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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Weyl Quantization Integral
I have some doubts when calculating the integral for Weyl Quantization symbol.
If I understand correctly, quantization using the Weyl symbol takes a function in phase space and takes it to an operator ...
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Extrinsic curvature, Gauss equation and Christoffel symbol contribution
This question is in the context of geometry of hypersurfaces and ADM formalism. In a $4$-dimensional manifold, we define a $3$-hypersurface with space-like tangent basis $e_a$, $a=1,2,3$, and a normal ...
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Evolution of volumes in Phase-Space
Liouville's theorem states that the volume occupied by an ensemble does not change as the ensemble evolves. My question regards the volume of the smallest sphere that contains the ensemble. Is there a ...
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Lagrangian equal to Hamiltonian - is it possible?
In one calculation, I got a result where the Hamiltonian equals the Lagrangian, with both values constant.
Eventually I found the error and corrected it, but I began to wonder whether there could even ...
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Question about Hamiltonian [closed]
In this question (1.a), you are asked to find the hamiltonian of the system described above. I understand where the Lagrangian comes from, but I don't understand why the generalized momentum related ...
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Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
In Lagrangian mechanics we have the Euler-Lagrange equations, which are defined as
$$\frac{d}{dt}\Bigg(\frac{\partial L}{\partial \dot{q}_j}\Bigg) - \frac{\partial L}{\partial q_j} = 0,\quad j = 1, \...
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Hamiltonian of the relativistic point particle [duplicate]
It is unclear to me why the Hamiltonian of a relativistic particle is zero. I know that, given a relativistic free particle, I can write the Lagrangian of it in this way: $$\mathcal{L} = mc^2\sqrt{1 - ...
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How does Poisson bracket vanish in this equation?
I am studying the book "Lectures on Quantum Field Theory", by Ashok Das. I am stuck in the last step of equation (10.36) as I will explain below.
Under section 10.2 (Dirac method and Dirac ...
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How to determine which coordinates to use for calculating the Hamiltonian? [closed]
In my classical mechanics course, I was tasked with finding the Hamiltonian of a pendulum of variable length $l$, where $\frac{dl}{dt} = -\alpha$ ($\alpha$ is a constant, so $l = c - \alpha t$.).
I ...
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Calculating Poisson brackets in classical non-relativistic Hamiltonian field theory
Summary of the question:
How can I prove the equal-time Poisson bracket relations for the classical Hamiltonian field theory? I.e
$$[q(x,t),H(t)]_\mathrm{PB}=\dot{q}(x,t)\tag{1}$$ for a field $q$ and ...
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Off-shell symmetry of the action and on-shell restrictions to this symmetry
In this question, it is shown that:
$$L=\frac{1}{2}(v^2_1+v_2^2-q_1^2-q_2^2)$$ and $$H=\frac{1}{2}(p^2_1+p_2^2+q_1^2+q_2^2)$$ both have a symetry group $U(2)$ despite their form that would suggest $O(...
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Dirac procedure for Wheeler De Witt equation
After computing the Hamiltonian constraint and the momentum constraint in general relativity the Hamiltonian constraint is turned into an operator equation and solved in a manner similar to a ...
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Symmetry group of a two-dimensional isotropic harmonic oscillator
The Lagrangian and the Hamiltonian of a two-dimensional isotropic oscillator (with $m=\omega=1$) are
$$L=\frac{1}{2}(v^2_1+v_2^2-q_1^2-q_2^2)\tag{1}$$ and $$H=\frac{1}{2}(p^2_1+p_2^2+q_1^2+q_2^2),\tag{...
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What would make the Legendre transformation interesting, from the graphical point of view?
The term "transformation" is often used in physics and mathematics for functions to denote a different (and useful) way of encoding the information in a function. In this question, I want to ...
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Questions on Symplectic approach to canonical transformations
Reading section 9.4 of Classical Mechanics by Goldstein, I got a question in my mind. That is, it says that for restricted canonical transformation, we have the new Hamiltonian equal to the old one. I ...
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How to explain the independence of coordinates from physics aspect and mathmetics aspect?
When I was studying Classical Mechanics, particularly Lagrangian formulation and Hamiltonian formulation. I always wondering how to understand the meaning of independence of parameters used of ...
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What does it mean for an operator to depend on position or momentum?
While trying to provide an answer to this question, I got confused with something which I think might be the root of the problem. In the paper the OP was reading, the author writes $$\frac{d\hat{A}}{...
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Non-formal expression for the classical propagator
I'm studying classical molecular dynamics and have come across an object called the classical propagator in the following context.
Let $\mathcal{A}(t) = \mathcal{A}(\vec{x}(t))$ be a function on phase-...
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Gauge symmetry and the volume in phase space
Recently, I am reading a paper about the soft theorem and large gauge symmetry which is non-zero in the boundary.
In section 6, the author introduces the covariant phase space method to illuminate why ...
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Perturbations of an integrable system with no resonant tori
Suppose I have a Hamiltonian $H_0$ which is just a collection of $N$ non-interacting harmonic oscillators. Written in action-angle coordinates $(J_i, \theta_i)$ we have $H_0 = \sum_{i=1}^N \omega_i ...
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How are Lie series used as canonical transformations in perturbation theory?
I have a few questions on how to use Lie series as a canonical transformation, which are widely used in perturbation theory (celestial mechanics).
I know that these series are related to a Taylor ...
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How does the Wigner function transform when the transformation is provided in $(x,p)$-form?
In quantum optics, when a transformation of canonical variables $\hat{x},\hat{p}$ is provided, how does the Wigner function change under the transformation? For example, when the transformation is
$$
\...
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Non-symplectic Hamiltonian systems
I'm wondering when the phase space of a Hamiltonian system looses its symplectic structure.
I think it happens when the Hamiltonian $H$ depends on a set of other variables $S_1,...,S_k$ as well as on ...
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What is the Hamiltonian of the Lagrangian $L=-mc^2\sqrt{1-\dfrac{v^2}{c^2}}$?
What is the Hamiltonian of the Lagrangian $$L=-mc^2\sqrt{1-\dfrac{v^2}{c^2}}.$$
So I know the answer is supposed to be $$H=\sqrt{m^2c^4 +p^2c^2}$$ but no matter how hard I try I can't substitute all ...
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Invariance of a volume element in phase space: What does it means?
I have been reading the third edition of Classical Mechanics by Goldstein, in particular, chapter 9 Poisson Brackets and Other canonical invariants. And it is shown that the magnitude of a volume ...
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(A modification to) Jon Pérez Laraudogoita’s "Beautiful Supertask" — What assumptions of Noether's theorem fail?
I am curious about the following (physically unrealizable) scenario involving a supertask described here: https://plato.stanford.edu/entries/spacetime-supertasks/#ClasMechSupe. The original paper is ...
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Equations of motion and infinitesimal canonical transformations
Currently, I'm diving into infinitesimal canonical transformations, with a particular focus on using the infinitesimal change $\epsilon=\delta t$ and $H$ as our generating function. So, in this ...
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Momentum $p = \nabla S$
My book mentions the following equation:
$$p = \nabla S\tag{1.2}$$ where $S$ is the action integral, nabla operator is gradient, $p$ is momentum.
After discussing it with @hft, on here, it turns out ...
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Hamiltonian energy $E=H = -\frac{\partial S}{\partial t}$ [closed]
In most of the cases, the Hamiltonian is equal to the total mechanical energy, which is usually conserved. They are equal and conserved when the potential energy is not dependent on velocity and there ...
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Deriving particle's wave equations [closed]
My book tries to derive two equations. Unfortunately the book is in a language you wouldn't understand, so I will try to translate it as much as I can.
$E = ℏw$
$p = ℏk$
It starts by lagrangian:
$ S ...
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Proof that canonical transformations implies symplectic condition [duplicate]
Goldstein's Classical Mechanics makes the claim (pages 382 to 383) that given coordinates $q,p$, Hamiltonian $H$, and new coordinates $Q(q,p),P(q,p)$, there exists a transformed Hamiltonian $K$ such ...
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What is the physical meaning of the minus sign in the Hamilton equation of momenta?
The Hamilton equations are
$$ \dot{q}_k=\frac{\partial H}{\partial p_k}~~~~-\dot{p}_k=\frac{\partial H}{\partial q_k}~~~~~-\frac{\partial L}{\partial t}=\frac{\partial H}{\partial t}.$$
Does the minus ...
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Canonical equations of motion
The Hamiltonian is obtained as the Legendre transform of the Lagrangian:
\begin{equation}
H(q,p,t)=\sum_i p_i \dot{q_i} - L(q,\dot{q},t)\tag{1}
\end{equation}
If the Hamiltonian is expressed in ...
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Canonical Commutation relations in gravity
The canonical commutation relations in gravity are sometimes written
$$
[\gamma_{ij}(x),\pi^{kl}(y)]=\frac{i\hbar}{2}(\delta_i^k\delta_j^l+\delta_i^l\delta_j^k)\delta^3(x-y),\tag{0}
$$
where $\gamma_{...
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Order $\epsilon^2$ mean rate of change in action variable for adiabatic oscillator
In adiabatic theory of classical mechanics, considering a linear oscillator with a slowly changing frequency $\omega=\omega(t)$, Percival and Richards's nice book (pp. 144-147) discusses how it is ...
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Noether Theorem in Terms of Symplectic Geometry
For the classical (=field theoretical) Noether theorem there exist a well known approach using symplectic geometry. Recall, that in classical terminology the conserved current (a $1+3$-vector time+...
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How Poisson developed Poisson bracket?
In textbooks, they just give Poisson bracket used for restating the Hamiltonian in a powerful and elegant way. But why it is useful, and how Poisson created it?
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Show that transformation mediated by $F$ is canonical
I am trying to solve this problem
If for some phase transformation $(p,q) \rightarrow (p',q')$ it holds $\sum p_i\dot{q_i} - H = \sum \dot{p_i}q_i + p_i\dot{q_i} - \dot{p_i'}q_i' - H' + \frac{d}{dt}F(...
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The right domain for Hamiltonians
This question came to me today, and I am now intrigued about it. For a system described by a Lagrangian $L$, the associated Hamiltonian is its Legendre transform. Suppose we consider a given ...
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Hamiltonian conservation in different sets of generalized coordinates
In Goldstein, it says that the Hamiltonian is dependent, in functional form and magnitude, on the chosen set of generalized coordinates. In one set it might be conserved, but in another it might not. ...
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Relation between old and new Hamiltonian for Canonical Transformations
The question I am asking was asked here but was never given a satisfactory answer and so I will rephrase it and add more detail.
In chapter 9 of Goldstein,when talking about a canonical transformatio $...
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Operators in quantum field theory
I am beginning to learn quantum field theory. I have a beginner level question. Please help me with it.
In quantum field theory, the operators $x$ at each point are demoted to just labels and every ...
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Complex Hamiltonian formalism [duplicate]
If
$x_k=\frac{1}{\sqrt{2}}(q_k+ip_k)$ and $\bar{x_k}=\frac{1}{\sqrt{2}}(q_k-ip_k)$
Show that the Hamilton's equation of motion can be expressed in the form:
$\frac{dx_k}{dt}+i\frac{\partial H}{\...
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Integrals of motion for a rotational symmetric 3D Hamiltonian $H=\frac{{\bf p}^2}{2m}+V(r)$ [closed]
A particle of mass $m$ moves in three dimensions under the action of the conservative force with potential energy $V(r)$. Using the spherical coordinates $r, \theta, \phi$ find the Hamiltonian of the ...
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BRST structure functions in gravity?
In the classical Hamiltonian BRST formalism, there arise structure functions $\Omega^{\beta_1...\beta_n}_{\alpha_1...\alpha_{n+1}}$ ($n\geq0$) --- see https://inspirehep.net/literature/221897 for ...
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Geometric interpretation of constrained dynamical systems
Below are two pictures from Bojowald's book Canonical Gravity. The author tries to present a geometrical picture of a constrained system, however, the description regarding this seems quite scant to ...
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Interpretation of this Hamiltonian
I'm studying a system with the following Hamiltonian $$ H = \frac{1}{2}P^TAP + \frac{1}{2}Q^T B Q$$ where $P,Q$ are canonical variables (4-vectors) and $A,B$ matrices such that $A = A^\dagger$ and $B =...
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Time evolution operator in classical mechanics?
Hamilton's equation can be written in terms of Poisson brackets, as follows:
$$\dot{q} = \{q,H\}$$
$$\dot{p} = \{p,H\}$$
where $H$ is the Hamiltonian of the system. Now, wikipedia says that the ...
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Legendre transformation of the potential in classical mechanics
we can write the differencial of the hamilton using its equations :
$$ d{H(q,p)} = -\dot{p}d{q} + \dot{q}d{p} $$
However we can write :
$$ d(\dot{p}q)=\dot{p}dq+qd\dot{p}$$ and we can replace
$$ -\...
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Hamiltonian function of a system of particles governed by Langevin equation of motion
I have a system of particles which interact among themselves via some pairwise additive potential ( position dependent ) and I am also considering the collision of the particles with background ...
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