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 ...

learn more… | top users | synonyms (1)

4
votes
1answer
140 views

Analytic proof that Lyapunov exponents in Hamiltonian systems pairwise sum to zero

I have read that in Hamiltonian systems, Lyapunov exponents come in pairs $(\lambda_i, \lambda_{2N-i+1})$ such that their sum is equal to zero. Is there a way of proving this analytically? EDIT: ...
3
votes
1answer
219 views

Deriving the Poisson bracket relation of the Ashtekar variables

I'm trying to figure out how to calculate the orthogonality of Ashtekar variables with respect to the ADM hypersurface metric and conjugate momentum. $$\{{A_a}^i(x), {E^b}_j(y)\} = 8 \pi \beta \delta^...
1
vote
0answers
43 views

Nabla Terms in the Energy Density of the Lagrangian for the Massive Spin 1 Field (Schwartz QFT 1st Ed. Eqn. 8.19)

The relevant part starts with a Lagrangian guess of, $$\mathcal{L}=-\frac{1}{2}\partial_{\nu}A_{\mu}\partial_{\nu}A_{\mu}+\frac{1}{2}m^2A_{\mu}^2$$ where the EOM's are, $$(\Box+m^2)A_{\mu}=0$$ The ...
1
vote
1answer
133 views

Can all symplectic-form preserving canonical transformations generated by generating functions

This question is related to this fascinating post and this post and this post, but more limited in scope in discussing the practical definition canonical transformations. Canonical transformation ...
1
vote
1answer
145 views

Hamiltonian of coupled oscillators

Let's say I have a system of coupled oscillators which are described by the coordinates $\{x_1,...,x_N\}$ and $\{\dot{x}_1,...,\dot{x}_N\}$. The equation of motion for each oscillator is $$\ddot{x}_n ...
0
votes
0answers
44 views

Finding Momentum in Parabolic Polar Coordinates

The problem I am doing necessitates the use of finding the momentum in parabolic polar coordinates. I need to transform the following Hamiltonian from cartesian to parabolic polar coordinates. $H=\...
3
votes
1answer
199 views

Is there any relation between Poisson Brackets and the Jacobian Matrix?

The Poisson brackets for $u,v$ can be written as, $$ \frac{\partial u}{\partial q} \frac{\partial v}{\partial p} - \frac{\partial u}{\partial p}\frac{\partial v}{\partial q}. $$ We can write this ...
1
vote
0answers
86 views

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 ...
0
votes
1answer
49 views

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 ...
2
votes
1answer
136 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 ...
0
votes
0answers
20 views

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 ...
1
vote
2answers
67 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 ...
3
votes
1answer
214 views

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 ...
0
votes
1answer
108 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 ...
1
vote
1answer
173 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 $$ R_{ij}=\frac{\partial\Gamma^{l}_{ij}}{\...
3
votes
1answer
179 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 ...
0
votes
1answer
124 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 ...
18
votes
2answers
488 views

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 ...
4
votes
1answer
102 views

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 ...
5
votes
2answers
314 views

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) $$...
-1
votes
1answer
56 views

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 [\dfrac{1}{...
3
votes
1answer
504 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 ...
1
vote
1answer
111 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 P_0(...
2
votes
1answer
279 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, ...
1
vote
1answer
83 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 ...
0
votes
0answers
49 views

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 ...
2
votes
1answer
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 _{X_{H}}\...
0
votes
0answers
47 views

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 $$ H = \frac{p^2}{2m} +V (|r|) . $$
2
votes
1answer
141 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 $...
2
votes
1answer
145 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 ...
3
votes
1answer
340 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 ...
1
vote
0answers
63 views

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 ...
3
votes
1answer
219 views

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 ...
7
votes
2answers
234 views

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 ...
1
vote
1answer
84 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?
5
votes
1answer
157 views

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 ...
3
votes
3answers
132 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 ...
1
vote
2answers
202 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 ...
2
votes
1answer
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 ...
1
vote
0answers
48 views

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?
2
votes
1answer
117 views

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 ...
2
votes
0answers
61 views

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 ...
1
vote
1answer
99 views

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 ...
3
votes
0answers
145 views

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 ...
4
votes
2answers
205 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 ...
0
votes
1answer
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, ...
2
votes
1answer
87 views

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 ...
3
votes
0answers
54 views

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)= T\exp(...
3
votes
1answer
112 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 ...
4
votes
3answers
393 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 ...