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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Defining the Hamiltonian and obtaining equations of motion for physical system [on hold]

I have a relative simple mechanical system, that is modeled with 3 point masses and 3 links. There is an input force F working in the vertical direction and 2 torques $\tau_1$ and $\tau_2$. All ...
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Proof of the conservation of the energy functional for the Gross-Pitaevskii equation?

From the Gross-Pitaevskii equation \begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\nabla^2+V+g|\psi|^2\right)\psi\end{equation} using the variational relation ...
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Is general relativity holonomic?

Is it meaningful to ask whether general relativity is holonomic or nonholonomic, and if so, which is it? If not, then does the question become meaningful if, rather than the full dynamics of the ...
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Finding conserved quantities from Hamiltonian when Symmetry is not evident [on hold]

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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Convert this to a Hamiltonian system and find Hamiltonian [on hold]

I tried solving this by using the conjugate momentum and position formulae: but could not get the answer. EDIT: I have used the same procedure as mentioned in this Phys.SE answer but the ...
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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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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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Derivation of (2.45) in Peskin and Schroeder

I'm having trouble understanding the step $$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int ...
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Classical mechanics textbook recommendation [duplicate]

I've just finished my first year of physics study and I'd like to learn some more classical mechanics. What textbook would you recommend as an introduction to Lagrangian and Hamiltonian mechanics? ...
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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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Hamiltonian reduction having constant of the motion

I have this $2^n*2^n$ matrix that represent the evolution of a system of $n$ spin. I know that I can have only one excited spin in my configuration a time. (eg: 0110 nor 0101 ar not permitted, but ...
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238 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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328 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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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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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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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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Understanding Poisson brackets

In quantum mechanics, when two observables commute, it implies that the two can be measured simultaneously without perturbing each other's measurement results. Or in other words, the uncertainty in ...
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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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199 views

Hamiltonian Noether's theorem in classical mechanics

How does one think about, and apply, Noether's theorem in the classical mechanical Hamiltonian formalism? From the Lagrangian perspective, Noether's theorem (in 1-D) states that the quantity ...
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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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About constraints of the first class and electrodynamics

Let's have some theory in hamilton formalism and let's assume that it has the constraints between canonical variables $Q, \pi$. By the Dirac terminology, the set of constraints $F_{a}(Q, \pi) \approx ...
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Quantum Liouville condition?

I understand there is the quantum-Liouville equation, namely the Von-Neumann equation, but does Liouville's theorem apply to the Wigner distribution too? What about the Moyal bracket, is that a ...
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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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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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Proof of Liouville's theorem: Relation between phase space volume and probability distribution function

I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of ...
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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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Dimension agreement in canonical transformation

In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical ...
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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} ...
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Is there useful information about normal modes/frequencies in the Hamiltonian matrix?

Suppose I have two $LC$ oscillators, one with $L_1$ and $C_1$, and the other with $L_2$ and $C_2$. If uncoupled, each oscillator has resonant frequency $\omega \equiv 1/\sqrt{LC}$. Using the flux in ...
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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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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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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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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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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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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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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
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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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1answer
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Poisson brackets for constrained system

Let's have some Hamiltonian which involves the set of first class constraints $\varphi$ and set of constraints $\kappa $, which play role of canonical conjugated momentums for $\varphi$,. They're ...
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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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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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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 ...
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A kind of Noether's theorem for the Hamiltonian

How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law using ...
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When do phase space functions' Poisson brackets inherit the Lie algebra structure of a symmetry?

I've seen several examples of phase space functions whose Poisson brackets (or Dirac brackets) have the same algebra as the Lie algebra of some symmetry. For example, for plain old particle motion in ...