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 ...

learn more… | top users | synonyms (1)

2
votes
2answers
154 views

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 ...
2
votes
1answer
268 views

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 ...
2
votes
1answer
317 views

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 ...
2
votes
1answer
260 views

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 $\mathbb R^{2n}$. ...
2
votes
3answers
187 views

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 ...
2
votes
1answer
286 views

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 ...
2
votes
2answers
100 views

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 ...
2
votes
1answer
258 views

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 ...
2
votes
1answer
206 views

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 ...
2
votes
0answers
47 views

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 ...
2
votes
0answers
45 views

Is there useful information about normal modes/frequencies in the Hamiltonian matrix?

Suppose I have two $LC$ oscillators, one with $L_1$ and $C_1$, and the other with $L_2$ and $C_2$. If uncoupled, each oscillator has resonant frequency $\omega \equiv 1/\sqrt{LC}$. Using the flux in ...
2
votes
1answer
57 views

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 ...
2
votes
0answers
50 views

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. ...
2
votes
0answers
91 views

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 ...
2
votes
0answers
76 views

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+ ...
2
votes
3answers
116 views

About constraints of the first class and electrodynamics

Let's have some theory in hamilton formalism and let's assume that it has the constraints between canonical variables $Q, \pi$. By the Dirac terminology, the set of constraints $F_{a}(Q, \pi) \approx ...
2
votes
1answer
118 views

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 ...
2
votes
0answers
144 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 ...
2
votes
0answers
176 views

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 ...
2
votes
0answers
186 views

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}, ...
2
votes
1answer
282 views

Hamiltonian reduction having constant of the motion

I have this $2^n*2^n$ matrix that represent the evolution of a system of $n$ spin. I know that I can have only one excited spin in my configuration a time. (eg: 0110 nor 0101 ar not permitted, but ...
2
votes
0answers
269 views

Gradient involved commutator in $\phi^4$ theory

In a phi fourth theory, the Hamiltonian density is: $$\mathcal{H}=\frac{1}{2}\pi^2+\frac{1}{2}(\nabla \phi)^2+\frac{1}{2}m^2\phi^2+\frac{\lambda}{4!}\phi^4$$ Now I impose the usual equal time ...
2
votes
0answers
38 views

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 ...
2
votes
0answers
353 views

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 ...
1
vote
3answers
577 views

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 ...
1
vote
2answers
2k views

How to find out whether a transformation is a canonical transformation?

We had a couple of examples where we were supposed to calculate the Canonical Transformation (CT), but we never actually talked about a condition that decides whether a transformation is a canonical ...
1
vote
1answer
897 views

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 ...
1
vote
3answers
187 views

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 ...
1
vote
1answer
380 views

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.
1
vote
1answer
114 views
+50

Help needed in finding the integral curves given by orbits of one-parameter groups

Equip $\mathbb{R}^2$ with standard symplectic structure and inner product. Consider a Hamiltonian $$H=(x,y)A(x,y)^t$$ where $$A=\begin{pmatrix} \alpha & \beta \\ \beta & \delta \end{pmatrix}$$ ...
1
vote
1answer
43 views

Practical Book on Hamiltonian and Lagrangians? [duplicate]

Are there any terse, accessible books that are geared specifically at learning these two formalisms and how to effectively use them? So far I've only see either topic introduced as a part of another ...
1
vote
2answers
48 views

Recommendation on ADM mass and Bondi mass

I want to learn some advanced topics in GR, such as ADM 4-momentum and Bondi 4-momentum. However nearly no textbooks on GR contain this area, such as Wald, MTW, Hawking, Carroll and Zee's. Can anyone ...
1
vote
2answers
217 views

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' ...
1
vote
3answers
287 views

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 ...
1
vote
1answer
457 views

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 ...
1
vote
1answer
224 views

interpretation of $\{H,L^2\}$

In Hamiltonian mechanics, we show $\{H,L_z\}=0$, which can be interpreted as the conservation of angular momentum around $Oz$. Following the same idea, how can we interprete $\{H,L^2\}$? Is the ...
1
vote
1answer
61 views

Symplectic notation proof?

Hello I am trying to work through a little proof of the symplectic condition for Hamilton's equations for a classical mechanics course. I am trying to understand the meaning of the relation ...
1
vote
1answer
45 views

How is force exerted on a wall equal to derivative of hamiltonian with respect to wall position?

I'm trying to understand a solution of a problem in Landau, Lifshitz "Quantum mechanis. Non-relativistic theory" in $\S22$ "The potential well": Determine the pressure exerted on the walls of a ...
1
vote
1answer
168 views

Hamilton's characteristic and principle functions and separability

Just hoping for some clarity regarding Hamilton's characteristic function (W). When we take a time independent Hamiltonian we can separate the Principle function (S) up into the characteristic ...
1
vote
1answer
91 views

Transforming a lagrangian to hamiltonian and vice versa

I am not refering to Legendre transform, but to something more simple. In analytical mechanics, the Lagrangian can be described as $L=T-V$, and the Hamiltonian is if the Lagrangian doesn't explicitly ...
1
vote
1answer
403 views

Angular momentum conservation in a central field through the Hamiltonian

In my teacher's notes there is a discussion of the Hamiltonian for a central force field with potential $V(r)$. The Hamiltonian is formulated in spherical polar coordinates: ...
1
vote
2answers
607 views

Canonical transformations and conservation of energy

I have an important doubt about the nature of canonical transformations in hamiltonian mechanics. Suppose I have a one-degree-of-freedom lagrangian system, whose hamiltonian depends explicitly on ...
1
vote
1answer
653 views

Conjugate Variables and Fourier Transforms in Classical Physics

Let q be a generalized coordinate with a conjugate momentum p and a potential resulting in a periodic motion of q. What is the meaning of the Fourier transform of q(t) over its period? Can this be ...
1
vote
2answers
313 views

Multiple classical paths from Hamilton's principle

Previous posts such as this ask about types of stationary point in Hamilton's Principle. There is, however, another aspect to discuss: the question as to whether the extremal path is unique. One ...
1
vote
1answer
64 views

Analytical solution of Liouville's equation for classic harmonic oscillator - which book?

So the past five hours I've spend fruitlessly searching the web for any materials containing the analytical solution of the simple PDE: $$\frac{\partial f}{\partial t} - m\omega^2x\frac{\partial ...
1
vote
2answers
40 views

Sign conventions in Devoret Les Houches course on quantum fluctuations in electrical circuits

In this article on p. 364 Devoret writes the "equations of motion" (using KCL) for the electric circuit shown on p. 363. He uses flux instead of voltages. The sign convention he uses, as shown on p. ...
1
vote
1answer
64 views

Hamilton's equations from the action with boundary conditions involving position and momentum

Generally, when you are given the action $$ S=\int_{t_1}^{t_2}\mathrm dt (p\dot q - \mathcal H )$$ the boundary conditions are $q(t_1)=q_1$ and $q(t_2)=q_2$. This is useful because to calculate ...
1
vote
2answers
155 views

How can I derive the Hamiltonian of simple harmonic oscillator from this Lagrangian?

I'm working through Leonard Susskind's Theoretical Minimum: Classical Mechanics and I can't seem to understand how the Hamiltonian of a simple harmonic oscillator is derived from the following ...
1
vote
1answer
89 views

Where can some worked problems in classical mechanics (and more specifically the Lagrangian and Hamiltonian formalisms) be found? [duplicate]

I've been looking for a textbook in classical mechanics that's readily available (like can be found in the library of James Cook University of Townsville, Australia) and full of fully-answered ...
1
vote
1answer
77 views

Darboux theorem and the canonical decomposition of a two-fermion wave function

It is a classical theorem in quantum mechanics or quantum chemistry or quantum information that a two-fermion wave function has a beautiful canonical expansion: $$f(x_1, x_2) = \sum_{j=1}^N ...