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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Can I use Hamilton-Jacobi equation here?
Suppose the Hamiltonian does not have a second impulse in it:
$$H = \frac{1}{2}(q_1^2+p_1^2)F(q_2, t).$$
Can I use Hamilton-Jacobi equation here or not? I used it as is and got the same answer, as if ...
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Drawing phase space diagrams from Hamilton's equations of motion
Consider a particle moving in the $(x,y)$-plane. Its motion can be described by the following Hamilton's equations of motion:
$$\dot x (t)=p_x+y$$
$$\dot p_x(t)=0$$
$$\dot y(t)=p_y$$
$$\dot p_y(t)=-y-...
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Infinitesimal transformation and weight of canonical momentum for the massless Klein-Gordon theory
Consider the massless Klein-Gordon theory in $3+1$ dimensions. Its Hamiltonian is
\begin{align}
H=\int d^3x\,\frac{1}{2}\left(\pi^2+\partial^i\phi \partial_i\phi\right)\,.
\end{align}
It is known ...
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Shadowing Lemma and Numerical Integration
Shadowing lemma tells that any pseudo-trajectory (numerically integrated trajectory) from some initial condition $x_0$ is the exact trajectory of a different initial condition $x_1$.
Q: Is this ...
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Invariance of the action under a symmetry of 2D isotropic harmonic oscillator
I have a question on the invariance of the action under symmetry transformation.
As the simplest example, here I consider two dimensional Harmonic oscillator. After some rescaling, the Hamiltonian can ...
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What is the meaning of this complex derivative with respect to a wave function?
In quantum optimal control papers such as (Loading a Bose-Einstein Condensate onto an Optical Lattice, https://arxiv.org/abs/cond-mat/0209195) and (Introduction to the Pontryagin Maximum
Principle for ...
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How to transform generalised (polar) coordinates into cartesian coordinates?
I have a set of observations $D=(q(t_i), p(t_i))$ for $i=1,...,n_{data}$, where $q(t_i), p(t_i) \in R^n$. It is known that the $(q(t_i), p(t_i))$ represent angles and angular momenta of a mechanical ...
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Symmetry and integrability in classical Hamiltonian
I am trying to understand the behaviour of an Hamiltonian system I'm simulating. I will give a quick context setting. The system is defined as
$$
\mathcal{H}(\mathbf{z};\mathbf{z}^*) = \sum_{i=1}^{M}...
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Physical significance of a constant of motion
In problem 21 of Chapter 9 of Goldstein's "Classical Mechanics," 3rd edition, it is given that if the Hamiltonian $H(q, p, t) $ satisfies the scaling condition
$$H (q\lambda, p/\lambda, t\...
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Canonical ensemble, ergodicity and Liouville’s equation
I understand that in Statistical Mechanics Liouville’s equation applies to the probability density of ensembles where microstates’ trajectories are governed by Hamiltonian dynamics. However I’m ...
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Classification of Hamiltonians having the same set of solutions
Assume that we are given $2n$ the functions $q_k(t),p_k(t);k=1,2...,n$ all "sufficiently" smooth and invertible in a finite interval $t\in [0,t_{max}]$. My question is what class of ...
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Problem with Dirac-Bergmann algorithm [closed]
This is from Lemos, Analytical Mechanics, Problem 8.24.
Consider the Lagrangian
$$L = (\dot{y}- z)(\dot{x} - y).$$
I found the Lagrangian equations of motion to be
$$\ddot{y} = \dot{z},$$
$$\ddot{x} = ...
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Solution of the classical 2D quartic oscillator
Consider a classical oscillator in two spatial dimensions with Hamiltonian
$$H=\frac{1}{2}\vec{p}^2 + \frac{1}{2}\omega r^2 + ar^4.$$
This system has two conserved quantities, the energy and angular ...
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How to do classical perturbation theory with Poisson brackets
I need some help computing the behavior of variables $c_i$ in a classical system with Hamiltonian $H$ using perturbation theory. There are as many $c_i$ variables as there are degrees of freedom. I'd ...
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Hamilton's characteristic function, wave-particle duality and constant-action surfaces
So, I'm currently doing some research about the way in which classical physics connects to quantum physics, and I came across the Hamilton-Jacobi equation, and the implications of Hamilton's ...
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Quantum mechanics with a classical Chern-Simons term
In this post, quantum mechanics falls under what is traditionally called "first quantization". This is in contrast to quantum field theory which traditionally falls under "second ...
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Preservation of exact equations of motion in time-dependent perturbation theory for the Hamilton-Jacobi equations
From the Hamilton-Jacobi formalism the solution for the unperturbed hamiltonian $H_0$ has a generating function $S(q,\alpha,t)$ such that
$$K_0 = H_0(q, \frac{\partial S}{\partial q},t) + \frac{\...
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Problem with asymptotic behavior of metric
In the Hamiltonian description of asymptotically flat spacetimes, the metric should deviate at infinity from the flat metric by terms of order 1/r
$$g_{ij} = \delta_{ij} + \frac{\overline{h}_{ij}}{r} +...
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Can we reduce the entropy of a system arbitrarily by sending out a photon after arbitrary delay?
I am not asking whether this is practically feasible given current technology. Rather I'm asking whether it is possible in principle given current laws of physics.
Suppose we have a system with a ...
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How does $\dot{q}_i p_i - H = \dot{Q}_i P_i - K + \frac{d}{dt}F$ will give the same EL and EoM for corresponding coords? [duplicate]
How does $\dot{q}_i p_i - H = \dot{Q}_i P_i - K + \frac{d}{dt}F$ give the same Euler-Lagrange equations and Equations of motion (EoM) for corresponding coordinates and allow us to determine a ...
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Exact solution of non-integrable Hamiltonian system?
Following this note (Introduction to Integrability – FS 2013,
Lecturers: Dr. Marius de Leeuw, Dr. Constantin Candu), a system is said to be solved by quadratures if the solution can be constructed by
•...
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Are the canonical momentum and the corresponding generalized coordinates independent? [duplicate]
I know that for a lagranian $L=L(q_i, \dot{q_i},t)$ the canonical momentum is given by $p_i = \frac{\partial L}{\partial \dot{q_i}}$. The lagrangian being a function of the generalized coordinate, I ...
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Deriving Lagrangian mechanics from Hamiltonian mechanics [duplicate]
I have just finished reading Susskind's Theoretical Minimum. I think I got it well, even found some mistakes in few equations🙈 The book starts from Newtonian Mechanics, then derives Lagrangian ...
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Lie algebra of Hamiltonian functions
Could anyone explain to me how to obtain the Corollaries 2 and 3 in the following paragraph taken by Arnold's book?
The Lie algebra of hamiltonian functions
The hamiltonian vector fields on a ...
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Hamiltonian is unbounded from below in only one coordinate system
I'm studying a complex scalar field theory in a spatially flat FLRW background. Using the standard conformal time metric $$ds^2 = dt^2 - a^2(dr^2 - r^2 d\Omega_2^2)$$ (where $a$ is the scale factor), ...
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Canonical Variables in Dirac Spinor Field Theory [duplicate]
In S.Weinberg [QFT V1][1] section 7.1, in eq (7.1.15) and (7.1.16), he states that in order to be consistent with the previous-derived anti-commutator relation, we should take $\psi_{\text{n}}$ and $\...
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Validity of $\mbox{d}H/\mbox{d}t=\partial H/\partial t$ for dissipative systems
It' well known that in Hamiltonian formalism one has
$$
\frac{\mbox{d}H}{\mbox{d}t} = \frac{\partial H}{\partial t}.\tag{*}
$$
One proof can be found here. Therefore, the total change of energy during ...
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Relation between energy and time
I would like help in understanding something that has been causing me a lot of trouble recently: Why is energy always related to time in physics?
Examples include the 4-momentum, the energy-time ...
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Are Landau-Lifshitz equations equivalent to Hamilton's equations for classical spins?
Let $\boldsymbol{s}_1$ describe a "classical spin", i.e. a point on the surface of a unit sphere embedded in $\mathbb{R}^3$. It can be parametrized, for example, as
$$ \boldsymbol{s}_1 = \...
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ADM Formalism for the Effective String Theory
We will consider the following effective action of string theory in leading order of $\alpha'$:
$$S=\frac{1}{2\kappa^2_0}\int d^{D}X\sqrt{-G}e^{-2\Phi}\left[R-2\Lambda-\frac{1}{12}H_{\mu\nu\lambda}H^{\...
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Is it possible to understand in simple terms what a Symplectic Structure is?
I would like to understand what a Symplectic Structure is, and its implications in Classical Mechanics (Phase Space), but in pre-grade terms (If that could be possible). I have not taken any ...
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Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
In Lagrangian mechanics the momentum is defined as:
$$p=\frac{\partial \mathcal{L}}{\partial \dot q}$$
Also we can define it as:
$$p=\frac{\partial S}{\partial q}$$
where $S$ is Hamilton's principal ...
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Is it possible there can be a non-Fourier model of string vibration? Is there an exact solution?
I am looking for a model of string vibration that does not assume the string is Fourier. Is there a Hamiltonian?
The equation of motion must be a function of length and tension, not time, and it must ...
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Volume preserving transformation in the Circular Restricted Three-Body problem
the Lagrangian of the planar circular restricted three-body problem in the rotating coordinate frame is:
$$\mathcal{L}(x,y,\dot{x},\dot{y})=\frac{1}{2}(\dot{x}-\Omega y)^2 + \frac{1}{2}(\dot{y}+\Omega ...
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What is the determinant of the Wheeler-DeWitt metric tensor constructed from spatial metrics in ADM formalism?
The Hamiltonian constraint of General relativity has the following form
\begin{align}
\frac{(2\kappa)}{\sqrt{h}}\left(h_{ac} h_{bd} - \frac{1}{D-1} h_{ab} h_{cd} \right)p^{ab} p^{cd} - \frac{\sqrt{h}}{...
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Action-angle variables for three-dimensional harmonic oscillator using cylindrical coordinates
I am solving problem 19 of ch 10 of Goldstein mechanics. The problem is:
A three-dimensional harmonic oscillator has the force constant k1 in the x- and y- directions and k3 in the z-direction. Using ...
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What does the optical Hamiltonian mean?
So I was trying to demonstrate Snell's law with Hamilton's equations, and when I got the Hamiltonian:
$$H = -\sqrt{n^2-p_{1}^2-p_{2}^2}.$$
I had a question about what this Hamiltonian indicates. I ...
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Canonical commutation relations of quantum fields in null coordinates
To quantize a scalar field, we impose the equal time commutation relations
$$
[\Phi(t,\mathbf{x}),\partial_t\Phi(t,\mathbf{x}')] = i\hbar\delta^{(3)}(\mathbf{x-x'}).
$$
This can also be generalized to ...
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When is the derivative of Hamilton flow respect to initial conditions independent of time?
Consider a Hamiltonian system with coordinates $\Gamma^A=(q^i,p_i)$ and let $X^A(s,\Gamma_0)$ be the Hamiltonian flow (i.e. a solution to Hamilton's equations) parametrized by time $s$ and initial ...
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Is it possible to derive Schrödinger's equation from Hamilton's equations?
Accepting the postulates of quantum mechanics, so promoting the classical dynamical variables to operators with appropriate commutation relations, is it possible to "derive" Schrödinger's ...
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How to get vector field from Poisson brackets?
Steinacker defines the Hamilton vector field as any field s.t.:
$$\{f,g\}=V_f[g].$$
I really can't understand this. The Poisson algebra is closed with respect to Poisson brackets (i.e. $\{\cdot ,\cdot\...
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Thermodynamic diagrams in Hamiltonian mechanics
If we compare the fundamental thermodynamic relation $(1)$ and the Hamilton's principal function $(2)$, than we have two practically identical equations:
$$dU=TdS-pdV \tag1$$
$$dS=pdq-Hdt\tag 2$$
In ...
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Confusing Goldstein Statement about Magnitude of the Lagrangian
On page 345 of Goldstein's Classical Mechanics 3rd Ed., he writes:
...the Hamiltonian is dependent both in magnitude and in functional form upon the initial choice of generalized coordinates. For the ...
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What are the "orbits" generated by a constraint?
I am currently reading the book "A First Course in Loop Quantum Gravity" by Gambini and Pullin. On page 55, they claim that the vanishing of the Poisson bracket between the smeared Gauss law,...
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Classical Hamilton’s equations in quantum mechanics [duplicate]
How can one derive what the position operator is in momentum space for a quantum wave function from the classical Hamilton’s equations?
Similarly, is a concept of an “angular momentum space” ...
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Derivation of Dirac Hamiltonian
In Minkowski spacetime with signature $(-,\;+,\;+,\;...,\;+)$ the Dirac Lagrangian reads
$$
L=\int d^dx\;\mathcal{L}=\int d^dx\;\psi^\dagger\left(i\gamma^0\gamma^\mu\partial_\mu-im\gamma^0\right)\psi.
...
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Where am I going wrong when obtaining the Hamiltonian density for the electromagnetic field?
I'm trying to verify that the Hamiltonian density for the classical electromagnetic field is equal to the energy density. But the electric field is disappearing and only the energy density of the ...
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How does this canonical transformation on a Schwarzschild black hole work?
In this paper "Holography of the Photon Ring" the authors use a canonical transformation in section 2.4 in eqs. (2.52)-(2.55).
It is basically a transformation from spherical coordinates for ...
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How do I formulate a quantum version of Hamiltonian flow/symplectomorphisms in phase space to have a "geometric", quantum version of Noether's theorem
I'm currently exploring how Noether's theorem is formulated in the Hamiltonian formalism. I've found that canonical transformations which conserve volumes in phase space, these isometric deformations ...
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Why can't we treat the Lagrangian as a function of the generalized positions and momenta and vary that? [duplicate]
Some background: In Lagrangian mechanics, to obtain the EL equations, one varies the action (I will be dropping the time dependence since I don't think it's relevant) $$S[q^i(t)] = \int dt \, L(q^i, \...
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