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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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How does a vanishing $[x, p]$ work with the group theoretical definition of $p \propto \frac{\partial}{\partial x}$?

Thought about this while I was looking at some stuff on quantum-classical correspondence and where precisely the difference between quantum and classical comes from. Usually it's said that the key/...
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Which Hamiltonian systems are intrisically linear?

What physical properties has a dynamical system whose equation of motion are linear? When does it exist a change of coordinates which turn the equation of motions in a linear system? My teacher says ...
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Hamiltonian time-independent, partial derivative always zero?

For conceptual simplicity, let's restrict the discussion to systems with a two-dimensional phase space $\mathcal P$ with generalized coordinates $(q,p)$. Hamiltonian is a function that maps a pair ...
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Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics

I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
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Show Solution to Hamilton's Equations are Given by Circular Paths

I am asked to compute Hamilton's equations and check that the solutions to said equations are circular paths, centered at the origin, with angular velocity $\xi \in R^3$. The Hamiltonian is given by $...
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Volume within equi-energetic surface of a classical harmonic oscillator in microcanonical ensemble

$$ V(E) = \int_{H\leq E} d\mu = \int_\Gamma d\mu\, \Theta\bigl(E-H(q,p)\bigr) . $$ To compute the volume within the equinergetic surface in the microcanonical ensamble, we use the formula above, ...
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Why are some symmetries invisible to the configuration space Lagrangian $L(q, \dot q,t)$?

Usually, when people talk about Lagrangians they are talking about a function of configuration space variables $q_i$ and their time derivatives $\dot q_i$. This is a function $L = L(q_i, \dot q_i,t)$. ...
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Liouville's volume theorem in differential forms language

I will cast my question mostly in words. I usually find Liouville's volume theorem cast in two forms: Easy way (without differential forms language): Phase space volume remains preserved under ...
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Does Liouville’s theorem hold for any ensemble?

I’m having some difficulties with my Statistical Mechanics course. In particular, I have some doubts about Liouville’s theorem and the various ensembles. Consider, for instance, the Canonical ...
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The classical limit of QM as a Hamilton-Jacobi equation?

I'am having difficulties to understand the so-called classical limit in quantum mechanics. There is a popular method to transform the Schrödinger equation into two coupled equations that are the ...
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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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What does a symmetry that changes the Lagrangian by a total derivative do to the Hamiltonian $H$?

A tiny symmetry transformation may change the Lagrangian $L$ by a total time derivative of some function $f$. This is a basic fact used in the proof of Noether's theorem. How can we see the effect of ...
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Why only 2 derivatives in classical mechanics? [duplicate]

The title conveys my true question, but for the sake of clarity I will now rephrase it in a more mathematical flavor using the Hamiltonian formalism of classical mechanics and the terminology of ...
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What if we set Hamilton-Jacobi mechanics as an axiom?

We postulate principle of least action then we get Lagrange mechanics, after we can get Hamilton mechanics either from postulate or lagrange mechanics. Then we get HJE. But what if we have HJE as ...
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Hamilton-Jacobi equation and method of solving it [duplicate]

So, this equation we get when we find canonical transformation that makes new hamiltonian=0. There are 4 main transformations: F1(q,Q,t), F2(q,P,t), F3(p,Q,t), F4(p,P,t). On practice and in every book ...
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Why do we use the Lagrangian and Hamiltonian instead of other related functions?

There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to ...
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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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Level set of Hamiltonian are the orbits?

Just a small question : If $x(t)=(p(t),q(t))$, then the position $x(t)$ of a particle is given by $$\dot p=-H_q(x(t))\quad \text{and}\quad \dot q=H_p(x(t)).$$ In particular, if $x$ solve the previous ...
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Different action-angle variables for a 2D harmonic oscillator

Consider a bidimensional harmonic oscillator. Ref. 1 says that, when the frequencies are commensurable, separating the variables in cartesian or polar coordinates leads to different action-angle ...
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Chirikov standard map derivation

This might be a stupid question, but I am having trouble understanding the derivation of Standard map by integrating Hamilton's equation of motion over one period. I am going through this dissertation ...
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393 views

Hamiltonian formulation of general relativity

Why is it not possible to find a Hamiltonian formulation of general relativity as easily as in classical mechanics? There was a remark to this in my lecture but no real explanation as to why this is. ...
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Energy in spherically symmetric space times

In deriving the equations of motion for geodesics in spherically symmetric spacetimes through Hamiltonian formalism, we can find some constants of motion, namely, $E$ and $L$, the energy per unit of ...
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An example of Hamiltonian which fails with canonical quantization

The Groenewold's theorem states that canonical quantization, regarded as a rule to replace $\{A,B\}$ by $\frac{1}{i\hbar}[A,B]$ is inconsistent for some 3rd order polynomials of canonical variables $p$...
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Constrained Hamiltonian systems: spin 1/2 particle

I am trying to apply the Constrained Hamiltonian Systems theory on relativistic particles. For what concerns the scalar particle there is no issue. Indeed, I have the action \begin{equation} S=-m\int ...
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Applications of Hamiltonians proportional to nth order momentum

Has anyone come across any physical applications of a linear theory where Hamiltonians are proportional to an arbitrary order of momentum? This paper https://michaelberryphysics.files.wordpress.com/...
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Hamiltonian description of a system

I know that phase space is the Hamiltonian description of a system, where we deal with position and momentum in equal footing. My question is in this phase space are those position and momentum are ...
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Canonical coordinate

Sorry for my broken English. I'm a physics undergrad and quite poor at math. While reading a mechanics textbook, I've found something I cannot understand. There are coordinates, $(q,p,t)$ $\...
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How to calculate Hamiltonian when Lagrangian has higher order derivatives? [duplicate]

If we have a Lagrangian density $\mathcal{L}$ for a scalar field $\phi$ depending on $\phi$, $\partial _{\mu} \phi$, and $\partial _{\mu} \partial _{\nu} \phi$, what is the Hamiltonian? Additionally, ...
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Classical method VS Hamiltonian method [duplicate]

I'm very confuse with the method using Hamiltonian to derive the equation of the movement. In example I have, it's easier to derive the equation of the movement using classical method (namely 2nd law ...
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Hamilton equations of motion for matter fields coupled to general relativity in ADM formalism

Do you know what are the Hamiltonian formalism analogs of the Klein-Gordon equation and/or the Maxwell equations in general relativity? Showing how these equations of motion for matter in the ...
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Phase space as differential manifold

Generally we "draw" phase space as typical coordinate system, where $q$s and $p$s are treated like perpendicular axes. Why do we then regard phase space as generall differential manifold while it ...
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Why do we care only about canonical transformations?

In Hamiltonian mechanics we search change of coordinates that leaves the Hamilton equation invariant: these are the canonical transformations. My question is: why we want to leave the equations ...
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Gaussian State Spread [closed]

A measurement device which can be represented by a 1D quantum system (with canonical observables $X$ and $P$) 'is prepared in a Gaussian state with spread $s$' $$\vert \psi \rangle = \frac{1}{(\pi^2s^...
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Conserved charge: partial or total derivative?

I want to obtain some clarification on the concept of Noether charge. Given conserved current $J^\mu$ e.g. in free scalar field theory in $(n+1)$ dimensional Minkowski spacetime $M$, i.e. $\partial_\...
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Hamiltonian for relativistic free particle is zero

One possible Lagrangian for a point particle moving in (possibly curved) spacetime is $$L = -m \sqrt{-g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu},$$ where a dot is a derivative with respect to a parameter $\...
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225 views

E.L. Equations in QFT

In QFT, we use the Lagrangian to construct the Hamiltonian, and in the Interaction Picture (with regards to the Free Field Hamiltonian) use the full Hamiltonian to calculate the changes in the field (...
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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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A short question regarding Hamiltonian [duplicate]

Can any one please tell me in what cases the hamiltonian is not Equal to Total energy. My guess, albeit educated, is if the potential is either a function of time explicitly or a function of velocity, ...
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1answer
131 views

Dirac Brackets in General Relativity

I want to calculate Dirac brackets of different phase space variables in gravity. In case of electrodynamics, one does the same using the following steps: Looking at the momenta to find that $\Pi^...
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1answer
148 views

Constraints in general relativity

In this review on inflation, on Pg. 135, Baumann talks about the energy and the momentum constraints for gravity. Are these equations the $G_{00} = T_{00}$ and $G_{0i} = T_{0i}$ components of the ...
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Building temperature into the Hamiltonian

Given a quantum Hamiltonian $H$ (e.g. the quantum Ising Hamiltonian $H= -h\sum_{i}X_i-\sum_{\langle i,j\rangle}Z_iZ_j$): we know that at temperature $T$, the system is in the state: $$\rho(T) = e^{-H/...
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Does the Heisenberg equation for fields and canonical momentums hold as well for the hamiltonian density operator instead of the Hamiltonian operator?

In quantum field theory, with the field $\phi$ and the momentum $\pi$ being operators, their time evolution is governed (in the Heisenberg-picture) by the Heisenberg equation: \begin{align} \dot{\phi}...
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158 views

Hamiltonian formalism and the phase space

In my book, it says that Hamilton's equations of motion are equations of the first order in the time and that they describe the motion of the system in the $2S$-dimensional phase space. Could someone ...
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Poincare return map as area-preserving map

I'm trying to get some intuition into how the Poincare return map is area-preserving (when there are two momenta and two positions). Suppose $H=H(q_1,q_2,p_1,p_2)$, and let's suppose the system is ...
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Coordinates change for separating the Hamiltonian of a quantum system

Are there general methods, tips or tricks for choosing the correct change of coordinates so that the Hamiltonian of a quantum system becomes separable? Referring to Shankar's Principles of Quantum ...
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54 views

Importance of analytic solutions to Hamiltonians

Why is it important to attempt to find an analytic solution for any theoretical model? It usually happens that many of the hamiltonians written to model the system may not usually have exact solutions....
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Why are canonical transformations made in the first place?

Is it to get cyclic coordinates in the Hamilton's equation?
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How can one construct a phase space for the time evolution trajectories in Hamiltonian?

I was wondering if conjugate momenta and position can be the variables of the phase space or not. I have $\frac{\mathrm{d}x_1}{\mathrm{d}t}$, $\frac{\mathrm{d}x_2}{\mathrm{d}t}$, $\frac{\mathrm{d}...
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Discrepancy in Lagrangian to Hamiltonian transformation results? [closed]

I know, $$ L=T-V \;\;\; \; \;\;\; [1]\;\;\; \; \;\;\; ( Lagrangian) $$ $$ H=T+V \;\;\; \; \;\;\;[2] \;\;\; \; \;\;\; (Hamiltonian)$$ and logically, this leads ...
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Diagonalization of a matrix with operators as elements

How to diagonalize a hamiltonian matrix that has differential operators as elements? My matrix looks something like: \begin{bmatrix} A \frac{d^{2}}{{d\theta}^{2}}+ B_{1} & a\cos{(b\theta +c)}\\ a\...