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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Is Hamilton-Jacobi equation valid for only conserved systems?

From derivation of Hamilton-Jacobi (HJ) equation one can see that it is only applicable for conserved systems, but from some books and Wikipedia one reads the HJ equation as ...
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Does a constant of motion always imply a Hamiltonian formulation?

If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
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Are Poisson brackets of second-class constraints independent of the canonical coordinates?

Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\{\chi_N(q,p),\chi_M(q,p)\}_P$ . Is ...
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Eikonal approximation for wave optics. Why follow the unit vector parallel to the Pointing vector?

The description of the passage from wave optics to geometrical optics claims that light rays are the integral curves of a certain vector field (the Pointing vector direction, normalized to 1). Here ...
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Find generating function $F_1$ for canonical trasformation

I'd like to know the steps to follow to find the generating function $F_1(q,Q)$ given a canonical transformation. For example, considering the transformation $$q=Q^{1/2}e^{-P}$$ $$p=Q^{1/2}e^P$$ ...
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Hamiltonian constraint in spherical Friedmann cosmology

I'm taking a GR course, in which the instructor discussed the 'Hamiltonian constraint' of spherical Friedmann cosmology action. I'm not quite clear about the definition of 'Hamiltonian constraint' ...
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Solution of motion in hamiltonian formalism

I have these canonical equations: $$\dot p = - \alpha pq$$ $$ \dot q =\frac{1}{2} \alpha q^2$$ I have to find $q(t)$ and p$(t)$, considering initial conditions $p_0$ and $q_0$. I thought to simply ...
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Find the Hamiltonian given $\dot p$ and $\dot q$

I have these equations: $$\dot p=ap+bq,$$ $$\dot q=cp+dq,$$ and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$. To answer to the first ...
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Factors of $c$ in the Hamiltonian for a charged particle in electromagnetic field

I've been looking for the Hamiltonian of a charged particle in an electromagnetic field, and I've found two slightly different expressions, which are as follows: $$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 ...
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Hamilton's equations for a simple pendulum

I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with $$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$ Now Hamilton's equations will be: $$\dot ...
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Hamiltonian or not?

Is there a way to know if a system described by a known equation of motion admits a Hamiltonian function? Take for example $$ \dot \vartheta_i = \omega_i + J\sum_j \sin(\vartheta_j-\vartheta_i)$$ ...
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Hamiltonian and non conservative force

I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$. I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is ...
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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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Canonical transformation and Hamilton's equations

I was trying to prove, that for a transformation to be Canonical, one must have a relationship: $$ \left\{ Q_a,P_i \right\} = \delta_{ai} $$ Where $Q_a = Q_a(p_i,q_i)$ and $P_a = P_a(p_i,q_i)$. Now ...
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Good book for Analytical Mechanics

What is a good book for Analytical Mechanics? To be more specific, I would prefer a book that: Is written "for mathematicians", i.e. with high mathematics precision (for example, with less emphasis ...
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Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation

In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
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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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Noise spectrum of two systems and interacting Hamiltonian

I've been discovering recently the concept of noise spectrum, defined as: $$S_{xx}[\omega] = \int dt \langle x(t)x(0)\rangle \text{e}^{-i\omega t}$$ Roughly the Fourrier transform of the two-point ...
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Operator Ordering Ambiguities

I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity. What does that mean? I tried googling but to no avail.
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Potential Energy tends to infinity on the N-Body Problem

I need help to solve this problem related with the N-Body problem, i dont understand quite well what I need to define or to express in order to solve it. We assume a particular solution to the N-Body ...
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Why don't we use the concept of force in quantum mechanics?

I'm a quarter of the way towards finishing a basic quantum mechanics course, and I see no mention of force, after having done the 1-D Schrodinger equation for a free particle, particle in an ...
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Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
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What makes an equation an 'equation of motion'?

Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint. For example, in the ...
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Relativistic Hamiltonian Formulations [duplicate]

Possible Duplicate: Hamiltonian mechanics and special relativity? The Hamiltonian formulation is beautifully symmetric. It's a shame that the explicit time derivatives in Hamilton's ...
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Poisson brackets: prove that they are canonical invariants

EDIT: I haven't forgotten to accept answer, the question is still open.. I need a clarification about Poisson brackets. I'm studying on Goldstein's Classical Mechanics (1 ed.). Goldstein proves ...
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What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)

What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)? I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ...
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Non-Integrable systems

Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities (n being the number of degrees of freedom), or n whose Poisson brackets with each other ...
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Yang Mills Hamiltonian: why do we use the Weyl's temporal gauge?

Do you know why in the quantization of SU(2) Yang Mills Gauge Theory, it is always chosen the Weyl (temporal) gauge to derive the Hamiltonian? Is it possible to fix another gauge?
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Dirac equation as Hamiltonian system

Let us consider Dirac equation $$(i\gamma^\mu\partial_\mu -m)\psi ~=~0$$ as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that ...
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Is the geometric formulation of Hamiltonian mechanics really necessary? [duplicate]

Possible Duplicate: Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics? I was sitting around with some friends the other day trying to come up with ...
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How to explain the different forms of the Hamilton-Jacobi equation?

In Arnold's Mathematical Methods of Classical Mechanics, he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get $$ H\left(\frac{\partial S_1(Q, q)}{\partial q}, ...
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When Hamiltonian and the total energy are the same

In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?
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Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?

I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude: $$F(r, t) = \frac{k}{r^2}e^{-at}$$ ...
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Deriving kinetic energy in cylindrical coordinate constraints

Consider a mass $m$ which is constrained to move on the frictionless surface of a vertical cone $\rho = cz$ (in cyclindrical polar coordinates $\rho, \theta, z$ with $z>0$) in a uniform ...
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Describing the movement of the object in a particular situation in Lagrangian way

Suppose there is a object M, (sliding motion) moving by the initial speed $v$ and the initial location $x_0$. Otherwise noted, friction is assumed to be nonexistent. It then meets a circular mold ...
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What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

I have a problem with one of my study questions for an oral exam: The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
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About turbulence modeling

I have some questions about this paper: Lagrangian/Hamiltonian formalism for description of Navier-Stokes fluids. R. J. Becker. Phys. Rev. Lett. 58 no. 14 (1987), pp. 1419-1422. After reading ...
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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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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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Ordering Ambiguity in Quantum Hamiltonian

While dealing with General Sigma models (See e.g. Ref. 1) $$\tag{10.67} S ~=~ \frac{1}{2}\int \! dt ~g_{ij}(X) \dot{X^i} \dot{X^j}, $$ where the Riemann metric can be expanded as, $$\tag{10.68} ...
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Lagrangian to Hamiltonian in Quantum Field Theory

While deriving Hamiltonian from Lagrangian density, we use the formula $$\mathcal{H} ~=~ \pi \dot{\phi} - \mathcal{L}.$$ But since we are considering space and time as parameters, why the formula ...
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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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A good example of a nonlinear symplectomorphism?

What is a good example of a simple, physically useful nonlinear symplectomorphism $\kappa: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$? I'm not much of a physicist, and all the examples I've worked ...
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Hamiltonian is conserved, but is not the total mechanical energy

I wondering about the interpretation for the energy difference between the Hamiltonian and the total mechanical energy for systems where the Hamiltonian is conserved, but it is not equal to the total ...
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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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What is the difference between manifest Lorentz invariance and canonical Lorentz invariance?

I often read that the Lorentz symmetry is manifest in the path integral formulation but is not in the canonical quantization - what does this really mean?
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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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An example of non-Hamiltonian systems

I am preparing for the exam. And I need to know the answer to one question which I can't understand. "Give an example of non-Hamiltonian systems: in case of infinite number of particles; for a finite ...
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Counting degrees of freedom in presence of constraints

In a $N$ dimensional phase space if I have $M$ 1st class and $S$ 2nd class constraints, then I have $N-2M-S$ degrees of freedom in phase space. How can I calculate the degrees of freedom in ...
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ADM Hamiltonian formalism and Quantum gravity

is there a Hamiltonian reformultion of gravity ?=? if so if we use the usual Quantization scheme we can not we quantizy the gravity ?? in terms of a Gauge Theory with the potential $ A_{\mu}^{i} $ ...