All Questions
39
questions with no upvoted or accepted answers
4
votes
0
answers
72
views
Classical "bird flocking" Hamiltonian with velocity-velocity interaction
Consider the following classical Lagrangian with an interaction between velocities:
$$\mathcal{L} = \sum_{i} \frac{1}{2}m \mathbf{v}_{i}^{2} + \sum_{i < j} J(r_{ij}) \hat{\mathbf{v}}_{i} \cdot \hat{...
- 1,330
3
votes
0
answers
101
views
Meaning of equations associated with Legendre transform
In the famous paper about semiclassical Bloch theory https://arxiv.org/abs/cond-mat/9511014, the Lagrangian
\begin{eqnarray}
L (\mathbf{k},\dot{\mathbf{k}}) = -e \delta \mathbf{A}(r,t)\cdot\dot{\...
3
votes
0
answers
130
views
What is the geometric interpretation of a general 'state space' in classical mechanics?
Let $\pmb{q}\in\mathbb{R}^n$ be some n generalized coordinates for the system (say, a double pendulum). Then the 'state space' is often examined using either the 'Lagrangian variables', $(\pmb{q},\dot{...
- 477
3
votes
0
answers
77
views
Is there a unique accepted Lagrangian formulation of Nambu mechanics?
In section 5 of their 2000 paper "Nambu Mechanics in the Lagrangian Formalism", Ogawa & Sagae critique previous attempts by Bayen & Flato and by Takhtajan to formulate the theory ...
- 3,684
3
votes
0
answers
396
views
Are all canonical transformations either a point transformation, gauge transformation or a combination of them?
It's regularly argued that in the Hamiltonian formalism, we have more freedom to choose our coordinates and that this is arguably its most important advantage.
To quote from two popular textbooks:
[S]...
- 10.1k
3
votes
0
answers
830
views
Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian
I want to show that the gauge transformation
$$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
141
views
Relativistic configuration space in classical mechanics
Okay so a couple of questions. Firstly I realise that in order to study the dynamics of one particle (classically), we define the Lagrangian and Hamiltonian to be the maps from the tangent and ...
- 504
3
votes
0
answers
714
views
Non-canonical transformation
I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
- 81
3
votes
0
answers
222
views
Does the additivity property of Integrals of motion and Lagrangians valid in all situations?
I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of ...
- 345
2
votes
2
answers
108
views
How to explain the independence of coordinates from physics aspect and mathmetics aspect?
When I was studying Classical Mechanics, particularly Lagrangian formulation and Hamiltonian formulation. I always wondering how to understand the meaning of independence of parameters used of ...
- 529
2
votes
1
answer
117
views
Do Legendre transformation form a group?
In my classical mechanics class, my professor asked if Legendre transformations form a group, and in my little knowledge about groups, I know that a transformation group consists of a set of ...
- 21
2
votes
0
answers
44
views
How are conjugate variables in mechanics and stat mech related to duality in convex optimization?
I recently studied duality in optimization where a primal optimization problem can be casted as a dual problem which provides meaningful lower bounds on the primal. There is also a notion of conjugate ...
- 131
2
votes
0
answers
226
views
Are first Integrals the same in Lagrangian and Hamiltonian formalism?
A first integral in Lagrangian formalism is defined as a function which is constant along the solutions $(q,\dot{q})$ where $q$ are the generalized coordinates; while a first integral in Hamiltonian ...
- 423
2
votes
0
answers
140
views
Deriving Hamilton's equations independently
The usual way to derive Hamilton's equations is to perform Legendre transformation of the Lagrangian and then use the stationarity principle. However, this procedure seems a little artificial to me ...
- 21
2
votes
0
answers
2k
views
Canonical Momentum Conjugate vs. Momentum
I stumbled upon this while reading about Legendre Transforms today. So consider an n-particle system. The Lagrangian is a function of $ q_i$'s and $\dot q_i$'s. If you consider the manifold $M$ where ...
- 2,997