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.

Filter by
Sorted by
Tagged with
2
votes
0answers
39 views

Are first Integrals the same in Lagrangian and Hamiltonian formalism?

A first integral in Lagrangian formalism is defined as a function which is constant along the solutions $(q,\dot{q})$ where $q$ are the generalized coordinates; while a first integral in Hamiltonian ...
1
vote
0answers
51 views

What exactly is the Legendre transformation? [duplicate]

Goldstein et al's Classical Mechanics states that: The Hamiltonian $H(q,p,t)$ is generated by the Legendre transformation $$ H(q,p,t) = \dot{q}_i p_i - L(q,\dot{q},t). \tag{8.15} $$ But I don't ...
1
vote
1answer
63 views

Question about using Liouville's theorem to calculate time evolution of ensemble average

With the Liouville's theorem $$\frac{{d\rho }}{{dt}} = \frac{{\partial \rho }}{{\partial t}} + \sum\limits_{a = 1}^{3N} {(\frac{{\partial \rho }}{{\partial {p_a}}}\frac{{d{p_a}}}{{dt}} + \frac{{\...
1
vote
0answers
53 views

Canonical Transformations that are Complex

I'm self studying through a book that has the following question. The book gives the answer, but I'm trying to understand why: Under what condition is the following transformation NOT canonical? $$Q =...
0
votes
1answer
79 views

A proof of Liouville’s theorem

I have found a proof of Liouville's theorem on the internet, which fits me very well except one step I don't understand, the derivation is as follows: In the derivative, it must have used the ...
2
votes
1answer
88 views

The complex form of Hamilton canonical equations

I found an excerpt on page 171 of "The variational principles of mechanics" written by Cornelius Lanczos stated that If, however, the conjugate variables $q_k$, $p_k$ are replaced by the complex ...
2
votes
1answer
182 views

Coupled oscillators in Hamiltonian formalism - problem with diagonalization

I have a problem with simple coupled oscillator system. I tried to solve single oscillator with Hamiltonian, and then coupled system of two, but when I try to put coupling constant $k^\prime=0$ in my ...
-1
votes
1answer
104 views

Hamilton's equation of motion with other momentum

I wrote here a problem couple days ago. I figured out what was the problem there, but now it made another problem. Sorry for similiar question. I'm trying to draw phase portrait for my ODE and for my ...
1
vote
2answers
51 views

Time-reversibility symmetry in classical mechanics

Newton's laws are invariant under time reversal transformation $$ t \longrightarrow -t $$ for time-independent potentials. But Hamilton-Jacobi equation is too an equivalent description of classical ...
4
votes
1answer
73 views

How does a Lagrangian with delta potential transform to a Hamiltonian?

Suppose the Lagrangian was given as: $$L = \frac{1}{2}\int_{-\infty}^{\infty} \underbrace{\left(\dot{A(}z)^2-A(z)^2\right)+\left(\dot{Q(}z)^2\delta(z)-Q(z)^2\delta(z)\right)+2\dot{Q(}z)\cdot A(z) \...
0
votes
0answers
52 views

Why are transformations $(q,p)\to (Q,P)$ that are canonical, more useful than any $(q,p)\to(Q,P)$?

I am facing a difficulty in understanding canonical transformations. It can be defined as a transformation $$(q_i,p_i)\to \big( Q_i(q_i,p_i,t),P_i(q_i,p_i,t)\big) \tag{1}$$ under which the Hamilton'...
0
votes
0answers
39 views

What is the Lagrangian for FRW universe?

A straightforward calculation to the Lagrangian for FRW as $\mathscr{L}=\sqrt{-g}R$ gets me the following result: $$\mathscr{L}\sim a^3\big(\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^2+\frac{k}{a^2} \big)...
0
votes
0answers
17 views

Legendre transform and coordinate system independence

I'm self-learning analytical mechanics. Consider a classical mechanical system. Even if it's clear to me that via (the usual) Legendre transform we can get a unique Hamiltonian function from a ...
1
vote
1answer
48 views

How is it possible to have four types of generating functions?

Since the Hamilton's equations of motion remain unchanged in form under a canonical transformation $(q,p)\to (Q,P)$, the Lagrangians must differ by a total time derivative of a function of $q,t$. In ...
1
vote
1answer
45 views

Why does the Hamiltonian of a Lagrangian that consists of only a coupled term become zero?

Let's say our Lagrangian looks something like this: $$L = \int dz\, Q\cdot \dot{A},\tag{1}$$ where $Q$ and $A$ are two generalized coordinates and $\dot{Q}$ and $\dot{A}$ would be the respective ...
0
votes
1answer
41 views

Justification of dropping term in Hamiltonian and expectation Values

While reading Sakurai's Modern QM, I was stuck at the point where he explains the absorption and emission of light quanta in atoms. He proceeds with Hamiltonian: $$H= p^2/2m + e\phi(x) -e/mc A\cdot p$...
1
vote
0answers
42 views

Book recommendation for relativistic classical mechanics

I need some good resource recommendations for the relativistic hamiltonian mechanics under special theory of relativity, with a good discussion on relativistic Hamilton-Jacobi formulation.
3
votes
1answer
43 views

Derivative with respect to vector

How in Lagrangian and Hamiltonian mechanics we take derivatives with respect to velocity and momentum respectively if they are vectors? Can we take derivative with respect to a vector?
0
votes
1answer
37 views

PDE from Hamiltonian density

For the wave equation Hamiltonian density is $2H=\phi_t^2+\phi_x^2$ while the Lagrangian density is $2L=\phi_t^2-\phi_x^2$. I can easily compute the pde from the Lagrangian density but how does one do ...
3
votes
1answer
76 views

Angular momentum and rotation symmetry

In my book, it is written that for any vector $\mathbf{v}$, we have $$\{\mathbf{v},\mathbf{L}\cdot \mathbf{n}\}=\mathbf{n}\times\mathbf{v}.\tag{1}$$ For me it is absurd... For example, if we take $\...
2
votes
0answers
34 views

Angular momentum of a cylindrical system in general relativity

The definition of energy and enegry flux of a cylindrical symmetric system in general relativity is given by Kip Throne in Phys. Rev. 138, B251 and generalized by Chandrasekhar in Proc. Roy. Soc. Lond....
1
vote
0answers
69 views

Liouville theorem and the ergodic assumption

I am following a course on statistical mechanics. My instructor presented us the following Liouville theorem in two (claimed) equivalent ways: Differential statement: The probability distribution $\...
6
votes
1answer
81 views

What does the Ostrogradsky instability have to do with stability?

Ostrogradsky's instability theorem says that under some conditions, a system governed by a Lagrangian which depends on time derivatives beyond the first is "unstable". In the proof, one computes the ...
1
vote
0answers
45 views

Open problems in Hamiltonian Dynamics and Symplectic Geometry [closed]

I'm starting an undergraduate research on Mathematical-Physics and the topic I chose is Hamiltonian mechanics (and its formalism, Symplectic Geometry). I'm looking for some interesting open problems ...
2
votes
0answers
65 views

Euler-Lagrange and Hamilton equations for $p$-forms [closed]

How are Euler-Lagrange equations and Hamilton equations modified (including higher order lagrangians) takinto into account $p$-forms? I think it should also depend on the $D$-dimensional target ...
0
votes
2answers
91 views

What is a Hamiltonian of a System?

What is a Hamiltonian of a System? When learning about Hamiltonian for first time it is an object introduced as Legendre Dual Transform of Lagrangian of the same system. And we learn further that it ...
2
votes
1answer
65 views

Choosing initial condition for Hamilton-Jacobi PDE

For separable solutions to Hamilton-Jacobi PDE (say in 2D), we treat the Hamilton's principal function $S$ as $$S= W(x) + W(y) - E*t$$ and treat the separate parts as constants and find $W(x)$, $W(y)$...
2
votes
1answer
88 views

The Killing vector at the bifurcation surface of a stationary black hole

In the paper Black Hole Entropy is Noether Charge, Wald related the black hole entropy to the Neother charge using the covariant phase space formalism. In proving this relation, Wald noticed that on ...
0
votes
0answers
29 views

Resonant Hamiltonian Mechanics

My question is regarding applying averaging theory to a perturbed Hamiltonian. Now, my Hamiltonian is of the form $$H=H_0 + R(q_i,p_i)$$ Where R is the disturbing potential which is a function of the ...
0
votes
1answer
78 views

Prove that a transformation is canonical by using $\mathbb{M}^T\cdot \mathbb{J}\cdot \mathbb{M}$ [closed]

So, I was given the following problem to solve: A system with two degrees of freedom is described by the following hamiltonian \begin{equation} H=p_1^2+p_2^2+\frac{1}{2}(q_1-q_2)^2+\frac{1}{8}(...
0
votes
0answers
40 views

Hamiltonian flows and Heisenberg picture of Quantum Mechanics

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial) In particular what I know is that: ...
6
votes
2answers
131 views

Compute the Legendre transform for a singular Lagrangian

I'm given the lagrangian: $$ L(q,\dot{q}) = \frac{1}{2}(\dot{q_1}^2+\dot{q_2}^2+2\dot{q_1}\dot{q_2})-\frac{k}{2}(q_1^4+q_2^4). $$ I have to compute the Legendre transformation associated to it. The ...
4
votes
1answer
56 views

Ehrenfest theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
2
votes
1answer
93 views

Kinetic energy always time independent?! Where is my mistake? [closed]

I have some problems understanding the Lagrangian and the Hamiltonian formalism. Those can be condensed in the following "derivation" of $\frac{\partial T}{\partial t} = 0$ from the equation $\frac{\...
1
vote
1answer
36 views

Asymmetry in Hamilton Equations

I noticed that in deriving Hamilton equations from the total deriveative of the Hamiltonian with respect to time, for the first equation $$\frac{dx_k}{dt}=\partial_{p_k}H$$ we do not need Lagrange's ...
0
votes
1answer
78 views

Goldstein expression for the Lagrangian

I was looking for help in order to proove 2 relations that Goldstein has put in his book. $$ L(q, \dot{q}, t)=L_{0}(q, t)+\tilde{\mathbf{\dot{q}}} \mathbf{a}+\frac{1}{2} \tilde{\boldsymbol{\dot{q}}} \...
5
votes
1answer
156 views

How to derive the Hamilton-Jacobi equation for the area of a minimal surface on a Riemannian manifold?

The action for a string in this background $$G_{IJ}\tag{1}$$ can be written as the Nambu-Goto action $$S_{NG}=\int d\sigma^1d\sigma^2\sqrt{g}\quad\quad\Rightarrow\quad\mathcal{L}=\sqrt{g}\tag{2}$$ ...
2
votes
1answer
73 views

Is there always a canonical transformation such that the new Hamiltonian only depends on the new momenta?

Given the Hamiltonian $H(x,p)$ of a system. Is there always a coordinate transformation such that the new Hamiltonian is $K(x',p')=K(p')$?
0
votes
0answers
37 views

Hamilton's equations for Dirac Hamiltonian [duplicate]

The Dirac Lagrangian $$\mathcal{L} = i\bar{\psi}\gamma^{\mu}\partial_\mu \psi - m \bar{\psi}\psi$$ gives a Hamiltonian $$\mathcal{H}(\Pi,\bar{\Pi},\psi,\bar{\psi})=\Pi \dot{\psi}-\mathcal{L}=-\bar{\...
2
votes
0answers
44 views

Why are only the Lagrangian and Hamiltonian used in mechanics? [duplicate]

Why is it that we have a closed set of four functions, connected by Legendre transforms, in thermodynamics but nobody ever mentions but two of the corresponding functions in mechanics? I've read that ...
3
votes
1answer
213 views

Couple of non-interacting, non-integrable Hamiltonian systems

I have two Hamiltonians, $H(p_1, q_1, \dots, p_n, q_n)$ and $H'(p_{n+1}, q_{n+1}, \dots, p_m, q_m)$. They are non-interacting, indeed they have different variables. Moreover, I know that they are both ...
0
votes
2answers
73 views

Rigourous formalism of Hamiltonian mechanics on Manifolds

I'm looking for books / articles that provide rigorous formulations of Hamiltonian mechanics on Manifolds. I found the book "Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds" [1]...
1
vote
0answers
55 views

Legendre transformation and correspondance between Noether charges and quasi-symmetries

I have been trying to understand the Legendre transformation (in mechanics, in the hyperregular case: when the Legendre transformation is one-to-one) and the correspondence between symmetry $\to$ ...
1
vote
0answers
30 views

What happens to the time evolution equations in canonical quantum gravity?

Many expositions on canonical quantum gravity start from a 3+1 type formalism, where spacetime is foliated along the time dimension. The Einstein equations then decompose into constraint equations on ...
2
votes
1answer
70 views

Do canonical transformations form a group?

In a course on classical mechanics, we barely touched upon canonical transformations via generating functions. Just like Lorentz transformations form a group, I want to know if canonical ...
2
votes
2answers
85 views

Liouville's integrability theorem: action-angle variables

For classical dynamical systems, let $I_{\alpha}$ stand for independent constants of motion which commute with each other. 'Remark 11.12' on pg 443 of Fasano-Marmi's 'Analytical Mechanics' suggest ...
1
vote
1answer
47 views

A paradox about canonical transform preserving Poisson bracket?

Let $q,p$ denote the position and momentum. Consider a transform generated by $g$: $q' = q + \epsilon \{q,g\}---(1a)$ $p' = p + \epsilon \{p,g\}---(1b)$ Then: $\{q',p'\} = \{q,p\}+o(\epsilon^2)+\...
3
votes
2answers
68 views

ADM formulation of GR derivative on the 3-metric

In the ADM formalism where the projector is given by ${P^\mu}_\alpha={\delta^\mu}_\alpha+n^\mu{n}_\alpha$ and $n^\alpha$ is a future pointing normal vector to the constant time hypersurface $\Sigma$. ...
3
votes
2answers
74 views

Derivation of Hamilton-Jacobi equation

I am trying my own way of deriving the Hamilton Jacobi equation $$\frac{\partial S}{\partial t} = -H \tag{1}$$ through direct variation. I think the difficulty of doing this is that the upper limit ...
0
votes
0answers
23 views

Hubbard Hamiltonian mean field decomposition in real and momentum space

Im working through some lecture notes on quantum field theory, and it gives the mean field hamiltonian in real space in terms of the spin operator: $$H_{int}^{MF}=\frac{3}{8U}\sum_{r_j}\vec{(M}(r_j))^...