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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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 ...
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Strange definition of microcanonical partition function

I always thought that the microcanonical partition function would measure the number of states that correspond to some fixed energy. Despite, I found in this paper (equation 3.4) that we integrate ...
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Why are Lagrangian subspaces called 'Lagrangian'?

I am wondering what the special role of Lagrangian subspaces (or submanifolds) are in mechanics. Do these subspaces have some sort of special property for which we have some sort of `Lagrangian ...
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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 ...
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Is general relativity holonomic?

Is it meaningful to ask whether general relativity is holonomic or nonholonomic, and if so, which is it? If not, then does the question become meaningful if, rather than the full dynamics of the ...
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Is there a Hamiltonian for the (classical) electromagnetic field? If so, how can it be derived from the Lagrangian?

The classical Lagrangian for the electromagnetic field is $$\mathcal{L} = -\frac{1}{4\mu_0} F^{\mu \nu} F_{\mu \nu} - J^\mu A_\mu.$$ Is there also a Hamiltonian? If so, how to derive it? I know how ...
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What's the value of the coupling constant in interacting field theories?

Consider this Lagrangian : $L = \frac{1}{2}(\partial_\mu \Phi)^2 - \frac{M^2}{2}\Phi^2 +\frac{1}{2}(\partial_\mu \phi)^2 -\frac{m^2}{2} \phi^2 -\mu\Phi\phi^2$ Its interaction term is given by : ...
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Who developed the phase space path integral?

The original path integral introducted by Feynman is $$ \lim_{N\to +\infty} \int \left\{\prod_{n=1}^{N-1} \frac{\mathrm{d}q_n}{\sqrt{2 \pi i \hslash \varepsilon}} \right\} ...
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Difference between Hamiltonian in classical Mechanics and in quantum Mechanics

I have a question about difference between Hamiltonian function (the description of system in classical physics) and the Hamiltonian operator (quantum mechanics). I think that there two different ...
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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 ...
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Hamilton-Jacobi equation with time dependent Hamiltonian

I was struggling with this exercise about Hamilton-Jacobi equation. I have to solve by menas of Hamilton's principal function the system with Hamiltonian: $$\tag{1} H=\frac{p^2}{2m}-mAtx $$ with $A$ ...
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Geometric mechanics - Symplecticity

I am just trying to wade through literature on classical mechanics and I really don't know where to start, everything is Fibre bundle this or manifold that, and doesn't really ease you in to the ...
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Can we explicitly solve the Hamilton–Jacobi equation for a particle in a uniform magnetic field?

HJE for nonrelativistic charged particle in an electromagnetic field is $$\frac{1}{2m}\left(\nabla S - q\mathbf{A}\right)^2 + q\phi + \frac{\partial S}{\partial t} = 0.$$ For a uniform magnetic ...
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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 ...
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Hamiltonian Noether's theorem in classical mechanics

How does one think about, and apply, Noether's theorem in the classical mechanical Hamiltonian formalism? From the Lagrangian perspective, Noether's theorem (in 1-D) states that the quantity ...
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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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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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What is canonical momentum?

What does the canonical momentum $\textbf{p}=m\textbf{v}+e\textbf{A}$ mean? Is it just momentum accounting for electromagnetic effects?
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Dimension agreement in canonical transformation

In this Physics.SE post, there is a transformation: $$Q = q,$$ $$P = \sqrt{p} - \sqrt{q}.$$ for Hamiltonian $H = \frac{p^2}{2}$. The post discusses the validity of this transformation as a canonical ...
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Proof of Liouville's theorem: Relation between phase space volume and probability distribution function

I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of ...
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Are Lagrangians and Hamiltonians used by Engineers?

Analytical Mechanics (Lagrangian and Hamiltonian) are useful in Physics (e.g. in Quantum Mechanics) but are they also used in application, by engineers? For example, are they used in designing bridges ...
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The FRW universe is NOT asymptotically flat? Its mass?

The Friedman-Robertson-Walker (FRW) metric in the comoving coordinates $(t,r,\theta,\varphi)$ which describes a homogeneous and isotropic universe is $$ ds^2\,= -dt^2+\frac{a(t)^2}{1-kr^2}\,dr^2 + ...
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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 ...
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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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What kind of manifold can be the phase space of a Hamiltonian system?

Of course it should have dimension $2n$. But any more conditions? For example, can a genus-2 surface be the phase space of a Hamiltonian system?
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Why can't we obtain a Hamiltonian by substituting?

This question may sound a bit dumb. Why can't we obtain the Hamiltonian of a system simply by finding $\dot{q}$ in terms of $p$ and then evaluating the Lagrangian with $\dot{q} = \dot{q}(p)$? Wouldn't ...
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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 ...
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Questions about the degree of freedom in General Relatity

I'm confused about the number of degrees of freedom in General Relatity. There are two ways to count it. However, they are contradictory. For simplicity, we consider vacuum solution. First, ...
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How can I derive this Hamiltonian?

I have a Lagrangian $L$, a momentum $p$ and a Hamiltonian $H$: $$L=\frac m 2(\dot z + A\omega\cos\omega t)^2 - \frac k 2 z^2$$ $$p=m\dot z + mA\omega\cos\omega t$$ $$H=p\dot z - L=\frac m 2 \dot ...
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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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name for 2D Electrostatics as Integrable System

I am trying to understand 2D electrostatics of $n$ point charges. Roughly, $$ H = \sum_{i=1}^N n_i \ln |z- z_i|$$ However, I keep bumping across the Gaudin model instead with this Hamiltonian $$ ...
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Rational ratio of frequencies leads to isolating integral of motion

Padmanabhan's discussion of dynamics mentions that in general the two dimensional harmonic oscillator fills the surface of a two torus. He further notes that there will be an extra isolating integral ...
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Why don't we experience dilatations in Minkowski spacetime?

The question This question follows on from the use of projective coords for spacetime in Notation for Translation Group Generators . Under Felix Klein's Erlangen Program, Minkowski spacetime starts ...
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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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Canonical transformation that contains the time as an explicit parameter

On the Page 385 of Goldstein's Classical Mechanics book (third edition), it starts to talk a bout the canonical transformation with time as an explicit parameter. But I don't quite under understand ...
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Recommendation on mathematical physics book of Symplectic geometry

I want to learn the applications of symplectic geometry in physics. Which mathematical physics textbook will have a detailed and heuristic explanation of this aspect?
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Conditions for hyperregularity

(Definitions from Marsden & Ratiu, ``Introduction to Mechanics and Symmetry''): A Lagrangian is regular if the Hessian $\partial^2 L/(\partial \dot{q}^i \partial \dot{q}^j)$ is weakly non ...
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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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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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Can all canonical transformation be obtained through generation function approaches?

The question can be formulated as following: Suppose $$\delta \int_{t_1}^{t_2}{[p\cdot \dot{q} - H(p,q,t) ]dt} = 0$$ $$\delta \int_{t_1}^{t_2}{[P\cdot \dot{Q} - K(P,Q,t) ]dt} = 0$$ in which $$P = ...
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Is there a physical system whose phase space is the torus?

NOTE. This is not a question about mathematics and in particular it's not a question about whether one can endow the torus with a symplectic structure. In an answer to the question What kind of ...
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Shouldn't the addition of angular momentum be commutative?

I have angular momenta $S=\frac{1}{2}$ for spin, and $I=\frac{1}{2}$ for nuclear angular momentum, which I want to add using the Clebsch-Gordan basis, so the conversion looks like: $$ \begin{align} ...
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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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Invariance of canonical Hamiltonian equation when adding the total time derivative of a function of $q_i$ and $t$ to the Lagrangian

The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's Classical Mechanics. Adding the total time derivative of a function of $q_i$ and t to the Lagrangian ...
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Eigenvalues of the Lagrangian?

It is often stated that the Lagrangian formalism and the Hamiltonian formalism are equivalent. We often hear people talk about eigenvalues of Hamiltonians but I have never ever heard a word about ...
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How to prove that a hamiltonian system is not integrable?

To show that a system is integrable, we just need to find $N$ independent functions $f_j$ such that $\{ f_i, f_j \} = 0$. But how to prove that such a set of functions do not exist? For example, ...
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Are there any hamiltonian systems without a periodic orbit?

Are there any hamiltonian systems without a periodic orbit? Can anyone give me an example? If such a system exists, does this fact have any implication on its quantum version?
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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 ...
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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 ...
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Derivation of (2.45) in Peskin and Schroeder

I'm having trouble understanding the step $$\left[\pi (\vec{x},t),\int d^{3}y ~(\frac{1}{2} \pi (\vec{y},t)^{2}+\frac{1}{2}\phi (\vec{y},t)(-\nabla^{2} +m^{2})\phi (\vec{y},t)) \right]$$ $$ =\int ...