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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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What happens to the time evolution equations in canonical quantum gravity?

Many expositions on canonical quantum gravity start from a 3+1 type formalism, where spacetime is foliated along the time dimension. The Einstein equations then decompose into constraint equations on ...
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Do canonical transformations form a group?

In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical ...
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Liouville's integrability theorem: action-angle variables

For classical dynamical systems, let $I_{\alpha}$ stand for independent constants of motion which commute with each other. Pg 323 of Jose-Saletan and also 'Remark 11.12' on pg 443 of Fasano-Marmi's '...
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A paradox about canonical transform preserving Poisson bracket?

Let $q,p$ denote the position and momentum. Consider a transform generated by $g$: $q' = q + \epsilon \{q,g\}---(1a)$ $p' = p + \epsilon \{p,g\}---(1b)$ Then: $\{q',p'\} = \{q,p\}+o(\epsilon^2)+\...
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ADM formulation of GR derivative on the 3-metric

In the ADM formalism where the projector is given by ${P^\mu}_\alpha={\delta^\mu}_\alpha+n^\mu{n}_\alpha$ and $n^\alpha$ is a future pointing normal vector to the constant time hypersurface $\Sigma$. ...
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Derivation of Hamilton-Jacobi equation

I am trying my own way of deriving the Hamilton Jacobi equation $$\frac{\partial S}{\partial t} = -H \tag{1}$$ through direct variation. I think the difficulty of doing this is that the upper limit ...
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Hubbard Hamiltonian mean field decomposition in real and momentum space

Im working through some lecture notes on quantum field theory, and it gives the mean field hamiltonian in real space in terms of the spin operator: $$H_{int}^{MF}=\frac{3}{8U}\sum_{r_j}\vec{(M}(r_j))^...
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Itzykson & Zuber: Conjugate momentum sign

I can't give myself peace on a confusion about the signs. I'm studying on Claude Itzykson & Jean-Bernard Zuber, Quantum Field Theory, Dover Publications. Metric convention $g_{\mu\nu}=diag(1,-1,-1,...
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Determining Whether a Given Hamiltonian is Conserved

in simple terms I'm looking to understand how we can tell whether a given Hamiltonian (or one that we've deduced) is conserved or not? I've tried looking at other similar questions but am not sure I ...
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The WKB approximation and the Cotangent bundle

When we say (see pag. 9 of Lectures on the Geometry of Quantization) that the image of the differential of the phase function lies in the level set of the classical Hamiltonian is it simply ...
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External force as defined in Landau's statistical mechanics

I was reading Landau's volume 5 on statistical mechanics and there in 5th Russian edition page 62 he mentions the following (translated very roughly) that assume we have Hamiltonian system which ...
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Can we do better than the Hamiltonian formalism? [closed]

I have been reading Sakurai's Modern Quantum Mechanics and I notice that every proposition that book has depends primarily on the Hamiltonian formalism of classical mechanics. Even the time evolution ...
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A mathematically illogical argument in the derivation of Hamilton's equation in Goldstein

In the book of Goldstein, at page 337, while deriving the Hamilton's equations (canonical equations), he argues that The canonical momentum was defined in Eq. (2.44) as $p_i = \partial L / \partial ...
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Theta-dependence of massive Schwinger model

I've read in Coleman's paper on the massive Schwinger model (and in other papers on the same topic, like this one) that the model's Hamiltonian contains a topological $\theta$-term. However, if I ...
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Hamiltonian Form Under Canonical Transformation

Let us consider a restricted canonical transformation such as $$ Q_{i} = Q_{i}(q,p) $$ $$ P_{i} = P_{i}(q,p) $$ Goldstein states that 'the Hamiltonian function does not change in such a ...
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Action variable integral

I am solving an action angle variable problem and I'm stuck at the point where I have the following expression for the integral $$ I = \frac{b\sqrt{mE}}{\pi} \int_\theta^{2\pi-\theta}\sqrt{(1-cos \...
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Finding period from action-angle variable in one dimensional potential [closed]

I want to calculate the period from the action-angle variable for a particle in a one dimensional potential $V = V_0 \tan^2(q \pi/2a)$. After doing some algebra I get $$I = \frac{\sqrt{2mE}}{2\pi} \...
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Why are 2 dimensions needed for every 1 dimension of space in order to determine the motion of a physical system?

In classical mechanics, the phase space of a mechanical system has twice the number of dimensions of "actual" space (i.e. position space). That is, in phase space, each particle has both a position ...
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Partial Legendre transform: understanding a simple example

Consider the following function: $$f(x_1, x_2) = x_1^2x_2+x_1x_2^3$$ $f$ is a function of $(x_1, x_2)$. The conjugate variables $(u_1, u_2)$ to $(x_1, x_2)$ are $$u_1 = \partial f/ \partial x_1 = 2 ...
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“Hidden” theta-term in Hamiltonian formulation of Yang-Mills theory

I've read in David Tong's lecture notes on gauge theory that the Hamiltonian of Yang-Mills theory does not depend on the angular parameter $\theta$, because it can be absorbed in the electric field: $...
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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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Connection between Classical and Quantum symmetries

I am an advanced undergraduate student.I would like to know about the importance of symmetry in classical and quantum mechanics.Also a good book concerning the connection between symmetries of ...
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Non-quadratic kinetic energy [closed]

Do you have examples of Lagrangians/Hamiltonians used in physics with non-quadratic kinetic terms? e.g. $\dot{x}^4$ What is the origin and the interpretation of such terms?
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Variation in Hamiltonian mechanics

I have a question about a property of variational calculus used in following bachelor thesis: http://users.physik.fu-berlin.de/~pelster/Bachelor/fraessdorf.pdf Here the excerpt: Why it is possible ...
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How are Dunkl operators used in Hamiltonian mechanics?

I am currently doing a math research project on the representation theory of Cherednik (double affine Hecke) algebras, specifically the algebra $\mathcal{H}_{t,c}(\mathfrak{S}_n,\mathfrak{h})$, which ...
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Hamiltonian with one constant of motion (besides the Hamiltonian itself)

The background of my question is a well known fact: a Hamiltonian system with $n$ degrees of freedom with $n$ constants of motion is integrable. My question is about the case in which there are only ...
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Reference for canonical quantization of E.M field (for quantum optics purposes in the end)

I am looking for a nice book or online course treating in a rigorous way the quantization of the E.M field. My needs in the end are to do quantum optics, but I would like to see the proof in a nice ...
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Generating function depending on $q$, $p$, $Q$ and $P$

If I have a generating function, say, $$G(q,p,P,Q)= qp - e^Q e^P\tag{1}$$ what are the equations that give me the transformations $Q=Q(p,q)$ and $P=P(q,p)$? I have only seen generating functions ...
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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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What does “substantial changes” in material properties mean in geometric optics?

We know that (I read it from Kip Throne's Modern Classic Physics) if a wave's wavelength is smaller than the length scale over which "substantial changes" of material properties occur, then the wave ...
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Why is the generating function in the Hamilton–Jacobi equation equal to the action? [duplicate]

The aim in Hamilton Jacobi formalism is to find a canonical transformation that generates a new Hamiltonian $H'$ which is equal to $0$. Therefor we find the equation: $$H(q_1,...,q_n,\frac{\partial F}{...
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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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What is the link between the rotating wave approximation and the algebraic representation of a dynamical system?

In analyzing a system of two coupled oscillators, I noticed a rather interesting correspondence between the so-called "rotating wave approximation" (RWA) for solving differential equations and the ...
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Name of the matrix that appears in matrix form of Hamilton's equations of motion

Consider a harmonic oscillator described by the second order differential equation $$\ddot{\phi} + \omega_0^2 \phi = 0 \, .$$ Defining $v \equiv \dot \phi$ we get two simultaneous equations \begin{...
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Can we also Legendre transform the Lagrangian using $\frac{\partial L}{\partial q}$ instead of $\frac{\partial L}{\partial \dot q}$? [duplicate]

We calculate the Hamiltonian as the Legendre transform of the Lagrangian $$H(q,p,t) = p \dot q - L(q,\dot q,t), $$ where $p$ is the slope function $$p \equiv \frac{\partial L}{\partial \dot q} .$$ ...
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Non-relativistic limit of Hamiltonian for a free particle in general relativity

The Hamiltonian for a particle moving in a gravitational field can be taken as $$\mathcal{H} = \frac12 \sum_{\mu,\nu=0}^3g^{\mu\nu}(x)p_\mu p_\nu\tag{1}$$ as long as the parametrization is affine. ...
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Solving time evolution equations in Hamiltonian formalism

I have 4 time evolution equations and the Hamiltonian $H(X_{1},X_{2},P_{1},P_{2})$ that generates the time evolution depends on 4 canonical coordinates but I would like to solve the differential ...
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Intuition between this construction of the sympletic form for classical fields

In this paper, Wald presents a quite general construction of a sympletic form for classical fields. If I understood (which I might have not, and in that case corrections are highly appreciated), the ...
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Integration of Poisson brackets by integration by parts [closed]

In the context of Statistical Mechanics I have to show that the following integral is zero: $$\int \sum_{i=1}^{3N}(\frac{\partial O}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial O}{\...
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What does conservation of probability mean in Classical Mechanics and why is it true?

In the context of the Liouville equation, regularly the conservation of probability is invoked. (Of course, the overall probability is always conserved but this is a truism and not what is meant here. ...
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What does $\frac{d\rho}{dt} =0$ but $\frac{\partial \rho}{\partial t} \neq 0 $ mean intuitively?

For concreteness, let's say that $\rho(q,p,t)$ describes the probability density in phase space. On a superficial level both $\frac{d\rho}{dt}$ and $\frac{\partial \rho}{\partial t} $ tells us how $\...
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Determinant of ADM metric

I am studying inflation and for the calculation of the bispectrum we are using the ADM formalism where the metric is the following form: $$g_{\mu\nu}=\begin{bmatrix}-N^2+N^iN_i&N_i\\N_i&h_{ij}...
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How does the Hamiltonian change if $L\to L + \frac{dF}{dt}$? [duplicate]

The Hamiltonian is defined as the Legendre transform of the Lagrangian $$H = p\dot{q} -L .$$ In the Lagrangian formalism we are free to add the total derivative of an arbitrary function $F=F(q,t)$ to ...
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What is the “special time dependence” that develops in an Ostrogradskian instability?

I've been reading papers that deal with Lagrangians containing second- and higher- order derivatives of field variables. In this paper in Section 3.1, I found this very interesting quote: The ...
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Derivative of $\nabla\times(\nabla\times A)$ by A

I'm trying to find out how to quantize EM field. It seems like $\vec{A}$ and $\vec{E}$ are it's canonical coordinates. For example: $$\mathfrak{H} = \frac12E^2 + \frac12(\nabla\times A)^2$$ $$H = \int ...
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Why is a single function sufficient to specify a canonical transformation?

Spivak argues at page 577 in his book Physics for Mathematicians: What are the $2n$ relations he is talking about?
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Are all canonical transformations either a point transformation, gauge transformation or a combination of them?

It's regularly argued that in the Hamiltonian formalism, we have more freedom to choose our coordinates and that this is arguably its most important advantage. To quote from two popular textbooks: ...
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Can all canonical transformations be generated using a generating function?

In Classical Mechanics, a gauge transformation is of the form \begin{equation} L \to L' = L + \frac{dF(q,t)}{dt} \, . \end{equation} Any transformation of this kind leaves the Euler-Lagrange equation ...
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Are point transformations necessarily canonical?

A point transformation $ Q(q,t)$ only modifies the location coordinates, while a canonical transformation, in general, also modifies the momentum coordinates $Q(q,p,t)$. In the context of the ...
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Deriving Canonical Transformation from Generating Function using Principle of Stationary Action

In Hamill's "A Student's Guide to Lagrangians and Hamiltonians", section 5.2, the equations for a canonical transformation $(q,p) \to (Q,P)$, induced by the generating function $F(q,Q,t)$ are derived ...