The Hamiltonian formalism is a formalism in Classical Mechanics. Besides Lagrangian Mechanics, it is an effective way of reformulating classical mechanics in a simple way. Very useful in Quantum Mechanics, specifically the Heisenberg and Schrodinger formulations. Unlike Lagrangian ...

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Derive the generating function for canonical transformation of type $F_3$

I'm working on some practice questions and I am a bit confused with this one: Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition ...
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If $(q,p)$ to $(Q,P)$ is a canonical transformation, then does this imply $(Q,P)$ to $(q,p)$ is also?

If $(q,p)$ to $(Q,P)$ is a canonical transformation, then does this imply $(Q,P)$ to $(q,p)$ is also, assuming Hamilton's equations hold for the coordinates $(q,p)$? This seems like it should be true ...
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Ostrogradski’s theorem's proof

I am looking for a proof of the next theorem: "If the higher order time derivative Lagrangian is non-degenerate, there is at least one linear instability in the Hamiltonian of this system." Where ...
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1answer
72 views

Determining the geometry of the phase space of a system [closed]

How do we check the geometry of the phase space ? I mean in classical mechanics we use position and conjugate momenta as a space of all possible states of the particle. How do we know that this phase ...
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1answer
50 views

In a rigid rotor, are there “elegant” orientation coordinates that are conjugate to angular momenta?

I just was looking at the big bag-of-math wikipedia article on rigid rotors, and the section on the Hamiltonian form bugs me a bit since they are using Euler angles to represent the orientation. As a ...
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59 views

How can I write the anderson hamiltonian as a matrix? [closed]

How can I write this Hamiltonian: $$ H = \sum E_d \hat{n}_d + \sum_k \epsilon_k\hat{n}_k + \sum_k V_{kd} (\hat{a}^\dagger_k \hat{a}_d + \hat{a}^\dagger_d \hat{a}_k) $$ in matrix form using its ...
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1answer
125 views

Noise spectrum of two systems and interacting Hamiltonian

I've been discovering recently the concept of noise spectrum, defined as: $$S_{xx}[\omega] = \int dt \langle x(t)x(0)\rangle \text{e}^{-i\omega t}$$ Roughly the Fourrier transform of the two-point ...
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73 views

Quantization conditions/ Real Scalar field

It is often written in books, the quantization conditions for classical field theory leading to Lagrangian of a real scalar field and thus to Klein Gordon equation. And these are introduced by ...
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113 views

Hamiltonian for a Lagrangian with coupling

I am dealing with the following Lagrangian density $$\mathscr{L}_{em}= -\frac{1}{2}\rho\omega^2 u^2 +\frac{1}{2}\nabla u:\Sigma :\nabla ...
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What are the eigenstates of the sum of the exchange and hyperfine hamiltonian in a quantum dot?

I am trying to find the eigenstates for the Hamiltonian for a quantum dot (QD). Here, I am considering the exchange and hyperfine Hamiltonian. The Zeeman Hamiltonian is not included. $\hat{H}$ = ...
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Any good resources for Lagrangian and Hamiltonian Dynamics?

I'm taking a course on Lagrangian and Hamiltonian Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics. So far at my university ...
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78 views

Practical Book on Hamiltonian and Lagrangians? [duplicate]

Are there any terse, accessible books that are geared specifically at learning these two formalisms and how to effectively use them? So far I've only see either topic introduced as a part of another ...
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Constant quantity associated to symmetry [closed]

I'm attending a subject in theoretical mechanics and we saw this fact that bugged me a little. It's by the way referenced in: John R Cary, Lie transform perturbation theory for Hamiltonian ...
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1answer
69 views

Symplectic notation proof?

Hello I am trying to work through a little proof of the symplectic condition for Hamilton's equations for a classical mechanics course. I am trying to understand the meaning of the relation ...
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1answer
524 views

Hamilton-Jacobi equation with time dependent Hamiltonian

I was struggling with this exercise about Hamilton-Jacobi equation. I have to solve by menas of Hamilton's principal function the system with Hamiltonian: $$\tag{1} H=\frac{p^2}{2m}-mAtx $$ with $A$ ...
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Connection between Poisson Brackets and Symplectic Form

Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ {q,p} $ basis are the same as those of the inverse of the symplectic matrix $ \Omega^{-1} $, whereas the matrix elements ...
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2answers
282 views

Bertrand's theorem

I found in Goldstein's Classical Mechanics that the condition for closed orbits is given by $\frac{d^2 V_{eff}}{dr^2}>0$.(bertrand's theorem). Can somebody explain to me, how this inequality is ...
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Contradiction in classical analysis of the hamiltonian $\mathcal{H}=xp$?

I am writing an essay on the Berry Keating article proposing to use the $\mathcal{H}=xp$ hamiltonian to get a correspondence between the nontrivial riemann zeros and the eigenvalues of an Hermitian ...
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What are resonant tori?

What is the definition of a resonant/invariant torus (in the phase space of a Hamiltonian system)? Are there non-resonant tori?
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Symplectic Structure without predefined Hamiltonian

Here there is a link which has helped me understanding the relationship between symplectic geometry and classical mechanincs. In short, the symplectic form transforms the derivative of the ...
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4answers
5k views

When is the Hamiltonian of a system not equal to its total energy?

I thought the Hamiltonian was always equal to the total energy of a system but have read that this isn't always true. Is there an example of this and does the Hamiltonian have a physical ...
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$N$ classical Harmonic Oscillators in microcanonical ensemble in three dimensions

Is my expression for the Hamiltonian for $N$ classical harmonic oscillators correct in three dimension as I am trying to solve it in microcanonical ensemble ...
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Finding conserved quantities from Hamiltonian when Symmetry is not evident [closed]

A particle is moving in 3D space, under a potential $$V = -\frac{\alpha}{r}-\frac{\vec{r} \cdot \vec{\mu}}{r^3 } $$ where $\vec{\mu}$ is some constant vector. I need to show there are three ...
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Hamiltonian is conserved, but is not the total mechanical energy

I wondering about the interpretation for the energy difference between the Hamiltonian and the total mechanical energy for systems where the Hamiltonian is conserved, but it is not equal to the total ...
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What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

I have a problem with one of my study questions for an oral exam: The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
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1answer
211 views

Lagrangian and Hamiltonian EOM with dissipative force

I am trying to write the Lagrangian and Hamiltonian for the forced Harmonic oscillator before quantizing it to get to the quantum picture. For EOM $$m\ddot{q}+\beta\dot{q}+kq=f(t),$$ I write the ...
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Introducing time-dependent drive into the Hamiltonian of quantised electric circuits

Suppose I have the schematic of a superconducting electric circuit composed of (quasi) lossless linear inductances and capacitances and some non-linear inductances, eg. Josephson junctions. The ...
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1answer
57 views

De Donder Weyl theory

Im trying to get my head around what the difference is between a symplectic and multisymplectic manifold is. My understanding currently is that on a symplectic manifold time is the parameter that ...
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1answer
295 views

Hamiltonian formulation of single particle in an electromagnetic field

Consider a charged particle in a static electromagnetic field. Suppose that the domain is simply connected so that the second law of Newton's dynamics reads: $$ ...
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Find port-Hamiltonian representation for DC motor connected to a load

We are modelling a DC motor connected to a rotating load via a rod that acts like a torsion spring, i.e., a rod that can have different angular velocities at its ends. Now the idea is to split the ...
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What is canonical momentum?

What does the canonical momentum $\textbf{p}=m\textbf{v}+e\textbf{A}$ mean? Is it just momentum accounting for electromagnetic effects?
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100 views

Analytical solution of Liouville's equation for classic harmonic oscillator - which book?

So the past five hours I've spend fruitlessly searching the web for any materials containing the analytical solution of the simple PDE: $$\frac{\partial f}{\partial t} - m\omega^2x\frac{\partial ...
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80 views

Reduced phase space density

I have a dimensional problem with the single particle phase space density The partition function in the microcanonical ensemble is of course dimensionless Thus $$ \rho ( q, p ) = ...
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Equilibrium in Stat Mech and Phase space density

I was wondering if there is any relationship between equilibrium in Stat Mechanics and the phase space density of a system? This does not seem to be completely independent, as Entropy is maximized in ...
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Meaning of phase space density

I am trying to understand Liouville's theorem physically. It says that $\frac{\partial \rho}{\partial t} + \{\rho,H\} = 0$. Thus, we have $\frac{d \rho(q(t),p(t),t)}{dt}=0$. I would like to ...
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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 ...
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Shouldn't we use a Hamiltonian that doesn't give special treatment to time? [duplicate]

If we have a Lagrangian $\mathcal L$ that depends on some scalar field $\phi$, we define the momentum as $\pi \doteqdot {\partial \mathcal L \over \partial \dot \phi}$. The Hamiltonian then is ...
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How to find canonical transformation for given $P_i$ which are constants

We have a 2-D harmonic oscillator with Hamiltonian $H(r,\theta,p_r,p_\theta)=\frac{p_r^2}{2m}+\frac{p_\theta^2}{2mr^2}+\frac{1}{2}kr^2$. I need a canonical transformation to $(Q_1,Q_2,P_1,P_2)$ with ...
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3answers
739 views

Liouville's theorem and conservation of phase space volume

It can be proved that the size of an initial volume element in phase space remain constant in time even for time-dependent Hamiltonians. So I was wondering whether it is still true even when the ...
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1answer
71 views

The Hamiltonian and Energy

if anyone can give assistance on this question it is much appreciated! Suppose I have a Hamiltonian $$H=\frac{p^{2}}{2m}+V(r)+F(r,t) $$ where $$F(r,t) = Q(r)\Re{(e^{{iT(t)}})}:\quad Q(r), T(t),T'(t) ...
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Poisson brackets of the Kepler Problem

For the hamiltonian of a particle of unit mass in a kepler potential: $$H = \frac{1}{2}\mathbf{p} \cdot \mathbf{p} - \frac{\mu}{r}$$ The angular momentum vector is given by: $\mathbf{L} = \mathbf{r} ...
5
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2answers
183 views

Can we explicitly solve the Hamilton–Jacobi equation for a particle in a uniform magnetic field?

HJE for nonrelativistic charged particle in an electromagnetic field is $$\frac{1}{2m}\left(\nabla S - q\mathbf{A}\right)^2 + q\phi + \frac{\partial S}{\partial t} = 0.$$ For a uniform magnetic ...
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377 views

Topology of phase space

Context: From Liouville's integrability theorem we know that: If a system with $n$ degrees of freedom exhibits at least $n$ globally defined integrals of motion (i.e. first integrals), where all ...
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2answers
95 views

Why aren't classical phase space distribution functions always delta functions?

The phase space distribution function (or phase space density) is supposed to be the probability density of finding a particle around a given phase space point. But, classically, through Hamilton's ...
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1answer
403 views

Non-integrability of the 2D double pendulum

Context: For a system with $n$ degrees of freedom (DOF), one has to deal with $2n$ independent coordinates ($2n$ dimensional phase space), of position $q$ and $\dot{q}$ in Lagrangian formulation, ...
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1answer
104 views

Landau's Problem - Poisson bracks of a spherical symmetry function and angular momuntum in z axis

In landau's Mechanics, there's a problem: I think, if the function has the property spherical symmetry, or: $\phi(r,p)=\phi(-r,-p)$ The form suggested by Landau follows this property, but I can't ...
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1answer
183 views

Solving the Liouville equation for classic harmonic oscillator via method of characteristics

I'm interested in solving Liouville's equation $$\frac{\partial W}{\partial t} + \{ H,W\}=0$$ with $$H=\frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$ using the method of characteristics. However I ...
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Sign conventions in Devoret Les Houches course on quantum fluctuations in electrical circuits

In this article on p. 364 Devoret writes the "equations of motion" (using KCL) for the electric circuit shown on p. 363. He uses flux instead of voltages. The sign convention he uses, as shown on p. ...
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129 views

Damped simple harmonic oscillator problem

I'm supposed to calculate and draw the phase space trajectory for this: for the two different cases when and . I've never done this sort of question before, how are they done? I've tried ...
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2answers
322 views

How can I derive the Hamiltonian of simple harmonic oscillator from this Lagrangian?

I'm working through Leonard Susskind's Theoretical Minimum: Classical Mechanics and I can't seem to understand how the Hamiltonian of a simple harmonic oscillator is derived from the following ...