All Questions
Tagged with classical-mechanics lagrangian-formalism
250
questions with no upvoted or accepted answers
8
votes
0
answers
335
views
Euler-Lagrange equations for chain fountain
Most of us are familiar with chain fountains.
I was wondering how this phenomenon is explained in the Lagrangian mechanics. I mean do we know how the Euler-Lagrange equations look like for this system?...
- 3,454
7
votes
0
answers
135
views
Variational principle with $\delta I \neq 0$
In Covariant Phase Space with Boundaries D. Harlow allows boundary terms in the variation of the action. If we have some action $I[\Phi]$ on some spacetime $M$ with boundary $\partial M = \Gamma \cup \...
- 36.4k
6
votes
0
answers
410
views
Is there a modified Least Action Principle for nonholonomic systems?
We know that one can treat nonholonimic (but differential) constraints in the same manner as holonimic constraints. With a given Lagrange Function $L$, the equations of motion for a holonomic ...
- 6,773
5
votes
0
answers
58
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Group theoretical approach to conservation laws in classical mechanics
I'm doing some major procrastination instead of studying for my exam, but I wanted to share my thought just to confirm if I'm right.
Suppose that the action, $S(\mathcal{L})$ forms the basis of a ...
- 362
5
votes
0
answers
335
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Lagrangian of the Euler equations - why are Lin constraints required?
The following equation describes the motion of a rigid body rotation, such as a gyroscope:
$$
\frac{d\textbf{L}}{dt} ={\bf{\tau}}= \textbf{r}\times m\textbf{g}= {\omega}\times \textbf{L}$$
where $...
- 185
5
votes
0
answers
948
views
Intuition behind the principle of virtual work
To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset \mathbb{R}^...
- 36.4k
5
votes
0
answers
470
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Naive questions on the classical equations of motion from the Chern-Simons Lagrangian
Consider a Chern-Simons Lagrangian $\mathscr{L}=\mathbf{e}^2-b^2+g\epsilon^{\mu \nu \lambda} a_\mu\partial _\nu a_\lambda$ in 2+1 dimensions, where the 'electromagnetic' fields are $e_i=\partial _0a_i-...
- 3,744
4
votes
0
answers
168
views
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
4
votes
0
answers
72
views
Classical "bird flocking" Hamiltonian with velocity-velocity interaction
Consider the following classical Lagrangian with an interaction between velocities:
$$\mathcal{L} = \sum_{i} \frac{1}{2}m \mathbf{v}_{i}^{2} + \sum_{i < j} J(r_{ij}) \hat{\mathbf{v}}_{i} \cdot \hat{...
- 1,330
4
votes
0
answers
81
views
Cases of various time symmetries
Is it possible to cook up three physically relevant examples where the Lagrangian has explicit time dependence but the system still has one of the following?
time-reversal invariance,
time ...
- 11.8k
4
votes
0
answers
178
views
Deriving the Lagrangian of a set of interacting particles only from symmetry
In section 5 of Landau and Lifshitz's Mechanics book, they show that the Lagrangian of a free particle must be proportional to its velocity squared, $\mathcal{L} = \alpha\mathbf{v}^2$ using only ...
- 2,157
4
votes
0
answers
720
views
Equilibrium points of three masses on a rigid spring ring with gravity
I'm trying to find the equilibrium points of a given system using Lagrangian mechanics (the system is still not rotating at the beginning).
should I find the diagonal matrix for the characteristic ...
4
votes
0
answers
230
views
Closed trajectories for Kepler problem with classical spin-orbit corrections?
Kepler problem explains closed elliptic trajectories for planetary systems or in Bohr's classical atomic model - let say two approximately point objects, the central one has practically fixed position,...
- 1,010
3
votes
0
answers
101
views
Meaning of equations associated with Legendre transform
In the famous paper about semiclassical Bloch theory https://arxiv.org/abs/cond-mat/9511014, the Lagrangian
\begin{eqnarray}
L (\mathbf{k},\dot{\mathbf{k}}) = -e \delta \mathbf{A}(r,t)\cdot\dot{\...
3
votes
0
answers
130
views
What is the geometric interpretation of a general 'state space' in classical mechanics?
Let $\pmb{q}\in\mathbb{R}^n$ be some n generalized coordinates for the system (say, a double pendulum). Then the 'state space' is often examined using either the 'Lagrangian variables', $(\pmb{q},\dot{...
- 477