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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13
votes
1answer
672 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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245 views

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 $\...
8
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1answer
363 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 ...
8
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2answers
347 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 ...
7
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1answer
618 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}\...
7
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1answer
468 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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247 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 ...
6
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169 views

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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450 views

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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180 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 ...
4
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1answer
67 views

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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85 views

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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255 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 ...
4
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191 views

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} = \...
4
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654 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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208 views

Some questions about spacetime topology, causality structures and other GR businesses

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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257 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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268 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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52 views

Is there a unique accepted Lagrangian formulation of Nambu mechanics?

In section 5 of their 2000 paper "Nambu Mechanics in the Lagrangian Formalism", Ogawa & Sagae critique previous attempts by Bayen & Flato and by Takhtajan to formulate the theory ...
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46 views

What is the process of finding a good canonical transformation to describe a system? How do I choose the correct generating function?

Supposedly, canonical transformations are used to provide a general procedure to transform a Hamiltonian such that all coordinates in the new frame are cyclic. I have done the proofs and derivations, ...
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25 views

Hamiltonian in gravity with $\Lambda =0$ and using it to generate time translations

Time translation is generated by Hamiltonian. In gravity, the bulk Hamiltonian for closed $d$ hypersurfaces (obtained by ADM decomposition of $d+1$ spacetimes) is 0. This is basically a constraint of ...
3
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1answer
108 views

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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113 views

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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132 views

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 ...
3
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90 views

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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368 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 ...
3
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387 views

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 ...
3
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138 views

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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212 views

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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610 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 ...
3
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2answers
487 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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224 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 ...
3
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1answer
369 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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26 views

Resonant and non-resonant tori density in non-degenerate system

I'm following the discussion on the page 290 of Mathematical Methods of Classical Mechanics by V. I. Arnol'd (you can download it here), and I've encountered the fact that in a nondegenerate system, ...
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52 views

Itzykson Zuber Quantum Field Theory: meaning of integrable system

Here is a part of the book Quantum Field Theory by Itzykson and Zuber: I have two questions: what does the author mean that equation (1-30) form and integrable system, and why? what is the ...
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58 views

Treat stochastically non-Hamiltonian perturbations

Let us consider a classical dynamical system whose obserbvables $A$ evolve according to the following equation of motion \begin{align} \dot A &= -\{H,A\}+f(q) \end{align} $f(q)$ is a non-...
2
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0answers
57 views

Frozen Formalism Problem

Before stating my question, let me say what I do understand: In the ADM formalism, the Hamiltonian density of the gravitational field can be written as, $$\mathcal{H} = h n + H_a N^a$$ where n is the ...
2
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1answer
42 views

Can Jacobi's formulation of Maupertuis' principle be derived in Riemannian geometry?

i want to arrive to hamilton-jacobi equation using the riemannian geometry. So let $\textbf{X}\in \mathfrak{X}(M)$, where $M$ is Riemannian manifold whose metric is $g:\textbf{T}M \times \textbf{T}M \...
2
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0answers
48 views

True Hamiltonian in Geometrodynamics

If I set $N=1$ and $N^a=0$ in the Einstein-Hilbert action $$S[N,N^a,q_{ab}]=\int\sqrt{\mathrm{det}(g)}\,R\,\mathrm d^4x \,\text{,}$$ expressed in terms of ADM variables, then $N$ and $N^a$ are no ...
2
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2answers
44 views

Tight-binding in a semi-infinite square lattice

I have a problem understanding how changing the boundaries from a periodic lattice to a finite lattice. For example, if we have a 2D square lattice of lattice constant $a$ whose $x$ axis has only $N_x$...
2
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1answer
63 views

Contradiction in canonical transformation

The problem I'm supposed to solve is finding $Q$, such that $(p,q)\rightarrow(P,Q)$ is a canonical transformation. In this case $\mathcal{H}=\frac{p^{2}+q^{2}}{2}$ and the new hamiltonian $\mathcal{K}$...
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0answers
32 views

Time-independent Canonical transformation conditions

A solution manual to Goldstein's $9.1$ states that for an explicit time-independent transformation (for a system with two degrees of freedom) to be canonical it must satisfy $$\frac{\partial Q}{\...
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0answers
51 views

Are electric and magnetic fields canonical conjugates?

When quantizing the electromagnetic fields in the context of quantum optics and quantum field theory, we often go for the vector potential $\mathbf{A}$ and its canonical momentum, which turns out to ...
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35 views

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.
2
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1answer
93 views

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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35 views

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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0answers
64 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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68 views

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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72 views

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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