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 formalism relies on a "Hamiltonian" instead of a Lagrangian, which differs from the Lagrangian through a Legendre transformation.

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How does a Lagrangian with delta potential transform to a Hamiltonian?

Suppose the Lagrangian was given as: $$L = \frac{1}{2}\int_{-\infty}^{\infty} \underbrace{\left(\dot{A(}z)^2-A(z)^2\right)+\left(\dot{Q(}z)^2\delta(z)-Q(z)^2\delta(z)\right)+2\dot{Q(}z)\cdot A(z) \...
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What is the Lagrangian for FRW universe?

A straightforward calculation to the Lagrangian for FRW as $\mathscr{L}=\sqrt{-g}R$ gets me the following result: $$\mathscr{L}\sim a^3\big(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^2+\frac{k}{a^2} \big)...
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Legendre transform and coordinate system independence

I'm self-learning analytical mechanics. Consider a classical mechanical system. Even if it's clear to me that via (the usual) Legendre transform we can get a unique Hamiltonian function from a ...
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81 views

How is it possible to have four types of generating functions?

Since the Hamilton's equations of motion remain unchanged in form under a canonical transformation $(q,p)\to (Q,P)$, the Lagrangians must differ by a total time derivative of a function of $q,t$. In ...
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Why does the Hamiltonian of a Lagrangian that consists of only a coupled term become zero?

Let's say our Lagrangian looks something like this: $$L = \int dz\, Q\cdot \dot{A},\tag{1}$$ where $Q$ and $A$ are two generalized coordinates and $\dot{Q}$ and $\dot{A}$ would be the respective ...
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Justification of dropping term in Hamiltonian and expectation Values

While reading Sakurai's Modern QM, I was stuck at the point where he explains the absorption and emission of light quanta in atoms. He proceeds with Hamiltonian: $$H= p^2/2m + e\phi(x) -e/mc A\cdot p$...
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Book recommendation for relativistic classical mechanics

I need some good resource recommendations for the relativistic hamiltonian mechanics under special theory of relativity, with a good discussion on relativistic Hamilton-Jacobi formulation.
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Derivative with respect to vector

How in Lagrangian and Hamiltonian mechanics we take derivatives with respect to velocity and momentum respectively if they are vectors? Can we take derivative with respect to a vector?
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PDE from Hamiltonian density

For the wave equation Hamiltonian density is $2H=\phi_t^2+\phi_x^2$ while the Lagrangian density is $2L=\phi_t^2-\phi_x^2$. I can easily compute the pde from the Lagrangian density but how does one do ...
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Angular momentum and rotation symmetry

In my book, it is written that for any vector $\mathbf{v}$, we have $$\{\mathbf{v},\mathbf{L}\cdot \mathbf{n}\}=\mathbf{n}\times\mathbf{v}.\tag{1}$$ For me it is absurd... For example, if we take $\...
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Angular momentum of a cylindrical system in general relativity

The definition of energy and enegry flux of a cylindrical symmetric system in general relativity is given by Kip Throne in Phys. Rev. 138, B251 and generalized by Chandrasekhar in Proc. Roy. Soc. Lond....
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Liouville theorem and the ergodic assumption

I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways: Differential statement: The probability distribution $\...
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What does the Ostrogradsky instability have to do with stability?

Ostrogradsky's instability theorem says that under some conditions, a system governed by a Lagrangian which depends on time derivatives beyond the first is "unstable". In the proof, one computes the ...
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Open problems in Hamiltonian Dynamics and Symplectic Geometry [closed]

I'm starting an undergraduate research on Mathematical-Physics and the topic I chose is Hamiltonian mechanics (and its formalism, Symplectic Geometry). I'm looking for some interesting open problems ...
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Euler-Lagrange and Hamilton equations for $p$-forms [closed]

How are Euler-Lagrange equations and Hamilton equations modified (including higher order lagrangians) takinto into account $p$-forms? I think it should also depend on the $D$-dimensional target ...
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What is a Hamiltonian of a System?

What is a Hamiltonian of a System? When learning about Hamiltonian for first time it is an object introduced as Legendre Dual Transform of Lagrangian of the same system. And we learn further that it ...
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Choosing initial condition for Hamilton-Jacobi PDE from initial $x$ and $p$

For separable solutions to Hamilton-Jacobi PDE (say in 2D), we treat the Hamilton's principal function $S$ as $$S= W(x) + W(y) - E*t$$ and treat the separate parts as constants and find $W(x)$, $W(y)$...
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The Killing vector at the bifurcation surface of a stationary black hole

In the paper Black Hole Entropy is Noether Charge, Wald related the black hole entropy to the Neother charge using the covariant phase space formalism. In proving this relation, Wald noticed that on ...
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Resonant Hamiltonian Mechanics

My question is regarding applying averaging theory to a perturbed Hamiltonian. Now, my Hamiltonian is of the form $$H=H_0 + R(q_i,p_i)$$ Where R is the disturbing potential which is a function of the ...
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Prove that a transformation is canonical by using $\mathbb{M}^T\cdot \mathbb{J}\cdot \mathbb{M}$ [closed]

So, I was given the following problem to solve: A system with two degrees of freedom is described by the following hamiltonian \begin{equation} H=p_1^2+p_2^2+\frac{1}{2}(q_1-q_2)^2+\frac{1}{8}(...
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Hamiltonian flows and Heisenberg picture of Quantum Mechanics

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial) In particular what I know is that: ...
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Compute the Legendre transform for a singular Lagrangian

I'm given the lagrangian: $$ L(q,\dot{q}) = \frac{1}{2}(\dot{q_1}^2+\dot{q_2}^2+2\dot{q_1}\dot{q_2})-\frac{k}{2}(q_1^4+q_2^4). $$ I have to compute the Legendre transformation associated to it. The ...
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Ehrenfest theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
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Kinetic energy always time independent?! Where is my mistake? [closed]

I have some problems understanding the Lagrangian and the Hamiltonian formalism. Those can be condensed in the following "derivation" of $\frac{\partial T}{\partial t} = 0$ from the equation $\frac{\...
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Asymmetry in Hamilton Equations

I noticed that in deriving Hamilton equations from the total deriveative of the Hamiltonian with respect to time, for the first equation $$\frac{dx_k}{dt}=\partial_{p_k}H$$ we do not need Lagrange's ...
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Goldstein expression for the Lagrangian

I was looking for help in order to proove 2 relations that Goldstein has put in his book. $$ L(q, \dot{q}, t)=L_{0}(q, t)+\tilde{\mathbf{\dot{q}}} \mathbf{a}+\frac{1}{2} \tilde{\boldsymbol{\dot{q}}} \...
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How to derive the Hamilton-Jacobi equation for the area of a minimal surface on a Riemannian manifold?

The action for a string in this background $$G_{IJ}\tag{1}$$ can be written as the Nambu-Goto action $$S_{NG}=\int d\sigma^1d\sigma^2\sqrt{g}\quad\quad\Rightarrow\quad\mathcal{L}=\sqrt{g}\tag{2}$$ ...
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Is there always a canonical transformation such that the new Hamiltonian only depends on the new momenta?

Given the Hamiltonian $H(x,p)$ of a system. Is there always a coordinate transformation such that the new Hamiltonian is $K(x',p')=K(p')$?
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Hamilton's equations for Dirac Hamiltonian [duplicate]

The Dirac Lagrangian $$\mathcal{L} = i\bar{\psi}\gamma^{\mu}\partial_\mu \psi - m \bar{\psi}\psi$$ gives a Hamiltonian $$\mathcal{H}(\Pi,\bar{\Pi},\psi,\bar{\psi})=\Pi \dot{\psi}-\mathcal{L}=-\bar{\...
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Why are only the Lagrangian and Hamiltonian used in mechanics? [duplicate]

Why is it that we have a closed set of four functions, connected by Legendre transforms, in thermodynamics but nobody ever mentions but two of the corresponding functions in mechanics? I've read that ...
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Couple of non-interacting, non-integrable Hamiltonian systems

I have two Hamiltonians, $H(p_1, q_1, \dots, p_n, q_n)$ and $H'(p_{n+1}, q_{n+1}, \dots, p_m, q_m)$. They are non-interacting, indeed they have different variables. Moreover, I know that they are both ...
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Rigourous formalism of Hamiltonian mechanics on Manifolds

I'm looking for books / articles that provide rigorous formulations of Hamiltonian mechanics on Manifolds. I found the book "Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds" [1]...
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Legendre transformation and correspondance between Noether charges and quasi-symmetries

I have been trying to understand the Legendre transformation (in mechanics, in the hyperregular case: when the Legendre transformation is one-to-one) and the correspondence between symmetry $\to$ ...
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What happens to the time evolution equations in canonical quantum gravity?

Many expositions on canonical quantum gravity start from a 3+1 type formalism, where spacetime is foliated along the time dimension. The Einstein equations then decompose into constraint equations on ...
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Do canonical transformations form a group?

In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical ...
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123 views

Liouville's integrability theorem: action-angle variables

For classical dynamical systems, let $I_{\alpha}$ stand for independent constants of motion which commute with each other. 'Remark 11.12' on pg 443 of Fasano-Marmi's 'Analytical Mechanics' suggest ...
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A paradox about canonical transform preserving Poisson bracket?

Let $q,p$ denote the position and momentum. Consider a transform generated by $g$: $q' = q + \epsilon \{q,g\}---(1a)$ $p' = p + \epsilon \{p,g\}---(1b)$ Then: $\{q',p'\} = \{q,p\}+o(\epsilon^2)+\...
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ADM formulation of GR derivative on the 3-metric

In the ADM formalism where the projector is given by ${P^\mu}_\alpha={\delta^\mu}_\alpha+n^\mu{n}_\alpha$ and $n^\alpha$ is a future pointing normal vector to the constant time hypersurface $\Sigma$. ...
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Derivation of Hamilton-Jacobi equation

I am trying my own way of deriving the Hamilton Jacobi equation $$\frac{\partial S}{\partial t} = -H \tag{1}$$ through direct variation. I think the difficulty of doing this is that the upper limit ...
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Hubbard Hamiltonian mean field decomposition in real and momentum space

Im working through some lecture notes on quantum field theory, and it gives the mean field hamiltonian in real space in terms of the spin operator: $$H_{int}^{MF}=\frac{3}{8U}\sum_{r_j}\vec{(M}(r_j))^...
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Itzykson & Zuber: Conjugate momentum sign

I can't give myself peace on a confusion about the signs. I'm studying on Claude Itzykson & Jean-Bernard Zuber, Quantum Field Theory, Dover Publications. Metric convention $g_{\mu\nu}=diag(1,-1,-1,...
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Determining Whether a Given Hamiltonian is Conserved

in simple terms I'm looking to understand how we can tell whether a given Hamiltonian (or one that we've deduced) is conserved or not? I've tried looking at other similar questions but am not sure I ...
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The WKB approximation and the Cotangent bundle

When we say (see pag. 9 of Lectures on the Geometry of Quantization) that the image of the differential of the phase function lies in the level set of the classical Hamiltonian is it simply ...
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External force as defined in Landau's statistical mechanics

I was reading Landau's volume 5 on statistical mechanics and there in 5th Russian edition page 62 he mentions the following (translated very roughly) that assume we have Hamiltonian system which ...
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Can we do better than the Hamiltonian formalism? [closed]

I have been reading Sakurai's Modern Quantum Mechanics and I notice that every proposition that book has depends primarily on the Hamiltonian formalism of classical mechanics. Even the time evolution ...
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A mathematically illogical argument in the derivation of Hamilton's equation in Goldstein

In the book of Goldstein, at page 337, while deriving the Hamilton's equations (canonical equations), he argues that The canonical momentum was defined in Eq. (2.44) as $p_i = \partial L / \partial ...
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Theta-dependence of massive Schwinger model

I've read in Coleman's paper on the massive Schwinger model (and in other papers on the same topic, like this one) that the model's Hamiltonian contains a topological $\theta$-term. However, if I ...
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Hamiltonian Form Under Canonical Transformation

Let us consider a restricted canonical transformation such as $$ Q_{i} = Q_{i}(q,p) $$ $$ P_{i} = P_{i}(q,p) $$ Goldstein states that 'the Hamiltonian function does not change in such a ...
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Action variable integral

I am solving an action angle variable problem and I'm stuck at the point where I have the following expression for the integral $$ I = \frac{b\sqrt{mE}}{\pi} \int_\theta^{2\pi-\theta}\sqrt{(1-cos \...
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1answer
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Finding period from action-angle variable in one dimensional potential [closed]

I want to calculate the period from the action-angle variable for a particle in a one dimensional potential $V = V_0 \tan^2(q \pi/2a)$. After doing some algebra I get $$I = \frac{\sqrt{2mE}}{2\pi} \...

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