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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Canonical commutation relations in Light-cone gauge

It seems that when trying to identify the physical degrees of freedom for the string some authors$^1$ use: $$ q^-=\frac{1}{\ell}\int_0^{\ell} X^-(\tau,\sigma)d\sigma$$ Then, the commutation relation ...
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How does the Hamiltonian change when going to a moving frame?

The Hamiltonian of a free particle in a rotating frame is given by $$ H = H_0 - \omega \cdot J, $$ where $H_0$ is the Hamiltonian in the non-rotating frame, $\omega$ is the angular velocity of the ...
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Poisson brackets in curved spacetime

The time evolution of any field $\phi$ is given in terms of the Poisson bracket with the Hamiltonian, $$ \frac{\partial\phi}{\partial t} = \{\phi, H\}. $$ How does this relation change in curved ...
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Is Liouville's theorem valid for dimensionally restricted systems?

Liouville's theorem states that the phase space volume of a system is conserved over time. Intuitively, this seems to imply that if a system is at some time constrained to, say, a curve in phase ...
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Poisson brackets for constrained system

Let's have some Hamiltonian which involves the set of first class constraints $\varphi$ and set of constraints $\kappa $, which play role of canonical conjugated momentums for $\varphi$,. They're ...
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Why the involution condition is imposed in the definition of integrability?

For an $N$-degree-of-freedom system to be integrable, the usual definition imposes the existence of $N$ independent conserved quantities, which must be in involution to each other, i.e., $$\{ F_i, ...
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What makes an abstract physical system describable by a “fluid” equations of motion?

We can describe (some of) the dynamics of many systems using fluid mechanics. Of course these include classical fluids like water, more exotic fluids like photon gases and the universe as a whole and ...
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Primary constraints for Hamiltonian field theories

I am currently trying to carry out the construction of the generalised Hamiltonian, constraints and constraint algebra, etc for a particular field theory following the procedure in Dirac's "Lectures ...
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Hamiltonian from Euclidean lagrangian?

Can somebody help me in deriving the Hamiltonian of system starting from Euclidean Lagrangian? Say we are given the Minkowski Lagrangian $$L_m = \frac{\dot{\phi}^2}{2} - V(\phi).$$ The Hamiltonian ...
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366 views

Why is $\{Q, P\} = 1$ for a canonical transformation?

Why is $\{Q, P\} = 1$ for a canonical transformation? Given $P(p,q)$ and $Q(p,q)$.
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The relation between Hamiltonian and Energy

I know Hamiltonian can be energy and be a constant of motion if and only if: Lagrangian be time-independent, potential be independent of velocity, coordinate be time independent. Otherwise ...
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Meaning of a canonical transformation “preserving” a differential form?

In Chapter 9 of Arnold's Mathematical Methods of Classical Mechanics, we find the following definition: Let $g$ be a differentiable mapping of the phase space $\mathbb R^{2n}$ to $\mathbb R^{2n}$. ...
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Quantizing first-class constraints for open algebras: can Hermiticity and noncommutativity coexist?

An open algebra for a collection of first-class constraints, $G_a$, $a=1,\cdots, r$, is given by the Poisson bracket $\{ G_a, G_b \} = {f_{ab}}^c[\phi] G_c$ classically, where the structure constants ...
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Equivalence of classical and quantized equation of motion for a free field

Suppose a classical free field $\phi$ has a dynamic given in Poisson bracket form by $\partial_o\phi=\{H, \phi\}$. If we promote this field to an operator field, the dynamic after canonical ...
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Commutation for constraints

Suppose from the Hamiltonian I got the Primary constraints $$(\Phi_m,\Phi)$$ And $\dot \Phi_m$ , $\dot \Phi$ leads to secondary constraints $$(\gamma_m,\gamma)$$ respectively. Now if the commutation ...
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A question regarding particle trajectories in the symplectic manifold formalism

How to solve a free particle on a 2-sphere using symplectic manifold formalism of classical mechanics ? Is there a way to get coriolis effect directly, without going into Newton mechanics? And is ...
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Question on 1st order Lagrangian Derivation in Faddeev-Jackiw Formalism

I'm looking at this reference (sorry it's a postscript file, but I can't find a pdf version on the web. This paper describes a similar procedure). The topic is the Faddeev-Jackiw treatment of ...
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108 views

Classical particle in a box [closed]

I'm trying to work out some of the details for this system. A particle with mass $\mu$, initial velocity $v_0$ at $x_0$ and moving freely between two walls located at $\pm L/2$, with which it bounces ...
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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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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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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 ...
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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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Shouldn't we use a Hamiltonian that doesn't give special treatment to time? [duplicate]

If we have a Lagrangian $\mathcal L$ that depends on some scalar field $\phi$, we define the momentum as $\pi \doteqdot {\partial \mathcal L \over \partial \dot \phi}$. The Hamiltonian then is ...
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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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Proof of the conservation of the energy functional for the Gross-Pitaevskii equation?

From the Gross-Pitaevskii equation \begin{equation}i\hbar\frac{\partial\psi}{\partial t}=\left(-\frac{\hbar^2}{2m}\nabla^2+V+g|\psi|^2\right)\psi\end{equation} using the variational relation ...
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Hamiltonian mechanics and conservation of energy?

Can anyone explain to me Hamiltonian mechanics relation to conservation of energy? I'm not very good at mathematics, and I know it's important into understanding Hamiltonian mechanics. However, can ...
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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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Can you give example of some problems with solutions in each of Newtonian, Lagrangian and Hamiltonian method? [closed]

I am a student from information system and just want to know about classical mechanics. I know Newtonian mechanics from high school and I have read about Lagrangian and Hamiltonian mechanics in ...
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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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Reference request: Classical Mechanics as an Application to Smooth Manifolds [duplicate]

Possible Duplicate: Classical Mechanics for Mathematician Last time I asked a question, but it does not sound specific. I am currently taking graduate topology class (using Lee's ...
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Square of Laplace–Runge–Lenz vector in Hydrogen atom [closed]

I have a problem. I've tried this question, but I don't get the correct expression. Can someone give me some ideas? Thanks! Consider the Hydrogen Atom Hamiltonian: $$ H = (\mathbf p^2/2 ...
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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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Noether's theorem and “translations” of the Hamiltonian function

In a nutshell, Noether's theorem states that for every continuous symmetry a corresponding conserved quantity exists. Now, the Hamiltonian equations of motion (let's talk about a classical system ...
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Help understanding what the Hamiltonian signifies for the action compared with the Euler-Lagrange equations for the Lagrangian?

Consider the Lagrangian for a simple harmonic oscillator \begin{equation} L (x,\dot{x}) = \frac{1}{2}m\dot{x}^2 - \frac{1}{2}kx^2 \end{equation} Obviously we have \begin{align} \frac{\partial ...
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Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), ...
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Find the error: If $L_x$ and $L_y$ are zero, then $L_z$ is conserved

From Goldstein's Classical Mechanics (2nd ed.), problem 38 of chapter 9 basically says the following: It's been shown that the Poisson bracket of two constants of the motion is also a constant of ...
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Phase space Lagrangian?

Reading out of this lecture series we define a phase space Lagrangian $\mathcal L$ to be a function of $4n+1$ variables namely $q,\dot q,p,\dot p,t$. My question is, what space is this function ...
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Deriving the Hamiltonian density for a free scalar field

I'm working through my old notes on QFT (cf. Ref 1) and I'm not quite sure how to approach the derivation of the Hamiltonian density for a free scalar field (question 2.3 on page 19) and the ...
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Cyclic Coordinates in Hamiltonian Mechanics

I was reading up on Hamiltonian Mechanics and came across the following: If a generalized coordinate $q_j$ doesn't explicitly occur in the Hamiltonian, then $p_j$ is a constant of motion ...
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Possible duality between Harmonic oscillator and free particle?

There is some connection between classical non-interacting harmonic oscillator (OH) and the free particle in higher dimensions? I was studying statistical mechanics and I came across the idea that ...
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Quantum field operators in HEP and CMT

For a real scalar field (which is a bosonic field) we have these commutation relations : $$ \left[\phi(x,t),\phi(y,t)\right]=0 \qquad \qquad \left[\phi(x,t),\pi(y,t)\right]=\delta(x-y).\tag{1}$$ But ...
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Classical Hamiltonian involving product of factors whose quantum analogues don't commute

Dirac remarked in his quantum mechanics book: One can usually assume that the Hamiltonian is the same function of the canonical coordinates and momenta in the quantum theory as in the ...
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Hamiltonian Flow Map

I'm reading this article and am struggling with some of the terminology. What is the flow map for a Hamiltonian system? I'm looking for a rigorous definition really! Many thanks in advance.
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Classical Field Theory: Physical meaning of various terms in total Hamiltonian

In a classical field theory problem Lagrangian density is given as $${\cal L}=\frac{1}{2}\dot{\phi }^{2}-\frac{1}{2}\left ( \bigtriangledown \phi \right )^{2}-\frac{1}{2}m^{2}\phi ^{2}\tag{2.6}$$ ...
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Can all symplectic-form preserving canonical transformations generated by generating functions

This question is related to this fascinating post and this post and this post, but more limited in scope in discussing the practical definition canonical transformations. Canonical transformation ...
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Why closed in the definition of a symplectic structure?

Why do we want the 2-form $\omega $ to be closed? What if it is not?