}x \right)}^{2}}-\frac{1}{2}k{{x}^{2}}$, for example.
The equation of motion of this system should be given by $m\frac{{{{\rm{d}}}^{...
- 41
2
votes
1
answer
72
views
Some doubts about action symmetry
We know that Symmetry of the Lagrangian ($\delta L = 0$) always yields some conservation law.
Now, if $\delta L \neq 0$, that doesn't mean we won't have conservation law, because if we can show action ...
- 525
1
vote
1
answer
136
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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
1
vote
1
answer
319
views
Spherical Potential and Angular Momentum Conservation
I have always found it clear that since a spherical potential has all components of angular momentum conserved since the entire system is symmetric under any rotation. However, I was trying to prove ...
2
votes
3
answers
84
views
All continuous symmetries and total number of independent conserved quantities for general classical free particle
Consider the following Lagrangian
$$L=\frac{1}{2}G_{ij}\dot{q}^i\dot{q}^j,$$
where $G_{ij}$ is symmetric and positive semi-definite and $i,j=1,\dots,n$. I want to determine all continuous symmetries ...
- 163
4
votes
0
answers
170
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Noether's Theorem in non-conservative systems
In most books on classical mechanics, Noether's Theorem is only formulated in conservative systems with an action principle. Therefore I was wondering if it is possible to also do that in non-...
- 470
1
vote
2
answers
77
views
Conservation of kinetic energy and external forces
In Goldstein's "Classical Mechanics", at page 360 below eq. (8.84) it is stated that:
"If, further, there are no external forces on the system (monogenic and holonomic), ..., then $T$ ...
- 402
-1
votes
2
answers
620
views
Conservation theorem for cyclic coordinates in the Lagrangian
Suppose $q_1,q_2,...,q_j,..,q_n$ are the generalized coordinates of a system.
$q_j$ is not there in the Lagrangian (it is cyclic).
Then $\frac{\partial L}{\partial\dot q_j}=constant$
In Goldstein, it ...
- 293
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
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
763
views
Conservation of angular momentum using symmetry properties
Goldstein pg 59
It can be shown that if a cyclic coordinate $q_{j}$ is such that $d q_{j}$ corresponds to a rotation of the system of particles around some axis, then the conservation of its ...
- 1,270
4
votes
3
answers
269
views
Newtonian vs Lagrangian symmetry
Suppose we have a ball of mass $m$ in the Earth's gravitational field ($g=const.$). Equation of motion reads as:
$$
ma = -mg
$$
From here we can conclude that we have translational symmetry of the ...
- 1,795
1
vote
1
answer
156
views
Number of conservation laws
I saw a discussion about the relation of symmetries of Lagrangian and conservation laws on a textbook of analytical mechanics. A part that was counterintuitive to me was that all the discussion was ...
- 63
0
votes
0
answers
124
views
Lagrangian and Friction
How does lagrangian mechanics explain loss of momentum conservation in presence of friction?
My try is this:
The lagrange equation would then include a generalized force term $Q_i$:
$$\frac{d}{dt}\...
- 132
0
votes
0
answers
71
views
Conservations for time or space translational invariance, why $\delta L\vert_{\text{time trans}}\neq 0$? but $\delta L\vert_{\text{space trans}}= 0$?
To summarize my question first,
Given a classical mechanics Lagrangian,
$L=L(x(t), \dot{x}(t); t)$,
Why the conservation law for time $t$-translational invariant system, under time variance $\delta ...
- 4,336
2
votes
3
answers
285
views
Can someone explain conservation laws in terms of state space?
"Whenever a dynamical law divides the state space into separate cycles, there is a memory of which cycle they started in. Such a memory is called a conservation law."
—What is meant by this ...
0
votes
0
answers
93
views
Which components of the linear and angular momentum are conserved in the following situation?
A particle moves on a gravitational potential produced by the distribution of mass on a sphere of radius R.
First I calculated the lagrangian:
$T=\frac{m}{2}(\dot{x}^{2}+\dot{y}^{2}+\dot{z}^{2})$
$...
0
votes
1
answer
164
views
Why is there a relationship between symmetries and conservation laws?
I am reading through my professor's notes and I am unsure as to what the intimate relationship between the symmetry property of a physical system and the conservation laws of energy, momentum, and ...
- 65
1
vote
2
answers
409
views
Can we detect a cyclic coordinate by just inspecting the Lagrangian?
I'm reading through Susskind-Hrabovsky's Theoretical Minimum. On page 126, where they are talking about cyclic coordinates, an example is given:
Suppose two particles moving on a line with a ...
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 ...
- 11.7k
0
votes
1
answer
50
views
Physical meaning of constants (momenta?) generated by Noether's theorem via an ${\rm SO}(3)$-action
Let $\Bbb R^3$ be our configuration space. Consider the Lagrangian $L\colon T\Bbb R^3 \cong \Bbb R^6 \to \Bbb R$ given by$$L(x,y,z,\dot{x},\dot{y},\dot{z}) = \frac{m}{2}(\dot{x}^2+\dot{y}^2+\dot{z}^3) ...
- 545
1
vote
1
answer
80
views
How do we define the quantity $Q$, in the conservation of energy? And what does it rely on?
Noether's theorem to me explains how a certain defined quantity (Q) is conserved (locally) in time due to the time translation symmetry, and to be more specific; if we had a ball that is placed in a ...
0
votes
1
answer
158
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Question about the concepts of Noether charge and Noether current
I read that a noether current occurs when the lagrangian assume vector values. Well, what are noether current and noether charge in comparison to elementary classical mechanics notions of Noether's ...
- 3,085
-2
votes
1
answer
151
views
Conservation laws for weird Lagrangian? [closed]
I am asked to find the conserved quantities for the following Lagrangian for a three-particle system in three dimensions
$$L = \left[\sum_{i=1}^{3} \frac{1}{2} m_i \left(|\dot{\bf{r}}|^2 - \omega^2 ...
- 620
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
5
votes
1
answer
2k
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Is there an "invariant" quantity for the classical Lagrangian?
$$
L = \sum _ { i = 1 } ^ { N } \frac { 1 } { 2 } m _ { i } \left| \dot { \vec { x } _ { i } } \right| ^ { 2 } - \sum _ { i < j } V \left( \vec { x } _ { i } - \vec { x } _ { j } \right)
$$
This ...
- 1,669
33
votes
3
answers
6k
views
Why is Noether's theorem important?
I am just starting to wrap my head around analytical mechanics, so this question might sound weird or trivial to some of you.
In class I have been introduced to Noether's theorem, which states that ...
- 490
14
votes
3
answers
1k
views
Justification of the Least Action Principle using conservation of information
In this Phys.SE question, one answer (by Ron Maimon) claims that one can make the assumption of a least action principle plausible using Liouville's Theorem as another starting point of the theory.
...
- 6,763
4
votes
4
answers
490
views
Action principle, Lagrangian mechanics, Hamiltonian mechanics, and conservation laws when assuming Aristotelian mechanics $F=mv$
Define a physical system when Aristotelian mechanics $F=mv$ instead of Newtonian mechanics $F=ma$.
Then we could have action $I=\int L(q,t)dx$ rather than $\int L(q',q,t)dx$.
Is there an action ...
- 417
1
vote
1
answer
67
views
Noethers Symmetries for a system in different cases
If the lagrangian is
$$L=\frac{m}{2}\left( \dot x_1^2+ \dot x_2^2\right)−b(x_1−x_2)^2+a( \dot x_1x_2− \dot x_2x_1).$$
What are the Noether symmetries of the system and the corresponding conserved ...
- 21
2
votes
1
answer
388
views
Noether's Theorem: form of infinitesimal transformation
Noether's theorem states that if the functional $J$ is an extremal and invariant under infinitesimal transformation,
$$ t' = t+ \epsilon \tau + ...,\tag{1}$$
$$ q^{\mu'} = q^{\mu} + \epsilon \zeta^{\...
- 328
7
votes
3
answers
11k
views
Constants of motion from a Lagrangian
If I have a Lagrangian (made up equation in this case):
$$L = \frac{1}{2}mr^2\dot\theta + \frac{1}{4}mg\ddot\theta \, ,$$
can I immediately conclude that the total energy is constant because $\...
4
votes
2
answers
501
views
Lagrange's Demon de-Conserves Angular Momentum
Monsieur Lagrange pulls a string down through a hole in a horizontal table thereby effecting a rotating (point) mass. A daemon sits on his shoulder and takes careful note of the proceedings. There is ...
- 41
8
votes
4
answers
2k
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Noether's theorem for space translational symmetry
Imagine a ramp potential of the form $U(x) = a*x + b$ in 1D space. This corresponds to a constant force field over $x$. If I do a classical mechanics experiment with a particle, the particle behaves ...
- 1,222
2
votes
1
answer
990
views
A mass hanging under a table: a problem from Goldstein [closed]
I'm trying to solve Problem 1.19 from Goldstein's Chapter 1 (2nd edition), and am getting bogged down in trigonometry (?). Please help me figure out what I'm doing wrong!
Two mass points of mass $...
- 433
3
votes
2
answers
821
views
Is there something more to Noether's theorem?
From the definition of Lagrangian mechanics, Noether's theorem shows that conservation of momentum and energy comes from invariance vs time and space. Is the reverse true? Are Lagrangian mechanics ...
- 4,256
4
votes
2
answers
635
views
Derivation of law of inertia from Lagrangian method (Landau)
I'm reading Landau's Book.
He tries to conclude the law of inertia from the Lagrange equations.
For that, he argues (by nice suppositions about space and time), that the lagrangian must depend only ...
- 285
4
votes
1
answer
143
views
Is it possible to project a problem of mechanics in a lower dimensionality?
I had the intuition that, in classical mechanics, when the trajectory
of a body is known, then analysis of its motion can be done in the
linear space of that trajectory, if all forces are projected on ...
- 3,798
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