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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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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

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 $\...
Ting-Kai Hsu's user avatar
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What is the relationship between phase space and Jacobian in Nakahara Eq.1.15 (under this equation)?

In Nakahara's Geometry, topology and physics, under Eq.1.15 they give an equation \begin{align*} \det\left(\frac{\partial p_i}{\partial\dot{q}_j}\right)=\det\left(\frac{\partial^2L}{\partial\dot{q}_i\...
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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 ...
Luessiaw's user avatar
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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 ...
Lucas's user avatar
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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 = \...
QuantumBrick's user avatar
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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^{\...
Daniel Vainshtein's user avatar
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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 ...
L. G. Romero's user avatar
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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 ...
Terence Allen's user avatar
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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}}{...
Faber Bosch's user avatar
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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 ...
SYD's user avatar
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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 ...
gordunox's user avatar
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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 ...
Ratul Thakur's user avatar
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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 ...
P. C. Spaniel's user avatar
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1 answer
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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 ...
Noumeno's user avatar
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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\...
polology's user avatar
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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” ...
TheorVHP's user avatar
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1 answer
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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 ...
MrClapton's user avatar
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1 answer
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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 ...
Geigercounter's user avatar
6 votes
1 answer
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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 ...
DingleGlop's user avatar
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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, \...
weirdmath's user avatar
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2 answers
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Lagrange Multipliers in ADM formalism

I'm following this lecture notes (https://javierrubioblog.com/wp-content/uploads/2017/08/adm1.pdf) on ADM formalism. After getting action as (see Eq.(5.41)) $$ S=\int\mathrm{d}^4xN\sqrt{\gamma}\left[\...
Photon's user avatar
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3 votes
1 answer
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Symplectic current, symplectic form, Hamiltonian and boundary conditions in Wald's formalism

I am learning Wald's covariant phase space (1, 2). Although this formalism is very clear on its purpose, in my opinion, I still have some questions regarding some particular aspects. We consider a ...
hyriusen's user avatar
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3 answers
351 views

Is invariance under rescaling of the Lagrangian lost during quantization?

In classical mechanics, a field theory can be described by a lagrangian involving the field and its derivatives, $\mathcal{L}=\mathcal{L}(\phi,\,\partial\phi,\,t).$ The equations of motion for the ...
TopoLynch's user avatar
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1 answer
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Generating function condition not satisfied?

We want to find a generating function $S(q_i,P_i,t)$ such that we get the best possible canonical transformations. So it must satisfy the Hamilton-Jacobi equation: $$H(q_i,\frac{\partial S}{\partial ...
Krum Kutsarov's user avatar
2 votes
1 answer
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What is the benefit of Hamiltonian formalism to promote ($q,\dot{q},t$) from Lagrangian to ($q,p,t$) despite getting the same EOM finally? [duplicate]

Hamiltonian formalism follows $$H(q,p,t)=\sum_i\dot{q_i}p_i-L(q_i,\dot{q}_i,t) $$ and $$\dot{p}=-\frac{\partial H}{\partial q}, \dot{q}=\frac{\partial H}{\partial p} $$ but finally these will get the ...
Kanokpon Arm's user avatar
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Relation between the Wheeler–DeWitt equation and string theory

Can we derive the Wheeler–DeWitt equations from string theory? Since they are both quantum gravity theory. A simple way seems to be the following logic: The Wheeler–DeWitt equation is the canonically ...
feng lin's user avatar
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Hamiltonian flows and Poisson Brackets confusion

I have been using results from this paper in calculations. In sections 2.4 and 3.4 they perform a canonical transformation into new coordinates consisting of constants of motion. My question is ...
Geigercounter's user avatar
1 vote
3 answers
151 views

Why $q,p,Q,P$ are Independent Variables when Using Generating Functions?

In Hamiltonian formalism, specifically generating functions, why do the variables $q, p, Q, P$ are treated as independent when finding the equations that arise from the generating function? I ...
R24698's user avatar
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Hamiltonian form for Yang-Mills theory

For trying to convert a Lagrangian system to Hamiltonian form, I've seen references to two different methods for dealing with constraints; the Dirac-Bergmann method (see reference here) or the Faddeev-...
Matt Dickau's user avatar
4 votes
2 answers
470 views

Hamiltonian of a complex scalar quantum field

Consider the Lagrangian density of a complex scalar quantum field: $$ \mathcal{L} = (\partial_\mu\varphi^\dagger)(\partial^\mu\varphi) - m^2\varphi^\dagger\varphi $$ With the conjugate momenta $\pi^\...
paulina's user avatar
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1 answer
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Why do we need a Poisson bracket structure?

Let me start by asking why we need a Poisson bracket like structure on the Hamiltonian phase space? Say we have a constraint, why do we go through the trouble of defining a Dirac bracket structure on ...
Spotless-hola's user avatar
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1 answer
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Changing variables from $\dot{q}$ to $p$ in Lagrangian instead of Legendre Transformation

This question is motivated by a perceived incompleteness in the responses to this question, which asks why we can't just substitute $\dot{q}(p)$ into $L(q,\dot{q})$ to convert it to $L(q,p)$, which ...
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Poisson bracket [duplicate]

The question starts with why Poisson brackets (in constrained systems) gives different relation if we substitute the constraints before or after expanding the bracket, and why this difference in ...
Spotless-hola's user avatar
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0 answers
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Regarding Poisson and Dirac brackets [duplicate]

The question starts with why Poisson brackets (in constrained systems) gives different relation if we substitute the constraints before or after expanding the bracket, and why this difference in ...
Spotless-hola's user avatar
1 vote
1 answer
45 views

Uncertainty due to assuming a variable is constant - Adiabatic Invariance

I am studying classical mechanics from Goldstein and I ran into a confusing equation in the textbook. In the third edition of the book, equation (12.92) calcucates the average change of the action ...
patrick7's user avatar
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BRST quantization of Hilbert-Palatini gravity, question regarding a choice of gauge fixing fermion

I am interested in the BRST quantization of the Hilbert-Palatini gravity theory. In the paper https://arxiv.org/abs/gr-qc/9806001, Alexandrov and Vassilevich write down the BRST procedure for defining ...
Jeanbaptiste Roux's user avatar
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1 answer
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Regarding Poission structure of Hamiltonian phase space

Why exactly do we need $$ \{q^i,p_j\}=\delta^i_j,$$ where $\delta^i_j$ is Kronecker delta and $\{\cdot,\cdot\}$ is the Poisson bracket? What happens to the phase space structure if these fundamental ...
Spotless-hola's user avatar
1 vote
1 answer
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Hamiltonian formalism (with symplectic form) for time-dependent Lagrangian

I have been working on some results that work for time-independent Lagrangians $L\Big(q,\dot{q}\Big)$ and return a Hamiltonian function $$ H(q,\dot{q})=\dot{q}^i \frac{\partial L}{\partial \dot{q}^i}-...
P. C. Spaniel's user avatar
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Constrained Hamiltonian problems [closed]

What happens to the poisson bracket structure of Hamiltonian phase space if We have some constraints in $p$ and $q$. What physical aspects this structure represents?
Spotless-hola's user avatar
1 vote
1 answer
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Regarding varying the coefficients of constraints in the pre-extended Hamiltonian in Dirac method

consider the following variational principle: when we vary $p$ and $q$ independently to find the equations of motion, why aren't we explicitly varying the Coeff $u$ which are clearly functions of $p$ ...
Spotless-hola's user avatar
1 vote
2 answers
213 views

Question about canonical transformation and generating functions

In Goldsteins' Mechanics, page 371 (relevant part appears below), it follows from what he states in the first yellow part that the equations of transformation: $$Q = Q(q, p,t), \quad P = P(q, p,t)\tag{...
R24698's user avatar
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Is it possible to formulate classical Hamiltonian mechanics without reference to a Lagrangian? [duplicate]

The typical way to arrive at Hamiltonian mechanics is through Lagrangian mechanics, defining canonical momentum and the hamiltonian itself in reference to the Lagrangian and its derivatives, but I'm ...
Strategist _'s user avatar
1 vote
0 answers
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Setting constraints to zero in extended Hamiltonian

I am dealing with a system that has some secondary constraints. I am trying to use Dirac-Bergmann procedure by following chapter 10 of Ashok Das, Lectures on Quantum Field Theory ( 2021, Second ...
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