Tagged Questions
1
vote
1answer
58 views
Is this a valid derivation of the Legendre transformation from the Euler-Lagrange condition
E-L condition:
$$\frac{d p}{dt}=\frac{\partial L}{\partial q}$$
Where $p=\frac{\partial L}{\partial \dot{q}}$
Are the following steps valid:
$$\frac{\partial q}{dt} dp=\partial L$$
$$\dot{q} \: ...
0
votes
3answers
101 views
How is a Hamiltonian constructed from a Lagrangian with a Legendre transform
many textbooks tell me that Hamiltonians are constructed from Lagrangians like
$$L=L(q,\dot{q})$$
with a Legendre transformation to obtain the Hamiltonian as
$$H=\dot{q}\frac{\partial L}{\partial ...
1
vote
1answer
85 views
Why do Lagrangians and Hamiltonians give the equations of motion? [duplicate]
I remember asking my second year Mechanics teacher about why do the Lagrangians give the equations of motion. His answer was that there is no answer to that, it is an empirical fact, and that asking ...
1
vote
0answers
27 views
Proving conservation of angular momentum in an elliptic billiard problem
This is for a course focusing on the connections between Newtonian, Lagrangian and Hamiltonian formalisms.
We're given an elliptic billiard table with foci 1 and 2, where $L_1$ and $L_2$ are the ...
0
votes
1answer
329 views
Hamilton's equations for a simple pendulum
I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with
$$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$
Now Hamilton's equations will be:
$$\dot ...
0
votes
2answers
150 views
Hamiltonian and non conservative force
I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$.
I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is ...
1
vote
2answers
165 views
Good book for Analytical Mechanics
What is a good book for Analytical Mechanics?
To be more specific, I would prefer a book that:
Is written "for mathematicians", i.e. with high mathematics precision (for example, with less emphasis ...
3
votes
3answers
214 views
Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation
In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
2
votes
4answers
315 views
Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations
Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression:
$\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$
where, ...
8
votes
4answers
233 views
What makes an equation an 'equation of motion'?
Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint.
For example, in the ...
2
votes
3answers
602 views
What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)
What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)?
I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ...
0
votes
0answers
181 views
Classical Mechanics: A particle move in one dimension under the influence of two springs [closed]
A particle of mass $m$ can move in one dimension under the influence of two springs connected to fixed points a distance $a$ apart (see figure). The springs obey Hooke’s law and have zero unstretched ...
3
votes
1answer
177 views
Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?
I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude:
$$F(r, t) = \frac{k}{r^2}e^{-at}$$
...
0
votes
0answers
77 views
Describing the movement of the object in a particular situation in Lagrangian way
Suppose there is a object M, (sliding motion) moving by the initial speed $v$ and the initial location $x_0$. Otherwise noted, friction is assumed to be nonexistent. It then meets a circular mold ...
6
votes
1answer
303 views
About Turbulence modeling
There is a paper titled "Lagrangian/Hamiltonian formalism for description of Navier-Stokes fluids" in PRL. After reading the paper, the question arises how far can we investigate turbulence with this ...
3
votes
4answers
645 views
Lagrangian to Hamiltonian in Quantum Field Theory
While deriving Hamiltonian from Lagrangian density, we use the formula
$$\mathcal{H} ~=~ \pi \dot{\phi} - \mathcal{L}.$$
But since we are considering space and time as parameters, why the formula
...
3
votes
3answers
280 views
What is the difference between manifest Lorentz invariance and canonical Lorentz invariance?
I often read that the Lorentz symmetry is manifest in the path integral formulation but is not in the canonical quantization - what does this really mean?
0
votes
2answers
204 views
Advice on classes: Theoretical Mechanics vs E&M II
So I'm having a tough time deciding between courses next semester. I'm a rising 3rd year undergrad math major whose goal is to get a solid understanding of theoretical physics through advanced math ...
5
votes
2answers
635 views
Hamilton-Jacobi Equation
In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and ...
3
votes
3answers
236 views
The number of independent variables in the Lagrangian and Hamiltonian methods in Classical Mechanics
It's told in Landau - Classical Mechanics, that in the Hamiltonian method, generalized coordinates $q_j$ and generalized momenta $p_j$ are independent variables of a mechanical system. Anyway, in the ...
4
votes
3answers
353 views
Generalizing Heisenberg Uncertainty Priniciple
Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$
$$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$
...
4
votes
2answers
914 views
Lagrangian mechanics vs Hamiltonian mechanics
First of all, what are the differences between these two: Lagrangian mechanics and Hamiltonian mechanics?
And secondly, do I need to learn both in order to study quantum mechanics and quantum field ...
9
votes
5answers
1k views
Why not using Lagrangian, instead of Hamiltonian, in non relativistic QM?
When we studied classical mechanics on the undergraduate level, on the level of Taylor, we covered Hamiltonian as well as Lagrangian mechanics.
Now when we studied QM, on the level of Griffiths, we ...
1
vote
1answer
126 views
Question on 1st order Lagrangian Derivation in Faddeev-Jackiw Formalism
I'm looking at this reference (sorry it's a postscript file, but I can't find a pdf version on the web. This paper describes a similar procedure).
The topic is the Faddeev-Jackiw treatment of ...
2
votes
1answer
322 views
Origins of the principle of least time in classical mechanics
Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least ...
7
votes
2answers
96 views
Group of symmetries of Lagrange's equations
Consider the following statements, for a classical system whose configuration space has dimension $d$:
Lagrange equations admit a smaller group of "symmetries" (coordinate change under which ...
5
votes
1answer
274 views
formal framework for talking about 'minimal couplings'
usually on physical theories one would have Lagrangians or Hamiltonians with multiple fields; say, a vector $A_{\mu}$ and a scalar $\phi$ and one would postulate ad hoc a coupling between the fields ...
11
votes
6answers
1k views
What is the symmetry which is responsible for conservation of mass?
According to Noether's theorem, all conservation laws originate from invariance of a system to shifts in a certain space. For example conservation of energy stems from invariance to time translation.
...
6
votes
9answers
2k views
Book about classical mechanics
I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...
