All Questions
231
questions
1
vote
2
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78
views
What is causing this sign difference in the centrifugal term between Lagrangian and Hamiltonian formalism? [duplicate]
Consider a central force problem of the form with the Lagrangian
$$
L(r, \theta, \dot{r}, \dot{\theta}) = \frac{1}{2} m \left( \dot{r}^2 + r^2 \dot{\theta}^2 \right) - V(r),
$$
where $r = |\vec{x}|$. ...
- 115
4
votes
0
answers
72
views
Classical "bird flocking" Hamiltonian with velocity-velocity interaction
Consider the following classical Lagrangian with an interaction between velocities:
$$\mathcal{L} = \sum_{i} \frac{1}{2}m \mathbf{v}_{i}^{2} + \sum_{i < j} J(r_{ij}) \hat{\mathbf{v}}_{i} \cdot \hat{...
- 1,330
0
votes
1
answer
68
views
Examples of solvable simple systems in relativistic mechanics [closed]
What examples are there of simple (special) relativistic systems in which the equations of motion are solvable? There are countless examples of these in non-relativistic mechanics, e.g. the simple ...
- 801
8
votes
2
answers
614
views
How is the Hamiltonian & Lagrangian non-relativistic & relativistic respectively?
I have read from the textbook of Matthew Schwartz on page 49 of the PDF viewer (or page 30 of the textbook) where he says:
I am interested in the last sentence of this paragraph where he says that ...
- 1,896
1
vote
0
answers
236
views
How to derive the canonical momentum of a single spin in the magnetic field in classical mechanics?
The classical spin can be denoted as $\vec{S}=S\, \vec{n}$, where $\vec{n}=(\sin{\theta}\cos{\phi},\sin{\theta}\sin{\phi},\cos{\theta})$. The magnet field $\vec{h}=(0,0,h)$.
The Hamiltonian is $H=-hS\...
- 868
0
votes
2
answers
99
views
Are there physical systems for which Hamiltonian is not defined? [duplicate]
Consider the following Lagrangian :
$$\mathcal{L}=\frac{1}{2}\dot{q}\sin^2q$$
It's easy to see for such a lagrangian we can't find the Hamiltonian since
$$p=\partial_{\dot{q}}\mathcal{L}=\frac{1}{2}\...
- 11.9k
3
votes
1
answer
326
views
Intuition about non-invariance of the Hamiltonian in canonical transformation
Suppose $q={\{q_i\}}_{i=1}^n$ is the set of generalized coordinates of a dynamical system. $L(q,\dot q,t)$ is the Lagrangian of the system. Now we make coordinate transformation $Q_i=Q_i(q,t)$. Then ...
- 436
1
vote
0
answers
45
views
How did Lagrange derive this Hamilton-like equation in Mechanique analytique
I was trying to understand how Hamiltonian formulation was derived in the history, and read Mechanicque Analytique, and found these pages from section V of this book, which derived the Hamiltonian-...
- 81
3
votes
1
answer
551
views
Limit Cycles of a Simple Pendulum
In this pdf file the dynamical behavior of a simple pendulum is discussed. The equation of motion for a pendulum with no dissipation is:
$$\dot{\theta}=\omega, \qquad \dot{\omega}=-\frac{g}{l}\sin\...
- 439
6
votes
2
answers
699
views
Contradiction between Lagrangian and Hamiltonian formalism
I am currently studying an introductory course to theoretical physics. I stumbled upon something I can't seem to understand, namely:
Suppose that we have a system without any potential energy, and ...
- 77
6
votes
1
answer
227
views
When are Hamiltonian and Lagrangian dynamics not globally Equivalent?
Let $Q$ be a smooth manifold, let $TQ$ be its tangent bundle and let $T^*Q$ be its cotangent bundle. Lagrangian mechanics takes place on level of the level of velocities $L:TQ\rightarrow \mathbb{R}$. ...
- 105
0
votes
0
answers
40
views
Are there any useful functions other than Hamiltonian and Lagrangian? [duplicate]
Besides Hamiltonian and Lagrangian, are there any other similar functions useful when studying mechanical system? (It is ok if the functions do not work well and lead to mistakes in some situation, ...
- 101
8
votes
1
answer
446
views
Hamiltonian systems without a corresponding Lagrangian system
I was playing around with a Hamiltonian model for the propagation of photons:
$$ H = c \sqrt{p \cdot p} + V(q) \tag{1}$$
which gives a meaningful set of equations of motion,
$$ \dot{q}_i = c \frac{p_i}...
- 183
2
votes
1
answer
750
views
Independence of position and momentum in action
Why are position and momentum independent with respect to the Hamiltonian Action $S_H$ given by
$$
S_H = \int_{t_1}^{t_2} (p \dot q - H) dt \ \ \ ? \tag{1}
$$
While deriving Hamilton's equations from ...
- 23
1
vote
1
answer
152
views
Significance of Lagrangian in Principle of Least Action?
I've been studying the Legendre transform and it's been a fun realization to see that the relationship between the Lagrangian and the Hamiltonian is simply a Legendre transform, i.e.,
$$\{H, p\}\...
- 802