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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Lapse and shift in ADM decomposition

Poisson in Relativist's Toolkit and also other authors in various papers state explicitly that after one does the 3+1 decomposition, the lapse and shift $N$ and $N^a$ are non-dynamical variables, and ...
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Quantization of a free field: Klein-Gordon case

I am a beginner and reading this course text on QFT. The author first introduces the KG equation: $$\partial_\mu\partial^{\mu}\phi+m^2\phi=0$$ [with Minkowski signature $(+,-,-,-)$]. Then the ...
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Why does choosing a time break covariance?

I'm reading that in EM theory, in hamiltonian formalism, we choose a specific reference frame with a specific time, and that this breaks covariance. Why? Surely it's simple because it's just stated ...
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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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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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Hamiltonian for forced systems

I am trying to learn Hamiltonian mechanics. While many textbooks treat closed systems, I have a hard time finding references for forced systems. For example, if I consider simple systems of masses ...
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871 views

Hamiltonian matrix off diagonal elements?

I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I ...
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Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?

I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude: $$F(r, t) = \frac{k}{r^2}e^{-at}$$ ...
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Ordering Ambiguity in Quantum Hamiltonian

While dealing with General Sigma models (See e.g. Ref. 1) $$\tag{10.67} S ~=~ \frac{1}{2}\int \! dt ~g_{ij}(X) \dot{X^i} \dot{X^j}, $$ where the Riemann metric can be expanded as, $$\tag{10.68} ...
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Elimination of velocities from momenta equations for singular Lagrangian

this doubt is related to Generalized Hamiltonian Dynamics paper by Dirac. Consider the set of $n$ equations : $p_i$ = $∂L/∂v_i$, (where $v_i$ is $q_i$(dot) = $dq_i/dt$, or time derivative of ...
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Hamilton's equations in terms of initial conditions

I'm trying to understand the way that Hamilton's equations have been written in this paper. It looks very similar to the usual vector/matrix form of Hamilton's equations, but there is a difference. ...
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How to formulate variational principles (Lagrangian/Hamiltonian) for nonlinear, dissipative or initial value problems?

Although this questions is very much math related, I posted it in Physics since it is related to variational (Lagrangian/Hamiltonian) principles for dynamical systems. If I should migrate this ...
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In a rigid rotor, are there “elegant” orientation coordinates that are conjugate to angular momenta?

I just was looking at the big bag-of-math wikipedia article on rigid rotors, and the section on the Hamiltonian form bugs me a bit since they are using Euler angles to represent the orientation. As a ...
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Hamiltonian for a Lagrangian with coupling

I am dealing with the following Lagrangian density $$\mathscr{L}_{em}= -\frac{1}{2}\rho\omega^2 u^2 +\frac{1}{2}\nabla u:\Sigma :\nabla ...
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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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Examples of Weyl transforms of nontrivial operators

I've been able to find examples of Weyl transforms of operators like $\hat{x}$,$\hat{p}$, and $\hat{1}$, but not anything more complicated. Are there derivations of the Weyl transforms of more ...
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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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Classical Mechanics & Coordinates [closed]

What is the meaning generalised coordinates in Classical Mechanics? How is Lagrangian formalism different from Hamiltonian formalism? How are they related to Hamilton's Principle? How are they ...
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Is symplectic form in Hamiltonian mechanics a physical quantity?

Is symplectic form $dp_i \wedge dq_i$ in Hamiltonian mechanics a physical quantity? It feels to me to be something different than say energy, momentum or mass. Like just certain structure. The real ...
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Is Hamilton-Jacobi equation valid for only conserved systems?

From derivation of Hamilton-Jacobi (HJ) equation one can see that it is only applicable for conserved systems, but from some books and Wikipedia one reads the HJ equation as ...
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When Hamiltonian and the total energy are the same

In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?
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Abstract, generic derivations of energy

How generic can be derivation of energy? In a system with gravity and masses – it is potential energy and kinetic energy. What if a constraint would be specified that no mass and velocity should be ...
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Is there useful information about normal modes/frequencies in the Hamiltonian matrix?

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 the flux in ...
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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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Hamiltonian Operator Interpretation of Quantum Anomaly

We can see the definition of quantum anomaly in terms of Lagrangian path integral formulation. What is the definition of quantum anomaly in terms of Hamiltonian operator approach or even more directly ...
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Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: ...
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Do primary first class constraints change the electric field in the Hamiltonian form of Maxwell's theory?

In my understanding of Dirac's theory of constrained Hamiltonians, the primary (and also the secondary) first class constraints are generators of canonical transformations that do not change the ...
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The Hamiltonian for clocks?

I am rather a theoretician and looking for a formalism to represent biological clocks by Hermitian operators. The simplest thought model I am looking for is a formal representation of a clock (for ...
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Poisson brackets: prove that they are canonical invariants

EDIT: I haven't forgotten to accept answer, the question is still open.. I need a clarification about Poisson brackets. I'm studying on Goldstein's Classical Mechanics (1 ed.). Goldstein proves ...
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Hamiltonian from a Lagrangian with constraints?

Let's say I have the Lagrangian: $$L=T-V.$$ Along with the constraint that $$f\equiv f(\vec q,t)=0.$$ We can then write: $$L'=T-V+\lambda f. $$ What is my Hamiltonian now? Is it $$H'=\dot q_i p_i ...
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EQUAL TIME commutation relations

Why is equal time commutation relation used in canonical quantization of free fields?
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Schroedinger equation. Why Potential energy instead of Force?

What is the reason Schroedinger equation is quoted in terms of potential energy instead of force?
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Why is it important that Hamilton's equations have the four symplectic properties and what do they mean?

The symplectic properties are: time invariance conservation of energy the element of phase space volume is invariant to coordinate transformations the volume the phase space element is invariant ...
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Hamiltonian mechanics really useful for numerical integration? Lagrangian can become 1st-order

(I'm talking about the classical mechanics.) Many texts say that Euler-Lagrange equations are difficult to treat numerically because they are second-order ODEs, ${f_i(\boldsymbol{q, \dot{q}, ...
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Bertrand's theorem

I found in Goldstein's Classical Mechanics that the condition for closed orbits is given by $\frac{d^2 V_{eff}}{dr^2}>0$.(bertrand's theorem). Can somebody explain to me, how this inequality is ...
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Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
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conservation of volume in phase space

I was reading through a proof of Liouville's theorem on conservation of volume in phase space from David Tong's lecture notes (Chapter 4: "Hamiltonian formalism") and on page 89 it says that ...
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Why aren't classical phase space distribution functions always delta functions?

The phase space distribution function (or phase space density) is supposed to be the probability density of finding a particle around a given phase space point. But, classically, through Hamilton's ...
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Schrödinger evolution for a Klein-Gordon equation

I have a problem with the transition from quantum relativistic wave equations (specifically Klein-Gordon equation) to QFT, since a lot of assumptions seem implicit. For example I have a problem with ...
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Conservation of Hamiltonian vs Conservation of Energy

What is the difference between conservation of the Hamiltonian and conservation of energy?
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Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

The question is about: (1) whether giving a Lagrangian is sufficient enough to (uniquely) well-define a Hamiltonianian quantum theory with a Hilbert space? The answer should be Yes, or No. If ...
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Probability density in Hamiltonian Mechanics

I am currently studying Liouville's theorem compare wikipedia and there this mysterious probability density $\rho$ appears and I was wondering how one can determine this quantity analytically for a ...
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Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0 = ...
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Canonical transformation generated by hamiltonian?

Someone told me that, in a hamiltonian system, the hamilonian function is the generating function of the canonical transformation given by time translation. However, this statement doesn't make any ...
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How is a Hamiltonian constructed from a Lagrangian with a Legendre transform

many textbooks tell me that Hamiltonians are constructed from Lagrangians like $$L=L(q,\dot{q})$$ with a Legendre transformation to obtain the Hamiltonian as $$H=\dot{q}\frac{\partial L}{\partial ...
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Hamiltonian of polymer chain

I'm reading up on classical mechanics. In my book there is an example of a simple classical polymer model, which consists of N point particles that are connected by nearest neighbor harmonic ...
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521 views

ADM Hamiltonian formalism and Quantum gravity

is there a Hamiltonian reformultion of gravity ?=? if so if we use the usual Quantization scheme we can not we quantizy the gravity ?? in terms of a Gauge Theory with the potential $ A_{\mu}^{i} $ ...
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Expectation of a commutation relation

Is there any significance to: $\langle[H,\hat{O}]\rangle =0$ (which can easily be shown) where $H$ is the Hamiltonian, $\hat{O}$ is an arbitrary operator? Thanks.
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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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Solution of motion in hamiltonian formalism

I have these canonical equations: $$\dot p = - \alpha pq$$ $$ \dot q =\frac{1}{2} \alpha q^2$$ I have to find $q(t)$ and p$(t)$, considering initial conditions $p_0$ and $q_0$. I thought to simply ...