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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Canonical Momentum Conjugate vs. Momentum

I stumbled upon this while reading about Legendre Transforms today. So consider an n-particle system. The Lagrangian is a function of $ q_i$'s and $\dot q_i$'s. If you consider the manifold $M$ where ...
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Adiabatic Invariance of 2 mass system

I am trying to solve the second part of Problem 10 from David Tongs CM notes. Specifically we have The neutron star is in a non-circular orbit with $E < 0$. Give an expression for the adiabatic ...
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132 views

Noether's Theorem for Hamiltonians and Lagrangians

Looking around I see one version of Noether's Theorem that creates conserved quantities from symmetries that preserve the Lagrangian (e.g. http://math.ucr.edu/home/baez/noether.html), and another ...
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What are the prerequisites needed to grasp analytical mechanics? [duplicate]

What are the mathematical prerequisites needed to grasp analytical mechanics conceptually and technically? What textbook is adequate for this purpose for an undergraduate student? To understand the ...
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66 views

Quantum field operators in HEP and CMT

For a real scalar field (which is a bosonic field) we have these commutation relations : $$ \left[\phi(x,t),\phi(y,t)\right]=0 \qquad \qquad \left[\phi(x,t),\pi(y,t)\right]=\delta(x-y).\tag{1}$$ But ...
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Area of phase space of Harmonic oscillator

We all know that the phase trajectory of an undamped linear harmonic oscillator is an ellipse. But when we calculate the area of the ellipse we find it does not depend of mass of the particle. Why is ...
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103 views

Determining which generating function to use for canonical tranformation

So I apologize if this vague, but I am just looking for steps to figure this out. I have the following CT $$Q_1(q_1)$$ $$Q_2(q_2,p_2)$$ $$P_1(p_1,p_2,q_1,q_2)$$ $$P_2(p_2,q_1)$$ Where I am just ...
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169 views

Ricci tensor as relativistic Hamiltonian

I am little bit dissapointment with action integral in General relativity. The action integral is: $$ \int Rd^{4}x=\int R_{ij}g^{ij}d^{4}x\tag{1} $$ Where $$ ...
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170 views

Relationship of symplectic group (Hamiltonian structure) to unitary group in quantum mechanics?

Wikipedia claims here that the 2 out of 3 property is the following relationship between unitary, orthogonal, symplectic, and general linear complex groups: $U(n)=Sp(2n,R)∩O(2n)∩GL(n,C)$ Intuitively ...
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123 views

Density of states of classical harmonic oscillator in phase space

Since all classical harmonic oscillators are ellipses in phase (position-momentum) space, and since the entire phase trajectory of a given system (with a fixed rigidity and mass factor) can be ...
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Does the poisson bracket $\{f,g\}$ have any meaning if neither of $f$ or $g$ is the system's Hamiltonian?

Say one has a mechanical system with hamiltonian $H$, and two other arbitrary observables $f,g$. $H$ is super useful since $\{H, \cdot\} = \frac{d}{dt}$. But does $\{f,g\}$ give any useful information ...
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Calculating the number of particles in phase space

I'm looking at the first part of question 7 here (I'm a mathematician trying to self teach some physics, this isn't a homework assignment so I'm just in need of hints)! I'm struggling to make sense of ...
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Quantum and Classical Liouville operators

In the Heisenberg picture of Quantum Mechanics, for an observable $\hat{A}$, we have the famous Heisenberg equation giving the time evolution of the operator: ($\hat{H}$ is the Hamiltonian operator) ...
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Vector calculus trouble in Hamilton's equations / particle in E.M.field [closed]

As part of applying Hamilton's equations to a particle in an electromagnetic field, one step is to take $\dot{\mathbf{p}} = - \dfrac{\partial H}{\partial \mathbf{r}} = -\nabla H = - \nabla ...
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461 views

Finding action-angle variables

Given a 1 d.o.f Hamiltonian $H(q,p)$ what is the general procedure for finding action angle variables $(I, \theta)$? I have read the Wikipedia page on action angle variables and canonical transforms ...
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110 views

Finding the Hamiltonian for sound vibrations in a gas in the momentum representation

I am working on a problem where I have been given the following Lagrangian density for describing sound vibrations in a gas: $\mathcal{L}=\frac{1}{2}[\rho_0\dot{\eta}^2+2P_0\nabla\cdot\eta-\gamma ...
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271 views

Any good textbook on the canonical perturbation theory for Hamiltonian systems?

My teacher of classical mechanics once told us, classical mechanics is more difficult than quantum mechanics in many ways. He used the perturbation theory as an example to illustrate this point. So, ...
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82 views

Independence of generalised coordinates and momenta in Hamiltonian mechanics [duplicate]

I am told that in Hamiltonian mechanics, we put the generalised coordinates $q_i$ and generalised momenta $p_i$ on equal footing, and treat them as being independent from one another. But I'm ...
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Formulation of Hamiltonian for an oscillating electron cloud?

How do I formulate the Hamiltonian of an electron cloud oscillating about a nanoparticle induced by an electromagnetic wave. Will the Hamiltonian be different if I consider the electron cloud as a ...
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82 views

How can we obtain equations of motion from $\iota_{X_{H}}\omega =dH$?

I don't know if this is an obvious result and I am just missing a trick, so please forgive me, but how do we obtain equations of motion from the following equation. \begin{equation} \iota ...
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First integrals for a particle in a central-force field

Consider an arbitrary dimension $n>3$. What are the independent first integrals for a particle? The Hamiltonian is $$ H = \frac{p^2}{2m} +V (|r|) . $$
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135 views

Intuition about Momentum Maps

I'm studying Classical Mechanics and there is one object that appeared recently on the book I'm not being able to get a physical intuition about it. The mathematical definition goes as follows: Let ...
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144 views

Poisson brackets and magnetic field [closed]

I'm a maths student trying to teach myself some physics so sorry if I'm missing something simple here. I think the main problem is lack of experience with the Levi-Cevita symbol. We have a particle ...
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284 views

The Liouville equation and the BBGKY hierarchy.

The Liouville equation of motion is written in terms of an $N$ particle distribution $f_N$. \begin{equation} \frac{\partial f_N}{\partial t}=\{H,f_N\} \end{equation} Where $\{\cdot ,\cdot \}$ is the ...
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Finding canonical transformation using type 3 generating function

Question: For a system with one degree of freedom, a canonical transformation $Q(q,p), P(q,p)$ obtained by a type 3 generating function satisfies $Q = e^t q^{1/2}\cos p$. Find the most general form ...
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Planck's constant and phase space in quantum mechanics

During my undergrad physics classes, I've come across several seemingly related phenomena dealing with $h$ and phase space in quantum mechanics. Let $T_x$ be a translation operator by $x$ in ...
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What canonical momenta are the “right” ones?

I'm doing some classical field theory exercises with the Lagrangian $$\mathscr{L} = -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$$ where $F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$. To find the ...
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82 views

Why closed in the definition of a symplectic structure?

Why do we want the 2-form $\omega $ to be closed? What if it is not?
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Chaos and integrability in classical mechanics

An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any ...
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130 views

Lagrangian from Path Integral

Suppose I somehow know propagator for a given quantum mechanical system but I don't happen to know either the Lagrangian or Hamiltonian. (For simplicity, assume that this is non-relativistic.) Is ...
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192 views

Why is the Hamilton-Jacobi equation important? [closed]

Someone may say it is related to the Schrodinger equation. Okay, let us forget about quantum mechanics. So, if we confine ourself to classical mechanics, why is the Hamilton-Jacobi equation important ...
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76 views

What is the utility of ADM decomposition of the space-time metric?

I know it's one of the possibility of quantization of gravitational field's degree of freedom but it is introduced also in other situation. My question is what is the use for this kind of ...
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Any good reference on Maslov index (or Morse index)?

Any good reference on Maslov index (or Morse index)? I have some basic knowledge of differential geometry, calculus of variation. So is there any good reference for me?
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Do time-invariant Hamiltonians define closed systems?

In classical mechanics, every time-invariant Hamiltonian represents a closed dynamical system? Can every closed dynamical system be represented as a time-invariant Hamiltonian? Or are there closed ...
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Foliation of the phase space

Consider an arbitrary classical Hamiltonian system. Given an initial state $(p_0, q_0)$, we can get a solution of the equation of motion, a curve in the phase space. Now the problem is, for a generic ...
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For infinitesimal Canonical Transformations, what functions are allowed for this to be a canonical transformation?

Consider two infinitesimal transformation: $$q_{i} \rightarrow Q_{i} =q_{i} + \alpha F_{i}(q,p) $$ $$p_{i} \rightarrow P_{i} = p_{i} + \alpha E_{i}(q,p) $$ where $α$ is considered to be ...
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What variable is the conjugate momentum for angular momentum?

From the definition of conjugate momentum for a generalized coordinate we get that the conjugate for angular momentum should be proportonal to its integral with respect to time. According to my ...
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202 views

Is the Legendre transformation a unique choice in analytical mechanics?

Consider a Lagrangian $L(q_i, \dot{q_i}, t) = T - V$, for kinetic energy $T$ and generalized potential $V$, on a set of $n$ independent generalized coordinates $\{q_i\}$. Assuming the system is ...
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35 views

from microscopic to kinetic transport theory

One way to model the dynamics of particles is to find the differential equation of motion of a particle. Of course, this will be nice and easy to do if we have only a few particles (like one-ish, ...
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Analytical mechanics with SR

Is there an analytical mechanics with SR? Of course you can write down the Lagrangian and Hamiltonian of a free particle. What about non-free? Are there any problems? To be specific: what would the ...
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Time evolution of a classical system [closed]

For a harmonic oscillator the Liouville operator is given by $$L = p \partial_q- q \partial_p.$$ Now I have a phase space distribution $f(t,q,p)$ for which it holds (in general) $$f(t+\tau,q,p)= ...
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106 views

Fermionic Poisson bracket

I'd like to understand the Poisson bracket for fermions in classical field theory defined on a cylinder (with coordinates $(t,x)$, $x$ being the compact direction) and propagating on $T^n$ with ...
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380 views

Constructing Lagrangian from the Hamiltonian

Given the Lagrangian $L$ for a system, we can construct the hamiltonian $H$ using the definition $H=\sum\limits_{i}p_i\dot{q}_i-L$ where $p_i=\frac{\partial L}{\partial \dot{q}_i}$. Therefore, to ...
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What exactly is the relationship between the symplectic 2-form and the frequency of leaves of integrable systems in classical mechanics?

In classical mechanics we equip a differential manifold with a closed symplectic 2-form $\omega$. The symplectic leaves of integrable systems also have a unique frequency, in literature denoted ...
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112 views

Canonical commutation relations in Light-cone gauge

It seems that when trying to identify the physical degrees of freedom for the string some authors$^1$ use: $$ q^-=\frac{1}{\ell}\int_0^{\ell} X^-(\tau,\sigma)d\sigma$$ Then, the commutation relation ...
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Taking a 'relative' limit

I am looking at Hamiltonians for specific physical situations. I have taken a given Hamiltonian $\vec{H}(\vec{p}, \vec{x})$ and have found the following Hamiltonian equations: $$\frac{d\vec{x}}{dt} = ...
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Derivation of an ordinary, Lagrangian/Hamiltonian and action formulation

I am confused as to how the different formulations in physics are derived. In many fields of physics, we usually begin with an ordinary formulation (e.g Newton's Laws in classical mechanics), and ...
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Using tensors on Lagrangian and Hamiltonian

We can write the Lagrangian (with $n$ generalized coordinates) using the following expression: ...
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Hamilton's Principle - achieving Hamilton equations

Consider the action function: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $\mathcal{L}$ is the Lagrangian of the system. The Hamiltonian is defined by the following ...
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How does the Hamiltonian change when going to a moving frame?

The Hamiltonian of a free particle in a rotating frame is given by $$ H = H_0 - \omega \cdot J, $$ where $H_0$ is the Hamiltonian in the non-rotating frame, $\omega$ is the angular velocity of the ...