All Questions
Tagged with hamiltonian classical-mechanics
134
questions
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Connection between dispersion relation and symmetries of the Hamiltonian
I am having trouble understanding intuitively the connection between the dispersion relation and the symmetries of the Hamiltonian. For example, suppose we have a lattice and there are four sub-...
- 324
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How do I formulate a quantum version of Hamiltonian flow/symplectomorphisms in phase space to have a "geometric", quantum version of Noether's theorem
I'm currently exploring how Noether's theorem is formulated in the Hamiltonian formalism. I've found that canonical transformations which conserve volumes in phase space, these isometric deformations ...
- 111
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45
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Uncertainty due to assuming a variable is constant - Adiabatic Invariance
I am studying classical mechanics from Goldstein and I ran into a confusing equation in the textbook. In the third edition of the book, equation (12.92) calcucates the average change of the action ...
- 121
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Is it possible to formulate classical Hamiltonian mechanics without reference to a Lagrangian? [duplicate]
The typical way to arrive at Hamiltonian mechanics is through Lagrangian mechanics, defining canonical momentum and the hamiltonian itself in reference to the Lagrangian and its derivatives, but I'm ...
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Classical mechanics: Hamiltonian perturbation theory. What if the perturbing parameter is < 0?
In Hamiltonian Perturbation theory, we have a Hamiltonian of the form $$H(q,p) = H_0(q,p) + \lambda H_1(q,p).$$
One proceeds by expanding the equations of motion in powers of $\lambda$, assuming $\...
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Confusion on Hamiltonian unbounded from below and Ostrogradsky Instability
This might be a silly question but I failed to get it. In Ostrogradsky Instability, we deduced that Lagrangian of higher-order derivatives leads to Hamiltonian linear to canonical momenta, and thus, ...
- 77
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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
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Changing the time parameter and finding the corresponding hamiltonian
I'm dealing with a problem where I have a (classical) Hamiltonian $H(q,p)$ such that, for any scalar function $f(p,q)$,
$$
\dot{f} = \frac{\mathrm{d} f}{\mathrm{d}t} =\{ f,H \}
$$
If I change the time ...
- 31
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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
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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
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Equations of motion and infinitesimal canonical transformations
Currently, I'm diving into infinitesimal canonical transformations, with a particular focus on using the infinitesimal change $\epsilon=\delta t$ and $H$ as our generating function. So, in this ...
- 81
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Hamiltonian energy $E=H = -\frac{\partial S}{\partial t}$ [closed]
In most of the cases, the Hamiltonian is equal to the total mechanical energy, which is usually conserved. They are equal and conserved when the potential energy is not dependent on velocity and there ...
- 525
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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
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94
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How to show that the Hamiltonian $H$ is invariant under flow generated by $F$?
I know usually if I have a transformation of phase space $Q(p,q), P(p,q)$ it is defined to be canonical if and only if its Jacobi matrix is part of the Symplectic Group or equivalently $\{Q^{i}, P_j \}...
- 512
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The Hamiltonian of a system under only the effect of an electric field
I have a maybe silly doubt: in quantum mechanics, we have the Hamiltonian as kinetic energy + potential energy. Now kinetic energy is obtained from the integral of force and displacement. Potential ...
- 306