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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About turbulence modeling

I have some questions about this paper: Lagrangian/Hamiltonian formalism for description of Navier-Stokes fluids. R. J. Becker. Phys. Rev. Lett. 58 no. 14 (1987), pp. 1419-1422. After reading ...
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Is Feynman talking about the Zeroth Law of Thermodynamics?

In Volume 1 Chapter 39 of the Feynman Lectures on Physics, Feynman derives the ideal gas law from Newton's laws of motion. But then on page 41-1, he puts a caveat to the derivation he has just ...
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Does the inverse of the Dirac conjecture hold?

In the theory of constrained Hamiltonian systems, one differentiates between primary and secondary constraints, where the former are constraints derived directly from the Hamiltonian in question and ...
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Hamiltonian function for classical hard-sphere elastic collision

I'm trying to find the Hamiltonian function for a system consisting of a single particle in one dimension colliding elastically with a wall at $x = 0$. Everything I've read on the topic (e.g. this ...
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The consistency conditions of constrained Hamiltonian systems

I am studying the Hamiltonian description of a constrained system. There are some questions puzzled me for days, which I have been stuck on it. From the lagrangian, we can obtain the primary ...
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How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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What exactly is the relationship between the symplectic 2-form and the frequency of leaves of integrable systems in classical mechanics?

In classical mechanics we equip a differential manifold with a closed symplectic 2-form $\omega$. The symplectic leaves of integrable systems also have a unique frequency, in literature denoted ...
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CM: Need to recover the Hamiltonian, knowing conserved quantities and information about the EOM, possibly via action-angle coordinates

Statement of the problem: I have a system with 2 degrees of freedom and I have found two independent conserved quantities, without knowledge of the Hamiltonian. I'm looking for a method to recover a ...
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Show that the Laplace-Runge-Lenz vector is conserved using poisson brackets

(I realise similar Phys.SE questions already exist but there is no answer with the Poisson bracket notation, I'll take this down if someone lets me know I should have commented in the existing ...
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Some questions about spacetime topology, causality structures and other GR businesses

1) What are the exact conditions required for the canonical transformation? Most papers just assume away with global hyperbolicity, but is there a more general condition for it? "Quantum gravity in ...
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207 views

The Hamiltonian for clocks?

I am rather a theoretician and looking for a formalism to represent biological clocks by Hermitian operators. The simplest thought model I am looking for is a formal representation of a clock (for ...
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96 views

What variable is the conjugate momentum for angular momentum?

From the definition of conjugate momentum for a generalized coordinate we get that the conjugate for angular momentum should be proportonal to its integral with respect to time. According to my ...
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101 views

Hamiltonian Operator Interpretation of Quantum Anomaly

We can see the definition of quantum anomaly in terms of Lagrangian path integral formulation. What is the definition of quantum anomaly in terms of Hamiltonian operator approach or even more directly ...
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Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: ...
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Do primary first class constraints change the electric field in the Hamiltonian form of Maxwell's theory?

In my understanding of Dirac's theory of constrained Hamiltonians, the primary (and also the secondary) first class constraints are generators of canonical transformations that do not change the ...
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Quasilocal stress tensor

I have been reading through the paper hep-th/9902121 and have a few questions about the first five lines of the introduction: 1) "In a generally covariant theory, it is unnatural to assign a local ...
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Foliation of the phase space

Consider an arbitrary classical Hamiltonian system. Given an initial state $(p_0, q_0)$, we can get a solution of the equation of motion, a curve in the phase space. Now the problem is, for a generic ...
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Hamiltonian linearly proportional to momentum

In this question, it is discussed why, in Lagrangians we usually stick to first derivatives and quadratic terms we never see higher derivatives. The selected answer shows that, if a Lagrangian $L(q, ...
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187 views

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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Why can any pair of master coordinates be used to calculate a nonlinear mode of a nonlinear dynamical system?

This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied. ...
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Interesting Hamiltonian System

The definition of a Hamiltonian system I am working with is a triple $(X,\omega, H)$ where $(X,\omega)$ is a symplectic manifold and $H\in C^\infty(X)$ is the Hamiltonian function. I am wondering if ...
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About the derivation of the Hamilton-Jacobi equation

It is an old question for me. In Goldstein's book, the H-J equation is derived in this way. We want to find a generating function $F(q,P,t)$ such that the transformed Hamiltonian vanishes identically, ...
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Interpretation of Poincare Map

I have been trying to interpret a Poincare Map. The Hamiltonian for the system is $$H=\frac{1}{4m}\left(p_r^2+p_z^2\right)+m\omega_\perp^2 r^2 +m\omega_z^2 z ^2+ ...
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Trying to solve 2D Toda Lattice Equation with Lax Pair Approach

I am working on this Hamiltonian: $$ H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch ...
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249 views

Optical Raytracing by using Adiabatic Hamiltonian Method

I'm looking into raytracing a Lüneburg Lens which is a gradient index (GRIN) optical element with a radially varying refractive index: $$ n(\rho)=n_0\sqrt{2-\left(\frac{\rho}{R}\right)^2}, ...
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Calculation of the non-Gaussity parameter for primordial cosmological perturbations by the ADM Formalism

Maldacena has used the ADM Formalism in one of his papers (http://arxiv.org/abs/astro-ph/0210603) in computing the the three point correlation function (i.e the non-Gaussianity) parameter for ...
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Recovering a symmetry transformation from a conserved charge

I'm going through some notes on how to apply the Hamiltonian formalism to systems with gauge invariance and I found a derivation of Noether's theorem I had never seen before. The idea is roughly that ...
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Is time-1 map of a Hamiltonian vector field on a cylinder always twist?

I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed ...
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Nabla Terms in the Energy Density of the Lagrangian for the Massive Spin 1 Field (Schwartz QFT 1st Ed. Eqn. 8.19)

The relevant part starts with a Lagrangian guess of, $$\mathcal{L}=-\frac{1}{2}\partial_{\nu}A_{\mu}\partial_{\nu}A_{\mu}+\frac{1}{2}m^2A_{\mu}^2$$ where the EOM's are, $$(\Box+m^2)A_{\mu}=0$$ The ...
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Canonical Momentum Conjugate vs. Momentum

I stumbled upon this while reading about Legendre Transforms today. So consider an n-particle system. The Lagrangian is a function of $ q_i$'s and $\dot q_i$'s. If you consider the manifold $M$ where ...
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Finding canonical transformation using type 3 generating function

Question: For a system with one degree of freedom, a canonical transformation $Q(q,p), P(q,p)$ obtained by a type 3 generating function satisfies $Q = e^t q^{1/2}\cos p$. Find the most general form ...
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Any good reference on Maslov index (or Morse index)?

Any good reference on Maslov index (or Morse index)? I have some basic knowledge of differential geometry, calculus of variation. So is there any good reference for me?
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Taking a 'relative' limit

I am looking at Hamiltonians for specific physical situations. I have taken a given Hamiltonian $\vec{H}(\vec{p}, \vec{x})$ and have found the following Hamiltonian equations: $$\frac{d\vec{x}}{dt} = ...
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Hamiltonian flow?

I was wondering what the Hamiltonian flow actually is? Here is my idea, I just wanted to know if I am correct about this. So let $(x(t),p(t))' = X_{H}(x(t),p(t))$ are the Hamilton's equations and ...
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What is the correct terminology for a “symplectic covariant” equation?

A Lorentz covariant equation is one that takes the same form even when a Lorentz transformation is applied to each variable. Lorentz covariance is generally made manifest by writing the equation with ...
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Advantages of having a first class system and possibility of transforming a system into a first class one

I have two questions regarding first class systems. What are the advantages of having a first class Hamiltonian (a Hamiltonian whose all constraints are first class) in a theory or having a first ...
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When to use Hamiltonian vs Lagrangian?

I currently studying the Lagrangian and Hamiltonian formalisms in classical mechanics, but something I'm not seeing is how do I know which one to use in a given problem? After I find the Lagrangian, ...
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127 views

Legendre Transformation for multiple variables

I need to show that for $F(x_1, .., x_n)$, the Legendre transformation is, $$G(s_1, ..., s_n) = \sum_{i}^{N} x_i s_i - F$$ where $$s_i = \frac{\partial F}{\partial x_i}$$ and has the property that ...
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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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41 views

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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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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170 views

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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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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Showing the Hamiltonian of the $\alpha$ FPU is real

I am studying the $\alpha$ FPU chain which is a model of coupled oscillators with small non-linearity. For these systems, I derived the following Hamiltonian $H$ which is given by $$ H=\sum_{j=1}^{N} ...
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79 views

Why are Lagrangian subspaces called 'Lagrangian'?

I am wondering what the special role of Lagrangian subspaces (or submanifolds) are in mechanics. Do these subspaces have some sort of special property for which we have some sort of `Lagrangian ...
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What's the value of the coupling constant in interacting field theories?

Consider this Lagrangian : $L = \frac{1}{2}(\partial_\mu \Phi)^2 - \frac{M^2}{2}\Phi^2 +\frac{1}{2}(\partial_\mu \phi)^2 -\frac{m^2}{2} \phi^2 -\mu\Phi\phi^2$ Its interaction term is given by : ...
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name for 2D Electrostatics as Integrable System

I am trying to understand 2D electrostatics of $n$ point charges. Roughly, $$ H = \sum_{i=1}^N n_i \ln |z- z_i|$$ However, I keep bumping across the Gaudin model instead with this Hamiltonian $$ ...
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Why don't we experience dilatations in Minkowski spacetime?

The question This question follows on from the use of projective coords for spacetime in Notation for Translation Group Generators . Under Felix Klein's Erlangen Program, Minkowski spacetime starts ...
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Are there any hamiltonian systems without a periodic orbit?

Are there any hamiltonian systems without a periodic orbit? Can anyone give me an example? If such a system exists, does this fact have any implication on its quantum version?
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Hamiltonian flow is volume preserving

I was reading about advantage of Hamiltonian over Lagrangian. One of the advantage is "Hamiltonian flow is volume preserving". Can anyone help me to understand this? Means what is advantage of ...