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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How can we obtain equations of motion from $\iota_{X_{H}}\omega =dH$?

I don't know if this is an obvious result and I am just missing a trick, so please forgive me, but how do we obtain equations of motion from the following equation. \begin{equation} \iota ...
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Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
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What canonical momenta are the “right” ones?

I'm doing some classical field theory exercises with the Lagrangian $$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the ...
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First integrals for a particle in a central-force field

Consider an arbitrary dimension $n>3$. What are the independent first integrals for a particle? The Hamiltonian is $$ H = \frac{p^2}{2m} +V (|r|) . $$
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Intuition about Momentum Maps

I'm studying Classical Mechanics and there is one object that appeared recently on the book I'm not being able to get a physical intuition about it. The mathematical definition goes as follows: Let ...
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Finding canonical transformation using type 3 generating function

Question: For a system with one degree of freedom, a canonical transformation $Q(q,p), P(q,p)$ obtained by a type 3 generating function satisfies $Q = e^t q^{1/2}\cos p$. Find the most general form ...
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Show that a Hamiltonian system is integrable [closed]

The following problem in my textbook is giving me some difficulty. Consider the Hamiltonian $H(q_1,q_2,p_1,p_2) = \frac{1}{2}(q_1^2 + q_2^2 + p_1^2 + p_2^2) + \frac{q_1^3}{3} - \frac{q_2^3}{3}. $ ...
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Poisson brackets and magnetic field [on hold]

I'm a maths student trying to teach myself some physics so sorry if I'm missing something simple here. I think the main problem is lack of experience with the Levi-Cevita symbol. We have a particle ...
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The Liouville equation and the BBGKY hierarchy.

The Liouville equation of motion is written in terms of an $N$ particle distribution $f_N$. \begin{equation} \frac{\partial f_N}{\partial t}=\{H,f_N\} \end{equation} Where $\{\cdot ,\cdot \}$ is the ...
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Chaos and integrability in classical mechanics

An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any ...
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What is the meaning of commuting Hamiltonians?

I have two quantum mechanical Hamiltonians such that \begin{equation} [\hat{H}_1,\hat{H}_2] = 0, \end{equation} where $\hat{H}_1$ and $\hat{H}_2$ act on the same set of states. What is there to ...
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Lagrangian from Path Integral

Suppose I somehow know propagator for a given quantum mechanical system but I don't happen to know either the Lagrangian or Hamiltonian. (For simplicity, assume that this is non-relativistic.) Is ...
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Why is the Hamilton-Jacobi equation important? [closed]

Someone may say it is related to the Schrodinger equation. Okay, let us forget about quantum mechanics. So, if we confine ourself to classical mechanics, why is the Hamilton-Jacobi equation important ...
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What is the utility of ADM decomposition of the space-time metric?

I know it's one of the possibility of quantization of gravitational field's degree of freedom but it is introduced also in other situation. My question is which is the powerful of this kind of ...
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Any good reference on Maslov index (or Morse index)?

Any good reference on Maslov index (or Morse index)? I have some basic knowledge of differential geometry, calculus of variation. So is there any good reference for me?
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Do time-invariant Hamiltonians define closed systems?

In classical mechanics, every time-invariant Hamiltonian represents a closed dynamical system? Can every closed dynamical system be represented as a time-invariant Hamiltonian? Or are there closed ...
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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 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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Jacobi energy function $h$ and the Hamilton $H$ and the Hamilton-Jacobi equation

My understanding of the Jacobi energy function $h$ as defined in Goldstein is that it is the total energy $T+V$ expressed as, \begin{equation} h(q,\dot q,t)=\sum \frac{\partial L}{\partial \dot q}\dot ...
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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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For infinitesimal Canonical Transformations, what functions are allowed for this to be a canonical transformation?

Consider two infinitesimal transformation: $$q_{i} \rightarrow Q_{i} =q_{i} + \alpha F_{i}(q,p) $$ $$p_{i} \rightarrow P_{i} = p_{i} + \alpha E_{i}(q,p) $$ where $α$ is considered to be ...
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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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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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Are there other less famous yet accepted formalisms of Classical Mechanics?

I was lately studying about the Lagrange and Hamiltonian Mechanics. This gave me a perspective of looking at classical mechanics different from that of Newton's. I would like to know if there are ...
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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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Is there useful information about normal modes/frequencies in the Hamiltonian matrix of a coupled system?

Single mode 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 ...
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What variable is the conjugate momentum for angular momentum?

From the definition of conjugate momentum for a generalized coordinate we get that the conjugate for angular momentum should be proportonal to its integral with respect to time. According to my ...
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Formalism to deal with discontinuous potentials in classical mechanics (hard wall, hard spheres)

It seems to me that Hamiltonian formalism does not suit well for problems involving instantaneous change of momentum, like particle collisions with hard wall or hard sphere gas model. At least I could ...
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Is the Legendre transformation a unique choice in analytical mechanics?

Consider a Lagrangian $L(q_i, \dot{q_i}, t) = T - V$, for kinetic energy $T$ and generalized potential $V$, on a set of $n$ independent generalized coordinates $\{q_i\}$. Assuming the system is ...
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What is the difference between configuration space and phase space?

What is the difference between configuration space and phase space? In particular, I notices that Lagrangians are defined over configuration space and Hamiltonians over phase space. Liouville's ...
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from microscopic to kinetic transport theory

One way to model the dynamics of particles is to find the differential equation of motion of a particle. Of course, this will be nice and easy to do if we have only a few particles (like one-ish, ...
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Analytical mechanics with SR

Is there an analytical mechanics with SR? Of course you can write down the Lagrangian and Hamiltonian of a free particle. What about non-free? Are there any problems? To be specific: what would the ...
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Time evolution of a classical system [on hold]

For a harmonic oscillator the Liouville operator is given by $$L = p \partial_q- q \partial_p.$$ Now I have a phase space distribution $f(t,q,p)$ for which it holds (in general) $$f(t+\tau,q,p)= ...
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Fermionic Poisson bracket

I'd like to understand the Poisson bracket for fermions in classical field theory defined on a cylinder (with coordinates $(t,x)$, $x$ being the compact direction) and propagating on $T^n$ with ...
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Constructing Lagrangian from the Hamiltonian

Given the Lagrangian $L$ for a system, we can construct the hamiltonian $H$ using the definition $H=\sum\limits_{i}p_i\dot{q}_i-L$ where $p_i=\frac{\partial L}{\partial \dot{q}_i}$. Therefore, to ...
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Wrong sign anticommutation relation for the Dirac field?

Consider the Dirac Lagrangian $$\mathcal{L}=\psi ^{\dagger }\gamma ^{0}\left( \mathrm{i}\gamma ^{\rho }\partial _{\rho }-m\right) \psi .$$ The conjugate momenta to $\psi ^{a}$ are defined, as usual, ...
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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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What exactly is the relationship between the symplectic 2-form and the frequency of leaves of integrable systems in classical mechanics?

In classical mechanics we equip a differential manifold with a closed symplectic 2-form $\omega$. The symplectic leaves of integrable systems also have a unique frequency, in literature denoted ...
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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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Symplectic leaves, tori and Poisson manifolds

For classical systems we can define a configuration manifold, whose cotangent bundle is a momentum phase space equipped with a closed, non-degenerate 2-form. Upon the commutative algebra of smooth ...
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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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Taking a 'relative' limit

I am looking at Hamiltonians for specific physical situations. I have taken a given Hamiltonian $\vec{H}(\vec{p}, \vec{x})$ and have found the following Hamiltonian equations: $$\frac{d\vec{x}}{dt} = ...
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Derivation of an ordinary, Lagrangian/Hamiltonian and action formulation

I am confused as to how the different formulations in physics are derived. In many fields of physics, we usually begin with an ordinary formulation (e.g Newton's Laws in classical mechanics), and ...
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How to find the rank of the matrix $\frac{\partial ^2\mathcal{L}}{\partial \dot{X^\mu} \partial \dot{X^\nu} }$ for the Nambu-Goto string Lagrangian?

In this case $$\mathcal{L}~=~-T\sqrt{-\dot{X^2}X'^2+(\dot{X}\cdot X')^2}.$$ I was reading some books and papers about the constraints in the Nambu-Goto action, and all of them say something like ...
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Reduction of Nambu Goto action to true degrees of freedom

First consider the particle $$S=m\int\sqrt{-\dot{X}^2}d\tau$$ if you choose the static gauge $\tau=X^0$ and replace it in the action you get $$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau$$ So now, you ...
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Amplitude-phase decomposition as a canonical transformation

I am studying a classical dynamical system defined on $\mathbb{CP}^2$: the phase space is parametrized in terms of three complex coordinates $\psi_i$ ($i=1,2,3$) and Hamilton's equations of motion ...
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Using tensors on Lagrangian and Hamiltonian

We can write the Lagrangian (with $n$ generalized coordinates) using the following expression: ...
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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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Hamilton's Principle - achieving Hamilton equations

Consider the action function: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $\mathcal{L}$ is the Lagrangian of the system. The Hamiltonian is defined by the following ...