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

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Why is the generating function in the Hamilton–Jacobi equation equal to the action? [duplicate]

The aim in Hamilton Jacobi formalism is to find a canonical transformation that generates a new Hamiltonian $H'$ which is equal to $0$. Therefor we find the equation: $$H(q_1,...,q_n,\frac{\partial F}{...
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Is there a physical connection between Lagrangian mechanics and Hamilton mechanics

I know that the Hamilton's equations results from the Legendre transformation of the Lagrange equation. It's also well-known that the Hamiltonian is equal to the energy of a system if the system is ...
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What is the link between the rotating wave approximation and the algebraic representation of a dynamical system?

In analyzing a system of two coupled oscillators, I noticed a rather interesting correspondence between the so-called "rotating wave approximation" (RWA) for solving differential equations and the ...
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Name of the matrix that appears in matrix form of Hamilton's equations of motion

Consider a harmonic oscillator described by the second order differential equation $$\ddot{\phi} + \omega_0^2 \phi = 0 \, .$$ Defining $v \equiv \dot \phi$ we get two simultaneous equations \begin{...
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Can we also Legendre transform the Lagrangian using $\frac{\partial L}{\partial q}$ instead of $\frac{\partial L}{\partial \dot q}$? [duplicate]

We calculate the Hamiltonian as the Legendre transform of the Lagrangian $$H(q,p,t) = p \dot q - L(q,\dot q,t), $$ where $p$ is the slope function $$p \equiv \frac{\partial L}{\partial \dot q} .$$ ...
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Non-relativistic limit of Hamiltonian for a free particle in general relativity

The Hamiltonian for a particle moving in a gravitational field can be taken as $$\mathcal{H} = \frac12 \sum_{\mu,\nu=0}^3g^{\mu\nu}(x)p_\mu p_\nu\tag{1}$$ as long as the parametrization is affine. ...
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Solving time evolution equations in Hamiltonian formalism

I have 4 time evolution equations and the Hamiltonian $H(X_{1},X_{2},P_{1},P_{2})$ that generates the time evolution depends on 4 canonical coordinates but I would like to solve the differential ...
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Intuition between this construction of the sympletic form for classical fields

In this paper, Wald presents a quite general construction of a sympletic form for classical fields. If I understood (which I might have not, and in that case corrections are highly appreciated), the ...
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Integration of Poisson brackets by integration by parts [closed]

In the context of Statistical Mechanics I have to show that the following integral is zero: $$\int \sum_{i=1}^{3N}(\frac{\partial O}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial O}{\...
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What does conservation of probability mean in Classical Mechanics and why is it true?

In the context of the Liouville equation, regularly the conservation of probability is invoked. (Of course, the overall probability is always conserved but this is a truism and not what is meant here. ...
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What does $\frac{d\rho}{dt} =0$ but $\frac{\partial \rho}{\partial t} \neq 0 $ mean intuitively?

For concreteness, let's say that $\rho(q,p,t)$ describes the probability density in phase space. On a superficial level both $\frac{d\rho}{dt}$ and $\frac{\partial \rho}{\partial t} $ tells us how $\...
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Determinant of ADM metric

I am studying inflation and for the calculation of the bispectrum we are using the ADM formalism where the metric is the following form: $$g_{\mu\nu}=\begin{bmatrix}-N^2+N^iN_i&N_i\\N_i&h_{ij}...
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How does the Hamiltonian change if $L\to L + \frac{dF}{dt}$? [duplicate]

The Hamiltonian is defined as the Legendre transform of the Lagrangian $$H = p\dot{q} -L .$$ In the Lagrangian formalism we are free to add the total derivative of an arbitrary function $F=F(q,t)$ to ...
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What is the “special time dependence” that develops in an Ostrogradskian instability?

I've been reading papers that deal with Lagrangians containing second- and higher- order derivatives of field variables. In this paper in Section 3.1, I found this very interesting quote: The ...
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Derivative of $\nabla\times(\nabla\times A)$ by A

I'm trying to find out how to quantize EM field. It seems like $\vec{A}$ and $\vec{E}$ are it's canonical coordinates. For example: $$\mathfrak{H} = \frac12E^2 + \frac12(\nabla\times A)^2$$ $$H = \int ...
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Why is a single function sufficient to specify a canonical transformation?

Spivak argues at page 577 in his book Physics for Mathematicians: What are the $2n$ relations he is talking about?
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Are all canonical transformations either a point transformation, gauge transformation or a combination of them?

It's regularly argued that in the Hamiltonian formalism, we have more freedom to choose our coordinates and that this is arguably its most important advantage. To quote from two popular textbooks: ...
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Can all canonical transformations be generated using a generating function?

In Classical Mechanics, a gauge transformation is of the form \begin{equation} L \to L' = L + \frac{dF(q,t)}{dt} \, . \end{equation} Any transformation of this kind leaves the Euler-Lagrange equation ...
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Are point transformations necessarily canonical?

A point transformation $ Q(q,t)$ only modifies the location coordinates, while a canonical transformation, in general, also modifies the momentum coordinates $Q(q,p,t)$. In the context of the ...
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Deriving Canonical Transformation from Generating Function using Principle of Stationary Action

In Hamill's "A Student's Guide to Lagrangians and Hamiltonians", section 5.2, the equations for a canonical transformation $(q,p) \to (Q,P)$, induced by the generating function $F(q,Q,t)$ are derived ...
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Symplectic and Euclidean structure invariance

Consider a $2n$ real symplectic space - the usual $\mathbb R^{2n}$. Suppose that the same space could be endowed too with an Euclidean structure, by which the vectors of the symplectic basis are ...
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Simple Hamiltonian in curved space-time

Consider the theory of a free massless scalar field with a non-trivial background metric: $$ \mathcal{L} = -\sqrt{-g} \left( \frac{1}{2}\partial_\mu \phi \partial^\mu\phi\right) .$$ (I prefer the '...
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Invariant density of Harmonic oscillator

In general, dynamical systems described by a pure Hamiltonian can have an infinite number of invariant densities. In fact, each initial state determines exactly a closed path in phase-space and the ...
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Canonical transformations preserve Hamilton's equations. Which transformation preserve the Euler-Lagrange equations?

An important aspect in the Hamiltonian formulation of Classical Mechanics are canonical transformations which provide maps between different sets of canonical coordinates. These canonical coordinates ...
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What's the difference between a generating function and a generator?

Usually in physics we use the notion generator to describe the infinitesimal elements associated with any finite Lie group transformation. But in the context of the Hamiltonian formalism, all authors ...
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Are symmetries necessarily canonical transformations?

A canonical transformation is defined as a transformation such that afterwards Hamilton's equations still hold. It can then be shown that this requirement implies that canonical transformations are ...
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How can we derive from $\{G,H\}=0$ that $G$ generates a transformations which leaves the form of Hamilton's equations unchanged?

In the Hamiltonian formalism, a symmetry is defined as transformation generated by a function $G$ is a symmetry if $$\{G,H\}=0 ,$$ where $H$ denotes the Hamiltonian. On the other hand, a symmetry is ...
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Why are symmetries in phase space generated by functions that leave the Hamiltonian invariant?

Hamilton's equation reads $$ \frac{d}{dt} F = \{ F,H\} \, .$$ In words this means that $H$ acts on $T$ via the natural phase space product (the Poisson bracket) and the result is the correct time ...
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Transformation of ADM parameters under diffeomorphisms

I am trying to prove the invariance of the ADM formalism under (infinitesimal) diffeomorphisms. I have checked Wald and other textbooks on the subject but have been unable to find expressions for how ...
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Hamiltonian ordering ambiguity in quantum cosmology/gravity

I am trying to study several quantum cosmology models. The standard procedure for quantization consists typically in several steps: People write the theory as an action, or Hamiltonian $H(p^i,q^j)$ ...
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How do anomalies affect the field equations of motion?

I find anomalies an extremely unintuitive subject, because they're studied so indirectly. In the standard textbook presentation, one computes an abstract quantity that should be zero classically (say, ...
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Looking for lecture videos that follow Arnold's Mathematical Methods of Classical Mechanics

I'm an undergrad and I'm looking for lecture videos (on youtube and such) that follow this textbook. My course roughly follows it, but glosses over some mathematical details that I feel would be ...
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Calculation of path integral in QFT

I am studing QFT using the text book of Srednicki's. And I am stuck on one of calculations of the integrals in his book. Consider a harmonic oscillator with hamiltonian: We can write the following ...
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Classical mechanics in coadjoint orbits

We know that coadjoint orbits are symplectic manifolds, and they can be used to find unitary representations of Lie groups and stuff, and it's also related to quantization. However, is it true that ...
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Physical meaning of theorem

This is the image of theorem from V.I Arnold's Mathematical method of mechanics. I understood the example given in text. But I want to know what is physical meaning of example? Can anybody help?
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Rigid Body Equations in terms of Body Coordinates by Hamilton's Principle

I sought-for the equations of motion of an unrestrained rigid body. The equations of motion are readily available in the literature, but my concern is to derive them by Hamilton's principle. ...
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Partition function in spherical coordinates

Suppose I write the Hamiltonian/energy of my system in spherical coordinates ($r,\theta,\varphi$) with conjugated momentums($p_r,p_\theta,p_\varphi$). How do I calculate the partition function? If ...
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Integral limits in phase space

If I am calculating the partition function for $H=cp$, ultrarelativistic gas in three dimensions. And by breaking down $d \Gamma$ into $dq$ and $dq$ and further using spherical coordinates I will get $...
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Liouville equation with Dirac delta as probability density

I would lke to know if the probability distribution given by $$\rho(q,p,t)=\delta(q-q(t),p-p(t)) $$ with the initial condition $\rho(t=0)=\delta(q,p), $ where $q(t)$ and $p(t)$ are trajectories ...
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Getting $h_x, h_y, h_z$ Components of Hamiltonian after Gauge Transformation

In Fruchart et al.'s An Introduction to Topological Insulators, the Bloch Hamiltonian for a two-band insulator is given in the general form $ H(k)= $ \begin{bmatrix} h_0+h_z & h_x-i h_y \\ ...
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Goldstein Classical Mechanics equation 8.20 clarification

In Goldstein (Third Edition, Page 339) the equation 8.20 is as follow: $$ H = \dot q_ip_i - L = \dot q_ip_i-[L_0(q_i,t)+L_1(q_i,t)\dot q_k+L_2(q_i,t)\dot q_k\dot q_m].\tag{8.20}$$ Can someone ...
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Are powers of the harmonic oscillator semiclassically exact?

The Duistermaat-Heckman theorem, although too complex for me to completely grasp, states that under some conditions, the partition function for a special class of Hamiltonians is semiclassically exact....
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Poisson bracket rhs coefficients only dependent on one variable

Dirac uses in his work The Conditions for a Quantum Field Theory to be Relativistic the trick that one can always write a localized field quantity $U$, which usually corresponds to the energy density,...
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How much information about a quantum operator is determined by its Poisson bracket Lie algebra?

Hamiltonian quantum mechanics is often built using many ideas from Hamiltonian classical mechanics like the Poisson bracket to determine the commutator between quantum operators, which is appropriate ...
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How does a vanishing $[x, p]$ work with the group theoretical definition of $p \propto \frac{\partial}{\partial x}$?

Thought about this while I was looking at some stuff on quantum-classical correspondence and where precisely the difference between quantum and classical comes from. Usually it's said that the key/...
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Which Hamiltonian systems are intrisically linear?

What physical properties has a dynamical system whose equation of motion are linear? When does it exist a change of coordinates which turn the equation of motions in a linear system? My teacher says ...
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Hamiltonian time-independent, partial derivative always zero?

For conceptual simplicity, let's restrict the discussion to systems with a two-dimensional phase space $\mathcal P$ with generalized coordinates $(q,p)$. Hamiltonian is a function that maps a pair ...
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Hamiltonian of quantum harmonic oscillator with $\psi(x)=\delta(x)$: comparison to classical mechanics

I was just reading the question Why can't $\psi(x)=\delta(x)$ in the case of a harmonic oscillator? The accepted answer says that $\psi(x)=\delta(x)$ is a mathematically valid state, though it's not ...
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Show Solution to Hamilton's Equations are Given by Circular Paths

I am asked to compute Hamilton's equations and check that the solutions to said equations are circular paths, centered at the origin, with angular velocity $\xi \in R^3$. The Hamiltonian is given by $...
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Volume within equi-energetic surface of a classical harmonic oscillator in microcanonical ensemble

$$ V(E) = \int_{H\leq E} d\mu = \int_\Gamma d\mu\, \Theta\bigl(E-H(q,p)\bigr) . $$ To compute the volume within the equinergetic surface in the microcanonical ensamble, we use the formula above, ...