Questions tagged [hamiltonian-formalism]

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 Mechanics, this formalism relies on a "Hamiltonian" instead of a Lagrangian, which differs from the Lagrangian through a Legendre transformation.

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1answer
616 views

Deriving the Poisson bracket relation of the Ashtekar variables

I'm trying to figure out how to calculate the orthogonality of Ashtekar variables with respect to the ADM hypersurface metric and conjugate momentum. $$\{{A_a}^i(x), {E^b}_j(y)\} = 8 \pi \beta \delta^...
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Does a good path integral exist in Loop Quantum Gravity?

The Hamiltonian operator of Loop quantum gravity is a totally constraint system $$H = \int_\Sigma d^3x\ (N\mathcal{H}+N^a V_a+G)$$ Here, $\Sigma$ is a 3-dimensional hypersurface; a slice of ...
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224 views

Is it possible to derive Liouville's Theorem purely from maximum differential entropy?

Typically in physics (at least the way I learned mechanics), this is derived using the multi-dimensional divergence theorem on the $2N$-dimensional phase space i.e. $0=\partial_t \rho + \sum\limits_{...
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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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2answers
288 views

Do unstable equilibria lead to a violation of Liouville's theorem?

Liouville's theorem says that the flow in phase space is like an incompressible fluid. One implication of this is that if two systems start at different points in phase space their phase-space ...
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1answer
546 views

Non-Relativistic Electron Hamiltonian

I have determined a Hamiltonian for an electron using an appropriate Lagrangian of the form $$ L=\frac{1}{2m}(m\overrightarrow{v}+\frac{q}{c}\overrightarrow{A})^2-\frac{q^2}{2mc^2}\overrightarrow{A}\...
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1answer
436 views

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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231 views

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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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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1answer
316 views

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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169 views

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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1answer
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Ehrenfest theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
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Significance of geodesics of a Hamiltonian surface in condensed matter physics?

Many Hamiltonians in 2D quantum systems can be parameterized as a surface (such as the Bloch sphere) by their k-space coordinates. Another example is given by the (kx,ky) points of the Brillouin torus ...
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230 views

Generating function of point transformation

I am asked to show that the generating function corresponding to a point transformation in Lagrangian mechanics can be taken as null. The point transformation consists of $$ Q_i=Q_i(q,t), $$ and ...
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QFT-Style Lagrangian for a system of two symmetrized bosons

I'm wondering if anybody may have suggestions regarding the following problem. The Hamilton operator of the quantum harmonic oscillator (QHO) can be written as follows: $$ \hat{\mathcal{H}}_{QHO} = \...
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Hamiltonian formulation of the semiclassical Model of electrons

I'm currently reading the book Solid State Physics by Neil W. Ashcroft and N. David Mermin. In Chapter 12 they introduce the "Semiclassical Model of Electron Dynamics". In short: After having solved ...
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648 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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190 views

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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238 views

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: $$\pi_n=\frac{\partial\...
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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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1answer
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Choosing initial condition for Hamilton-Jacobi PDE from initial $x$ and $p$

For separable solutions to Hamilton-Jacobi PDE (say in 2D), we treat the Hamilton's principal function $S$ as $$S= W(x) + W(y) - E*t$$ and treat the separate parts as constants and find $W(x)$, $W(y)$...
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Fluid mechanics, symplectic structure, the Hamiltonian, and vorticity

Consider an inviscid irrotational fluid in two dimensions. There are some explicit connections with symplectic geometry that I do not understand. I am not well versed in the later topic, so please ...
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Using action-angle variables in non-periodic system

I'm a little confused by the discussion in the last section $\S 50$ of Landau and Lifshitz's (Classical) Mechanics (1960, first English ed.). Here, they consider finite motion of a system whose ...
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How are action variables linked to first integrals of a Hamiltonian?

Suppose I have an integrable Hamiltonian system $H(q_{1}, p_{1},..., q_{n}, p_{n})$, with first integrals $F_{1} = H, F_{2},..., F_{n}$. Excluding certain singular level sets (i.e. separatrices), one ...
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352 views

Breakdown of the Legendre transform for the complex scalar field

Suppose we wish to obtain the energy density of the free complex scalar field $\varphi$ as a Legendre transform of the corresponding action. From Wikipedia, writing the action of a free complex scalar ...
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Non-canonical transformation

I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
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How do I obtain the SUSY Transformations from Poisson Brackets?

In Friedman's and Van Proyen's Supergravity textbook it is explained how one can get the supersymmetry transformations using the conserved currents. Specifically this is in section 6 where we are ...
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1answer
313 views

How to scale variables in a classical Hamiltonian?

So I looked at some research articles where one has a classical Hamiltonian $H(p,q,t) = p^{2}/2 + V(q,t)$. If one introduces the scaling transformation $$t \mapsto t/\sqrt{s}, \quad H \mapsto Hs, \...
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What are the implications of integrating the Poisson bracket?

Reading Ref. 1 I admit I am a little lost in some places. I was hoping that someone in this area could explain the basic premise of integrating the Lie bracket and further is it connected to nlab's ...
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591 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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2answers
443 views

Spherical phase space dynamics

I have a hamiltonian of the form $$H(\phi,z) = (1-z^2)\cos(2\phi) + \chi z^2$$ with position $\phi$ and conjugate momentum $z$. It has this form provided that $z \in [-1,1]$ and we have natural ...
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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 formulation of classical spin chains

Is there a way to construct a Lagrangian formulation of the classical dynamics of a spin chain - say a Heisenberg or XY chain? The Hamiltonians here are obvious.
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1answer
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How to find the canonical transformation if given the new Hamiltonian we want?

If we know the new Hamiltonian $H^{\prime}$ we want, and $H^{\prime} \neq 0$, how can we find the canonical transformations? For example, we want to transform the $$H(p,q)=\frac{p^2}{2m}+\frac{1}{2}...
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The radiative modes of an asymptotically flat spacetime and the symmetries

In Ashtekar's paper, the radiative degrees of freedom of an asymptotic flat spacetime in general relativity are obtained. These degrees of freedom live on the null infinity $\mathscr I$, given by the ...
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57 views

Canonoid transformation that is NOT canonical

Canonoid transformations are defined here as preserving the hamiltonian structure of the dynamical system for a particular hamiltonian, but not necessariy for every hamiltonian, in such a way that a ...
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Reference request: Scattering from action

Consider a separable solution to a Hamilton-Jacobi equation of an $n$-dimensional autonomous system of the form $$W(\alpha_1,...,\alpha_n,x^1,...,x^n) = \sum_j \int_0^{x^j} w^j(\alpha_1,...,\alpha_n,x'...
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Are first Integrals the same in Lagrangian and Hamiltonian formalism?

A first integral in Lagrangian formalism is defined as a function which is constant along the solutions $(q,\dot{q})$ where $q$ are the generalized coordinates; while a first integral in Hamiltonian ...
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Angular momentum of a cylindrical system in general relativity

The definition of energy and enegry flux of a cylindrical symmetric system in general relativity is given by Kip Throne in Phys. Rev. 138, B251 and generalized by Chandrasekhar in Proc. Roy. Soc. Lond....
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Symplectic Manifolds in General Relativity for Integrable Systems

To solve the geodesic equations for a specific metric in General Relativity I can find conserved quantities $F = \xi_{\mu}\frac{\mathrm{d}x^{\mu}}{\mathrm{d}\lambda}$ along geodesics by using Killing ...
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Hamiltonian directly expressed in $(q,\dot{q})$ : how to find what is $p$?

I am reading a book about non relativistic quantization of E.M field. But first we do classical field theory. We directly wrote the Hamiltonian of our study, and a part of our Hamiltonian is the ...
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Is there a physical connection between Lagrangian mechanics and Hamilton mechanics

I know that the Hamilton's equations results from the Legendre transformation of the Lagrange equation. It's also well-known that the Hamiltonian is equal to the energy of a system if the system is ...
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Looking for lecture videos that follow Arnold's Mathematical Methods of Classical Mechanics

I'm an undergrad and I'm looking for lecture videos (on youtube and such) that follow this textbook. My course roughly follows it, but glosses over some mathematical details that I feel would be ...
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1answer
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Classical mechanics in coadjoint orbits

We know that coadjoint orbits are symplectic manifolds, and they can be used to find unitary representations of Lie groups and stuff, and it's also related to quantization. However, is it true that ...
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Getting $h_x, h_y, h_z$ Components of Hamiltonian after Gauge Transformation

In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $ H(k)= $ \begin{bmatrix} h_0+h_z & h_x-i h_y \\ ...
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Deriving Hamilton's equations independently

The usual way to derive Hamilton's equations is to perform Legendre transformation of the Lagrangian and then use the stationarity principle. However, this procedure seems a little artificial to me ...
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1answer
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Quantizing first class constraints

Let $\gamma$ denote a first class constraint. Then if there exists a function on phase space $f(q,p)$ for which the Poisson bracket with the constraint does not vanish $\lbrace f, \gamma\rbrace \neq 0$...
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Lapse and shift inside or outside the Poisson bracket?

For general relativity in the 3+1 ADM formulation, one has $H=\int dx [N{\cal H}+N^a{\cal H}_a]$ with $N$ and $N^a$ the lapse and shift which are undetermined Lagrange multipliers. The dynamical ...
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Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian

I want to show that the gauge transformation $$L(q,\dot{q},t)\mapsto L^\prime(q,\dot{q},t):=L(q,\dot{q},t)+\frac{d}{dt}f(q, t)$$ corresponds to a canonical transformation of the Hamiltonian $H(p, q, ...
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How are the endpoints of motion determined?

The Hamilton's principle tells us which path is taken by the system in going from one point of the configuration space at time t1 to another point of the configuration space at time t2. My question is ...

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