All Questions
Tagged with hamiltonian lagrangian-formalism
69
questions
2
votes
1
answer
115
views
Derivation of Dirac Hamiltonian
In Minkowski spacetime with signature $(-,\;+,\;+,\;...,\;+)$ the Dirac Lagrangian reads
$$
L=\int d^dx\;\mathcal{L}=\int d^dx\;\psi^\dagger\left(i\gamma^0\gamma^\mu\partial_\mu-im\gamma^0\right)\psi.
...
- 503
0
votes
0
answers
54
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Why the kinetic term of the Hamiltonian has to be positive definite for well-posed time evolution?
I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12:
$$\frac{\mathcal{S}}{\mathcal{T}}= \int \mathrm{d}t\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)...
- 193
8
votes
1
answer
2k
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If the Lagrangian depends explicitly on time then the Hamiltonian is not conserved?
Why is the Hamiltonian not conserved when the Lagrangian has an explicit time dependence? What I mean is that it is very obvious to argue that if the Lagrangian has no an explicit time dependence $L=L(...
- 227
0
votes
3
answers
106
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Lagrangian equal to Hamiltonian - is it possible?
In one calculation, I got a result where the Hamiltonian equals the Lagrangian, with both values constant.
Eventually I found the error and corrected it, but I began to wonder whether there could even ...
- 9
1
vote
0
answers
83
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Independence of variables in Lagrangain and Hamiltonian mechanics - Rigorous Mathematical approach
I am trying to self-learn the Hamiltonian and Lagrangian mechanics and I came across thoughts to which I could not find an answer therefore I would like to try and ask them here.
My questions are as ...
- 23
0
votes
0
answers
64
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What would make the Legendre transformation interesting, from the graphical point of view?
The term "transformation" is often used in physics and mathematics for functions to denote a different (and useful) way of encoding the information in a function. In this question, I want to ...
- 2,599
2
votes
1
answer
861
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Proof that Hamiltonian is constant if Lagrangian doesn't depend explicitly on time
I know that on solutions of motion we have $\frac{dH}{dt}=\frac{\partial H}{\partial t} $ and i understand the proof for this fact. Then, we have that $$\frac{\partial H}{\partial t}=-\frac{\partial L}...
- 49
0
votes
1
answer
148
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Why is the hamiltonian density written in terms of $\phi$ and $\pi$ only? [duplicate]
Why is the hamiltonian density defined as:
$$\mathcal{H}=\dot{\phi}\pi-\mathcal{L}$$
Where $\pi \equiv \frac{d\mathcal{L}}{d\dot{\phi}}$ and $\mathcal{L}(\dot\phi, \nabla \phi, \phi)$ is a function of ...
- 1,324
0
votes
2
answers
42
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Is there a way to understand which variable is more influential in the dynamics of a system?
Is there any known way to identify which variable has the most impact in the dynamics of a system given its lagrangian or hamiltonian formulation? Let's say i have a system with 3 variables, two ...
0
votes
1
answer
202
views
How to find the ground state of a system via its Hamiltonian density?
I am trying to find the ground state of the following Lagrangian (with $\lambda> 0 , g > 0$):
$$\tag{1} \mathcal{L}= -\frac{1}{2}(\partial_\mu \partial^\mu \sigma + \partial_\mu \pi \partial^\mu ...
- 23
2
votes
2
answers
215
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Hamiltonian from Lagrangian defined as an integral
I need to derive the Hamiltonian from the Lagrangian defined in the following way:
$$L[x, \dot{x}] = \int_{t_0}^{t_1} f(x(t), t) \sqrt{1 + \dot{x}^2} \mathrm{d}t.\tag{1}$$
The usual method is to ...
- 131
7
votes
2
answers
2k
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What is "gradient energy" in classical field theory?
For the simple theory of a single real scalar field $\phi$ in 1+1D, the Lagrange density is $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-U(\phi)\tag{1}$$
with Minkowski signature $(+,-)$, ...
- 661
2
votes
2
answers
156
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How to derive the fact that $p\sim d/dx$ and $H\sim d/dt$ from classical mechanics?
I am trying to understand Noether's conserved quantities to shifts in time and or position. I have seen the derivation of the operators for Schrodinger's equation but not for classical mechanics.
Is ...
- 381
3
votes
1
answer
324
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What is the difference between gauge transformation and canonical transformation？
Recently I have been studying theoretical mechanics, then I have this question, Is the gauge transformation the same as the canonical transformation？
3
votes
2
answers
209
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Scalar field Hamiltonian $H = 0$ from parameterization independence
This question is related (but not similar) to this old one of mine:
How to derive the two Friedmann-Lemaître equations from a Lagrangian?
Consider the Lagrangian of an isotropic-homogeneous ...
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