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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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Book reference on variational principles of mechanics [on hold]

Which book is a better choice for self study of variational principles of mechanics and the Lagrangian and Hamiltonian formulation , The variational principles of mechanics by Cornelius Lanczos or ...
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33 views

Meaning of the uniform ensemble

In statistical physics (and viewing this from a classical point of view) for an isolated system we can say that a system will have an energy constant in time. People define the following uniform ...
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40 views

Derivation of Hamilton-Jacobi theory using canonical transformations

The derivation of the Hamilton-Jacobi equation using canonical transformations is typically done involving a type-2 generating function. Is it possible to use a another type of generating function, ...
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69 views

Action angle variables and Action

Action given by principle of least action ($S$) and action variable given by action angle variable theory ($J$) are same?
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Generating function in action-angle method and Hamilton-Jacobi theory

I think that in action angle method, generating function which generates such a canonical transformation does not explicitly depend on time, so new and old hamiltonians are equal. But in H-J method, ...
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Action angle variables and Hamilton jacobi theory

Action P= Closed integral of Pdq , Why would we choose a closed integral, we know that hamiltonian flows preserve volume and area , so can we replace this closed integral with a definite integral ...
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Decomposition of the metric tensor to gauge transformations

I am working in a 3+1 space with linearized gravity. I here try to decompose the spatial metric tensor perturbation $\gamma_{ij}$ into a longitudinal, transverse and transverse traceless part. These ...
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Significance of geodesics of a Hamiltonian surface in condensed matter physics?

Many Hamiltonians in 2D quantum systems can be parameterized as a surface (such as the Bloch sphere) by their k-space coordinates. Another example is given by the (kx,ky) points of the Brillouin torus ...
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Question about ergodicity and the evolution of the probability distribution under Liouville's theorem

According to Liouville's theorem, the probability distribution function $\rho$ evolve in phase space with $$ \frac{d \rho}{d t} = \frac{\partial \rho}{\partial t}+\left\{\rho,H\right\}_{P.B} =0 $$ ...
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Using fourier analysis of the Klein Gordon equation

This question is more about a mathematical detail, and I am undoubtedly missing something very obvious. And note, I have sifted through the numerous questions on Fourier transform (FT) and the Klein-...
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Must a classical Lagrangian or a Hamiltonian be a real function?

$\bullet$ Is it fair to assume that the classical Hamiltonian or Lagrangian of a system (a particle or a field) is always a real-valued function? $\bullet$ If not, can you provide counter-examples? ...
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Do outside factor affect the Hamiltonian?

This is probably a fairly obvious question, but I'm unsure of whether the environment outside a system effects the Hamiltonian. Clearly temperature should affect it, since it is a measure of the total ...
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81 views

Derive Lorentz Equation from Relativistic Hamilton-Jacobi Equation

Consider a ralativistic particle of rest mass $m$ and electric charge $e $ moving in electromagnetic field with four-potential ${\displaystyle A^{\mu}=(\phi ,\mathrm {A} )} $ in vacuum, then the ...
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Additive constant in Hamilton-Jacobi theory?

In Hamilton-Jacobi theory Hamilton's principal function S is a function of n+1 constants , But we take one of the n+1 constants as an additive constant . I don't get this step?
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Lapse and shift inside or outside the Poisson bracket?

For general relativity in the 3+1 ADM formulation, one has $H=\int dx [N{\cal H}+N^a{\cal H}_a]$ with $N$ and $N^a$ the lapse and shift which are undetermined Lagrange multipliers. The dynamical ...
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Gauge freedom in Lagrangian corresponds to canonical transformation of Hamiltonian

I want to show that the gauge transformation $$L(q,\dot{q},t)\mapsto L^\prime(q,\dot{q},t):=L(q,\dot{q},t)+\frac{d}{dt}f(q, t)$$ corresponds to a canonical transformation of the Hamiltonian $H(p, q, ...
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Poisson Brackets and Angular Momentum using Poisson Bracket algebra

I've been trying to prove the identity for $\left\{L_i,L_j\right\}$ using only the algebra for the Poisson Brackets (of course I could do it by the definition with the derivatives and such, but where'...
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Hamiltonian formulation of the geodesic equation [duplicate]

I am using the Hamilton's formulation of the geodesic equation in order to obtain geodesic equations in terms of the particle's coordinates and the conjugate momenta. Now, I am figuring out a way to ...
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25 views

Cauchy problem for Hamilton-Jacobi equation

In Arnol'd V.I, "Mathematical methods of classical mechanics" p.257, I was asked to find a solution for the Cauchy problem $$H=\frac{p^2}{2},\ \ \ S_0=\frac{q^2}{2}$$ of the Hamilton-Jacobi equation ...
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Asymptotic behavior of canonical perturbation theory for the classic anharmonic oscillator

What do we know about the asymptotic behavior of the perturbative expansion for the classical anharmonic oscillator? The Hamiltonian is $$ H = \frac{p^2}{2m}+\frac{1}{2}m\omega_0^2 q^2 +\mu q^4 $$ ...
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Problem on deriving canonical transformation condition

I'm trying to compute how a canonical transformation should be, given that preserve the symplectic form and trying to recover the condition on the Poisson Bracket. I then start with $$\omega=\stackrel{...
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Difference between conserved quantities and constants of motion?

In Hamiltonian mechanics, consider extended phase space, the trajectory followed by a particle in that space is formed by an intersection of different 2n dimensional surfaces, all of these surfaces ...
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Hamilton-Jacobi theory vs Hamiltonian formalism

I'm writing some notes on Hamilton-Jacobi Theory and I'd like to find an example of a system that is quite difficult to integrate in the usual Hamiltonian formalism, but quite easy in the Hamilton-...
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Hamiltonian Structure of Chern Simons Electrodynamics

I am reading the review paper "Aspects of Chern-Simons Theory" by Gerald Dunne https://arxiv.org/abs/hep-th/9902115 Starting from p. 17, Dunne works on the Hamiltonian structure of the CS ...
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Expansion to show $g$ is conserved if $H$ is invariant

On Shankar QM page 99 it says that If $H$ is invariant under the following infinitesimal transformation $$q_i\rightarrow\bar{q_i}=q_i +\epsilon\frac{\partial{g}}{\partial{p_i}}$$ $$p_i\...
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Application of Darboux's theorem to magnetic field line flows

In this famous paper by Cary and Littlejohn on noncanonical Hamiltonian mechanics and its application to magnetic field line flow, they claim that as a result of Darboux's theorem, it is always ...
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Calculus of Variations. Finding the extremals of a perturbed Lagrangian [closed]

Im trying to solve the following problem: Approximate with an error of $O(\epsilon ^3)$ the extremals of the Lagrangian $$L(y,y',x) = y^2 + (y')^2 - 2y \sin(x) + \epsilon y^3$$ with $y(0)=1$ ...
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Boundary conditions for calculus of variations in phase space and under canonical transformations

This might be a stupid question, but I just don't get it. In Hamiltonian mechanics when examining conditions for a $(\boldsymbol{q},\boldsymbol{p})\rightarrow(\boldsymbol{Q},\boldsymbol{P})$ ...
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Why does the Hamiltonian represent something different after plugging in the solution?

so I am beginning to learn Hamiltonian mechanics. We have learned that the Hamiltonian is a function of q, p, and t. Once we have a Hamiltonian, we can use the Hamiltonian equations to derive the ...
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How can momentum and position be combined into a phase space when they have different units?

Elaboration of the question: What is the geometrical interpretation of units? As in, a unit of length is a choice of scaling of the coordinate systems i.e. it is a choice of diffeomorphism, but then ...
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Self-modifying Hamiltonians (Lagrangians) and emerging intelligence? [closed]

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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37 views

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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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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170 views

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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73 views

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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87 views

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 ...