All Questions
44
questions
5
votes
1
answer
2k
views
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 \, ,$$
can I immediately conclude that the total energy is constant because $\...
4
votes
2
answers
501
views
Lagrange's Demon de-Conserves Angular Momentum
Monsieur Lagrange pulls a string down through a hole in a horizontal table thereby effecting a rotating (point) mass. A daemon sits on his shoulder and takes careful note of the proceedings. There is ...
- 41
8
votes
4
answers
2k
views
Noether's theorem for space translational symmetry
Imagine a ramp potential of the form $U(x) = a*x + b$ in 1D space. This corresponds to a constant force field over $x$. If I do a classical mechanics experiment with a particle, the particle behaves ...
- 1,222
2
votes
1
answer
990
views
A mass hanging under a table: a problem from Goldstein [closed]
I'm trying to solve Problem 1.19 from Goldstein's Chapter 1 (2nd edition), and am getting bogged down in trigonometry (?). Please help me figure out what I'm doing wrong!
Two mass points of mass $...
- 433
3
votes
2
answers
821
views
Is there something more to Noether's theorem?
From the definition of Lagrangian mechanics, Noether's theorem shows that conservation of momentum and energy comes from invariance vs time and space. Is the reverse true? Are Lagrangian mechanics ...
- 4,256
4
votes
2
answers
635
views
Derivation of law of inertia from Lagrangian method (Landau)
I'm reading Landau's Book.
He tries to conclude the law of inertia from the Lagrange equations.
For that, he argues (by nice suppositions about space and time), that the lagrangian must depend only ...
- 285
4
votes
1
answer
143
views
Is it possible to project a problem of mechanics in a lower dimensionality?
I had the intuition that, in classical mechanics, when the trajectory
of a body is known, then analysis of its motion can be done in the
linear space of that trajectory, if all forces are projected on ...
- 3,798
0
votes
1
answer
191
views
Non-relativistic Kepler orbits
Consider the Newtonian gravitational potential at a distance of Sun:
$$\varphi \left ( r \right )~=~-\frac{GM}{r}.$$
I write the classical Lagrangian in spherical coordinates for a planet with mass $...
- 307