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Questions tagged [hamiltonian-formalism]

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 Mechanics, this ...

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Self-modifying Hamiltonians (Lagrangians) and emerging intelligence? [on hold]

Are there dynamical physical systems that are governed by self-modifying Hamiltonians (Lagrangians), i.e. Hamiltonians (Lagrangians) determine not only the next point in phase space, but also the form ...
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Mixed canonical transformation

Wikipedia and most authors denote four types of canonical transformations: $F_1(\mathbf{q},\mathbf{Q})$ , $F_2(\mathbf{q},\mathbf{P})$, $F_3(\mathbf{p},\mathbf{Q})$ and $F_4(\mathbf{p},\mathbf{P})$. ...
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Hamiltonian for a variable length pendulum

This question is taken from the book "Classical Dynamics of Particles and Systems" - Marion, problem 7.24. The problem is about a pendulum that is set into motion, it's length varies at a constant ...
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What happens to time component of metric in Wheeler-de-Witt equation?

In the Wheeler-DeWitt equation, space-time is "foliated" and the metric $g_{\mu\nu}$ is decomposed into a metric on the surface of a 3D slice of space-time. The Wheeler-DeWitt equation is then written ...
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Mechanical dynamics of self-modifying systems? [closed]

Let's assume that there is a rigid pendulum whose length somehow depends functionally on the mean velocity of the end-point. (The connection is functional, but it can be an inferential process as ...
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Time-independent canonical transformations

Lie's criterion tells us that $(q,p) \to (Q,P)$ is a canonical transformation, for a system with Hamiltonian $H$ and "Kamiltonian" $K$, if and only if the identity $$\sum_k p_k dq_k -Hdt = \sum_k P_k ...
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Relationship between complex analysis and Hamilton's canonical equation

I recently came across a mathematical field called complex analysis. There was an important equation called Cauchy-Riemann equation. When I saw it at first, I recalled a book's sentence stating ...
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When can action-angle variables be defined?

According to Goldstein, "We can define action-angle variables for [a separable Hamiltonian] system when the orbit equations for all of the $(q_i, p_i)$ pairs describe either closed orbits (...
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Why do we prefer “Lagrangian and Hamiltonian mechanics ” over the newtonian mechanics? [duplicate]

Basically, I just want to know the advantages of Lagrangian and Hamiltonian mechanics over Newtonian mechanics, that made it much more preferable and widely used!
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Relationship between the Galilei Group and the Phase Space

This question is kind of a follow up question to my last question on the need for canonical commutation relations and conjugate observables. A comment from Valter Moretti suggested that, given a ...
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Hamiltonian definitions in the presence of boundary term [duplicate]

Consider a Lagrangian of the form \begin{equation} L(q,\dot{q})=L_1(q,\dot{q})+\frac{d L_2(q,\dot{q})}{dt} \end{equation} I understand that $\dot{L_2}$ does not modify the equations of motion, ...
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Do compact symplectic manifolds play a role in physics?

In classical mechanics, the phase space is the cotangent bundle of the configuration space, and it is a symplectic manifold, but not compact. Do compact symplectic manifolds have physical meaning? ...
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Can theoretical mechanics be derived from differential geometry?

I've heard this somewhere, I don't know how True it is, can somebody clarify the relationship between theoretical mechanics and differential geometry? Can theoretical mechanics really be derived from ...
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How are the endpoints of motion determined?

The Hamilton's principle tells us which path is taken by the system in going from one point of the configuration space at time t1 to another point of the configuration space at time t2. My question is ...
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What is the Lagranian correponding to this Hamiltonian?

The Hamiltonian is $$H=\vec{S}\cdot\vec{B},$$where $\vec{S}$ is spin, and $\vec{B}$ is external magnetic field. My question is what is the corresponding Lagranian?
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Solving Hamilton-Jacobi via canonical transformations

Given a solution to a Hamilton-Jacobi equation in $(X, P)$ variables and a canonical transformation from $(x, p)$ to $(X, P)$, how does one write down the solution to the Hamilton-Jacobi in terms of ...
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Arithmetic of Hamiltonian in canonical transformation

I have the following Hamiltonian: $$ \mathcal{H} = \frac{p^2}{2m} + V(q-X(t)) + \dot{X}(t)p, $$ and I make the usual canonical transformation for the momentum: $$ p \rightarrow p' = p + m\dot{X},$$ ...
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Canonical quantization of time-dependent lagrangians

I have a lagrangian $$ L(x^{a}, \dot{x}^{a}, t), $$ which is non-degenerate, quadratic in the fields, and contains an explicit dependence on the evolution parameter $t$. If $L$ was time-independent,...
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why is a Lagrangian submanifold a semi-classical state and not a classical state?

I read that the Lagrangian submanifold can be regarded as a semi-classical state when classical mechanics is formulated using symplectic geometry. Does anyone know why it would be a semi-classical ...
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$q\mathbf{A}\cdot\mathbf{v}$ term in potential energy

In the famous Goldstein mechanics book, there is an example about a single (non-relativistic) particle of mass m and charge q moving in an E&M field. It says the force on the charge can be ...
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A question over Liouville’s Theorem

I have some doubts about Liouville theorem, probably its just something conceptual. So: I know that for a system in which Liouville’s theorem holds, the volume in the phase space is conserved. But ...
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Why any operator is specified by this characteristic function?

On the paper "Tutorial Notes on One-Party and Two-Party Gaussian States", arXiv:quant-ph/0307196, the author states on section 2: Any operator referring to a harmonic oscillator — position operator ...
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Why is dynamics first order in phase space?

I have watched some lectures in which the lecturer said that system dynamics are (generally?) first order in phase space, forming a system of coupled differential equations. At a basic level I see ...
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Intuition for Hamilton-Jacobi equation derived from least action

I am trying to understand the Hamilton-Jacobi equation without the framework of the canonical transformations. Even on the case of a 1D free particle I'm getting stuck. The system starts at fixed ...
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Averaging a tensor in NMR Hamiltonian for infinite speed limit MAS

I'm trying to figure out if there is a simple way to compute what the average Hamiltonian for an NMR experiment at ideal infinite speed MAS would look like. I keep failing, but I'm not sure why. I'll ...
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Dirac bracket for the Madelung (polar) form of the Schrodinger field

I'm having an issue with obtaining the Dirac bracket in the Madelung (polar) representation of the Schrödinger field: \begin{equation} \Psi=\sqrt{\rho}e^{i\theta/\hbar}. \label{eq:...
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A confusing point of the Hamiltonian for a particle interacting with electromagnetic fields

In non-relativistic quantum theory the Hamiltonian for a particle interacting with electromagnetic fields is $$H=\frac{(\mathbf{p}-\mathbf{A}*e/c)^2}{2m}+e\phi+\int\,d^3x \frac{\mathbf{E^2}+\mathbf{B^...
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Legendre Transformation [closed]

Can someone explain how the Legendre transformation for multivariable functions work? I have been trying to find Legendre transformation for a function like $$F(x, y) = x^{2}+y^{2}+xy.$$ But I don't ...
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Energy of classical Inverted Harmonic Oscillator

Quick one. Does the energy of inverted harmonic oscillator $$H(x, p) = \frac{p^2}{2} - \frac{x^2}{2},$$ remain conserved?
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Determining the stress-energy tensor from the equations of motion

I have a question on finding the stress-energy tensor from the equations of motion in general relativity. Given the Einstein-Hilbert action+matter Lagrangian, it is straightforward to then determine ...
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Why isn't it important, after which coordinates the Variation of the action integral is done?

I often read,that if the lagrangian $L=p\dot{q}-H$ of a pair of coordinates in phase space $(q,p)$ and $P\dot{Q}- K $, for some new pair of coordinates $(Q,P)$ only differ by a total time derivative $...
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Is there a physical interpretation of symplectic manifolds which are not cotangent bundles?

The inspiration for symplectic geometry was from Hamiltonian mechanics. However, I am wondering how close the ties are between arbitrary symplectic manifolds and real physical systems. In ...
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How does the Hamiltonian changes after rotating the coordinate frame

A basic question in rotation. Initially I have a Hamiltonian $$ H = \frac{-1}{2}(4E_c(1-n_g)\sigma_z + E_J\sigma_x) \, . $$ It's said that, if I rotate the coordinate system by mixing angle, $$ \...
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Boundedness of general relativity Hamiltonian [duplicate]

When one consider a lagrangian and construct hamiltonian, we expect to be bounded below. While looking to the Hamiltonian formulation of general relativity, I have difficulties to see how it can be ...
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Boundedness of general relativity Hamiltonian

When one consider a lagrangian and construct hamiltonian, we expect to be bounded below. While looking to the Hamiltonian formulation of general relativity, I have difficulties to see how it can be ...
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Boundary contributions to the BTZ black hole hamiltonian

I've been trying to understand every step of the (2+1 dimensional) BTZ black hole paper (https://arxiv.org/pdf/hep-th/9204099.pdf ) but I'm stuck in the obtention of the boundary terms in the ...
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Showing time-dependent transformations are canonical

In Goldstein's mechanics book he defines a canonical transformation as a transformation from $q, p$ to $Q, P$ with $$ Q = Q(q,p,t) \tag{9.4a} $$ and $$ P = P(q,p,t) \tag{9.4b} $$ such that if $H$ ...
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Hamilton or Hamilton-Jacobi formalism with Hamiltonian equal to zero

I have the Lagrange function: $$L=\sqrt{\frac{\dot{x}^2+\dot{y}^2}{-y}}.\tag{1}$$ The energy is then: $$H=\dot{x}\frac{\partial L}{\partial \dot{x}}+\dot{y}\frac{\partial L}{\partial \dot{y}}-L=0.\...
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What does it mean to say a 'double' Legendre transform?

Ref. 1 mentons that you can achieve the momentum space Lagrangian by doing a so called double Legendre transform. It goes on to write:$$ K(p,\dot{p},t) ~=~ L(q,\dot{q},t) - p\dot{q} - q\dot{p},\tag{5....
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Hamiltonian for particle moving in a sphere

Context: I know that if we have a particle (say, with unit mass) moving in the plane $\Bbb R^2$ subject to a spherically symmetric potential $V\colon \Bbb R^2 \to \Bbb R$, it will move along the ...
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Apparent emergence of conserved quantities in non-integrable systems

This question arises from the comments relevant to the post When is the ergodic hypothesis reasonable? Consider a Hamiltonian system having more effective degrees of freedom than conserved quantities....
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When is the ergodic hypothesis reasonable?

Consider an Hamiltonian system. In which circumstances is it possible to assume that all the states belonging to the hypersurface $H=E_0$ are equally visited? Is it necessary to have a very high ...
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Action of conjugate momentum on $TM$ and explicit form

In hamiltonian mechanics the phase space of a particle is a symplectic manifold. In the case we have a configuration space $M$, that is the manifold describing the possible positions of the particle, ...
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1answer
231 views

Is the Hamiltonian of a relativistic charged particle in an electromagnetic field only an approximation?

Consider a system of two relativistic charged point particles 1 and 2 which interact through their electric and magnetic fields. The equation of motion for the first particle is then given by the ...
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How do contact transformations differ from canonical transformations?

From Goldstein, 3rd edition, section 9.6, page 399 after equation 9.101: [...] The motion of a system in a time interval $dt$ can be described by an infinitesimal contact transformation generated ...
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Symplectic Standard map [closed]

I have come across this map, which the notes call standard symplectic map. Why is it symplectic? How do I show it? Are those action-angle variables? $I(t+1)=I(t)+K\sinθ(t)$ $θ(t+1)=θ(t)+I(t+1) \quad ...
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A fundamental question about Time-dependent Hamiltonians

I have a fundamental question about Quantum Mechanics or even mechanics in general. I am aware that there are stationary solutions and non-stationary solutions. The stationary solutions solve ...
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1answer
64 views

Lagrangian and Hamiltonian dynamics, momentum and canonical transformations

I am relatively new to Lagrangian and Hamiltonian dynamics. I am aware of how to form the equations of motion using the Legendre Transformation. I, however, have one fundamental question and I was ...
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Linear canonical transformation represented by a unitary operator

I am reading a paper on Squeezed states which mentions the following fact "a linear canonical transformation can be represented by a unitary transformation" and then used a operator $\hat{U}$ for ...
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Hamiltonian formalism, polynomial of a certain rank and velocity-independent potential

In page 88 of Shankar's Principles of Quantum Mechanics, we have the following lines: $$\mathcal{H}(q,p) = \sum_{i=1}^{n}p_i \dot{q_i} - \mathcal{L}(q, \dot{q}) \tag{2.5.8 } $$ where the $\dot{q}$'...