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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${\rm 2D}$ isotropic oscillator: Is ${\rm SO(4)}$ a subgroup of ${\rm Sp}(4,{\rm R})$?

Consider the ${\rm 2D}$ isotropic oscillator. The hamiltonian is $$H=\frac{1}{2}(p_x^2+p_y^2+x^2+y^2)$$ and the phase space is $4$ dimensional. In this case, the set of all linear canonical ...
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How does one find the Hamiltonian corresponding to a pendulum equation of motion? [duplicate]

How can a Hamiltonian-function be derived out of a differential equation of second order describing a pendulum motion as for example: $$\theta^{(2)}(t) = A*\theta(t) + C*cos(B*\omega)(t),$$ where the ...
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How can one derive the Hamiltonian-function out of a differential equation of second order that describes the motion of a pendulum?

How can a Hamiltonian-function be derived out of a differential equation of second order describing a pendulum motion as for example: $$\theta^{(2)}(t) = A*\theta + cos(B*\omega),$$ where the ...
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Legendre Transformation of Lagrangian with constraints

I have problems with obtaining a Hamiltonian from a Lagrangian with constraints. My overall goal is to find a Hamiltonian description of three particles independent of any Newtonian Background and ...
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How to find the canonical transformation?

Knowing that the Hamiltonian of a system is $H=\frac{1}{2}(q^{4}p^{2}+\frac{1}{q^{2}})$ The Hamiltonian after the canonical transformation is $K=\frac{1}{2}(P^{2}+Q^{2})$ How do I know what ...
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Conjugate momentum in the vacuum functional for the fermionic oscillator

The vacuum functional for the fermionic oscillator is given by $$ Z[0] = N\int\mathcal{D}\overline{\psi}\mathcal{D}\psi \exp\left(i\int_0^Tdt\left(i\overline{\psi}\psi-w\overline{\psi}\psi \right)\...
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How are the two definitions of Canonical Transformations related/equivalent? [duplicate]

I am aware of two definitions of canonical transformations which I state below. Definition $1$ We go from old set of $\{q_i,p_i,t\}$ of $2n$ phase space variables to a new set $\{Q(q_i,p_i,t),P(q_i,...
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How the Hamiltonian of a classical system expressed in quantum mechanics?

I was dealing with a problem, which said that, Supposedly Hamiltonian of a conservative system in classical mechanics is $\omega xp$, where $\omega$ is a constant, and $x$ and $p$ are the position ...
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Are symmetry operations necessarily only transformations on the configuration space?

Main question When we talk about symmetry operation in classical mechanics, do we necessarily mean transformations on the configuration space (e.g. translations, rotations etc) or could it also be ...
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Itzykson Zuber Quantum Field Theory: meaning of integrable system

Here is a part of the book Quantum Field Theory by Itzykson and Zuber: I have two questions: what does the author mean that equation (1-30) form and integrable system, and why? what is the ...
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Hamiltonian formalism of the massive vector field

I am currently working through a problem concerning the massive vector field. Amongst other things I have already calculated the equations of motion from the Lagrangian density $$\mathcal{L} = - \frac{...
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Modified Hamilton's Principle overconstraining a system by imposing too many boundary conditions

In Hamiltonian Mechanics, a version of Hamilton's principle is shown to evolve a system according to the same equations of motion as the Lagrangian, and therefore Newtonian formalism. In particular, ...
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Determinism and frame-relativity

It's a well known fact that classical mechanics isn't a deterministic theory if you only include the positions and masses of various particles as part of the initial conditions. You also need to ...
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Treat stochastically non-Hamiltonian perturbations

Let us consider a classical dynamical system whose obserbvables $A$ evolve according to the following equation of motion \begin{align} \dot A &= -\{H,A\}+f(q) \end{align} $f(q)$ is a non-...
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Frozen Formalism Problem

Before stating my question, let me say what I do understand: In the ADM formalism, the Hamiltonian density of the gravitational field can be written as, $$\mathcal{H} = h n + H_a N^a$$ where n is the ...
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Analogues to Hamilton's equations in Infinitesimal Canonical Transformations

This is from chapter 4 of David Tong's notes on Classical Dynamics (Hamiltonian Formalism). Let's say you make an infinitesimal canonical transformation (with $\alpha$ as the infinitesimal parameter) ...
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Change in Hamiltonian under infinitesimal canonical transformation [duplicate]

Consider an infinitesimal canonical transformation from the (symplectic) coordinates $z$ with Hamiltonian $H(z, t)$, to the coordinates $$ Z = z + \epsilon J \frac{\partial G}{\partial z} $$ with the ...
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Mathematical prerequisites for classical (Lagrangian and Hamiltonian) mechanics [duplicate]

I have just discovered the ideas of Lagrangian and Hamiltonian formulation of mechanics. I wish to self-study further. I (believe I) grasp the very basic idea that both approaches are based on the ...
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Can Jacobi's formulation of Maupertuis' principle be derived in Riemannian geometry?

i want to arrive to hamilton-jacobi equation using the riemannian geometry. So let $\textbf{X}\in \mathfrak{X}(M)$, where $M$ is Riemannian manifold whose metric is $g:\textbf{T}M \times \textbf{T}M \...
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Momentum and infinitesimal translation

My problem is all about this previous question. I'm trying to understand the reasoning behind the definition of the momentum operator in quantum mechanics. Sakurai tells me that for the infinitesimal ...
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One third of Lyapunov exponents are zero? What does it mean?

This may be quite a straightforward question, but I have a dynamical system with a high dimensional phase-space. I calculated the Lyapunov spectrum for it and saw that one third of my Lyapunov ...
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Symmetry transformation of canonical coordinates of the EM field

The EM free field can be written with the Hamiltonian formalism as: $$H=\sum_\lambda\frac 1 2(P_\lambda^2+\omega^2 Q_\lambda^2)$$ In this expression $Q_\lambda$ and $P_\lambda$ are canonical variables....
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Deriving the Gauss Constraint in Free Maxwell Theory

In section 6.2 (page 128) of David Tong's Lectures on QFT, Gauss' law is derived for the free Maxwell theory. The result of computing the Hamiltonian of the theory is (eq. 6.17), $$H = \int d^{3} x \...
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Deriving the quantum Hamiltonian from the expression of classical energy

I am currently learning about the Dirac formalism in quantum mechanics, but don't quite understand how we derive the expression of the quantum Hamiltonian, given the value of energy in classical ...
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Different results for the Hamiltonian of a disc rolling on an inclined plane

$\hskip2in$ Starting from a Lagrangian of a disc rolling down on a inclined plane without slipping, given by: $$ \mathcal{L}=\frac{M}{2}\dot{x}^2+\frac{MR^2}{4}\dot{\theta}^2+Mg(x-L)\sin(\alpha) \...
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Motivating momentum without Newton's laws [closed]

I have been thinking recently about how one could introduce and motivate elementary mechanics starting from the hamiltonian (or lagrangian, but that is not mostly what I'm thinking about) point of ...
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Solving Liouville's Equation for the Harmonic Oscillator and Fluctuating energy

I am trying to solve the Liouville eqaution of the classic harmonic oscillator with fluctuating energy and arbitrary initial condition $\rho_0$. I want to approach the problem using the method of ...
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True Hamiltonian in Geometrodynamics

If I set $N=1$ and $N^a=0$ in the Einstein-Hilbert action $$S[N,N^a,q_{ab}]=\int\sqrt{\mathrm{det}(g)}\,R\,\mathrm d^4x \,\text{,}$$ expressed in terms of ADM variables, then $N$ and $N^a$ are no ...
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How are asymptotic states obtained from a Lagrangian?

I am trying to obtain the formula for a scattering process but I don't quite understand the concept of asymptotic states. I know that: For a Lagrangian, such as $(1)$, where the last term represents ...
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Jacobian rules with canonical transformations

If we consider a canonical transformation from $(q,p)$ to $(Q,P)$, it is stated in several sources that by Jacobian rules, $$ \frac{\partial(Q,P)}{\partial(q,p)} = \frac{\partial(Q,P)/\partial(q,P)}{...
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Hellmann-Feynman theorem for nearly degenerate states

If we have: $$\tag{1} E(P)=\int \psi_a(P)^{*} H(P) \psi_b(P)\ \mathrm{d} \tau $$ Then taking the derivative wrt. the parameter P yields: $$ \begin{aligned} \frac{\mathrm{d} E}{\mathrm{d} P} &=\int\...
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Is $H = T + U$ for a pendulum on a circle movement?

I have this problem: Obtain Hamilton's equations of motion for a plane pendulum of length $l$ with mass point $m$ whose radius of suspension rotates uniformally on the circumference of a vertical ...
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Tight-binding in a semi-infinite square lattice

I have a problem understanding how changing the boundaries from a periodic lattice to a finite lattice. For example, if we have a 2D square lattice of lattice constant $a$ whose $x$ axis has only $N_x$...
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Mathematical Construction: ADM Formulation in General Relativity

I'm doing my undergraduated thesis and now I'm looking for references that presents ADM Formulation in General Relativity mathematically. I studied the basic of General Relativity theory by O'Neill ...
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Non-hamiltonian systems which evolve into hamiltonian by change of coordinates

I am very new to the subject, so please forgive my very naïf question. I learned that there are some non-hamiltonian systems which can become hamiltonian, just by a change of coordinates. I was given ...
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Hamiltonian of a quantum field that is minimally coupled to gravity

The action for the gravitational field is known as the Einstein-Hilbert action: $$\begin{equation} S_{G}=\int d^4 x \sqrt{|g|} R \end{equation}$$ where $R$ is the Ricci scalar. The ...
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Quantum dissipation when coupling a system with a heat bath

In quantum mechanics, a good way to model dissipation of energy is to couple your system with a heat bath. This heath bath can be represented by a chain of harmonic oscillators. The Hamiltonian of the ...
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When are the derivatives applied over the trajectories in Classical Mechanics?

Given a dynamical system with canonical variables $q$ and $p$. The equations of motion are given by Hamilton's equations \begin{equation} \dot{q}=\frac{\partial H}{\partial p} \\ \dot{p}=-\frac{\...
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Physical interpretation of the symplectic property for particle systems

This this Wikipedia page states that symplecticness (being symplectic) is a property of particle systems governed by Hamilton's equation. I am thinking of classical-mechanics point particles described ...
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Why/Do Hamilton's equations Hold with Complex Variables?

I am investigating the problem of taking a hamiltonian of bilinear terms, and converting them into a bunch of uncoupled oscillators, such as in a periodic lattice. To do this, you have to introduce ...
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Issue with a derivation of the Hamilton-Jacobi equation

I'm trying to derive the HJ the easiest way I can but some issues come up. $$\mathrm{dS}=\dfrac{\partial S}{\partial q}\mathrm{d}q+\dfrac{\partial S}{\partial t}\mathrm{d}t\Rightarrow\displaystyle{S=\...
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Legendre transform in classical mechanics and statistical entropy maximization

I am trying to get some understanding of convex optimization, and in particular of why the Legendre transform appears in certain optimization problems. I am particularly interested in two examples, ...
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Hamiltonian in gravity with $\Lambda =0$ and using it to generate time translations

Time translation is generated by Hamiltonian. In gravity, the bulk Hamiltonian for closed $d$ hypersurfaces (obtained by ADM decomposition of $d+1$ spacetimes) is 0. This is basically a constraint of ...
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Dirac action momenta conjugate to conjugate field

Consider the Dirac action $S=\int d^4x\bar{\psi}(x)(i\not\partial-m)\psi(x)$. Since there are no time derivative of $\bar{\psi}$, we get the constraint that its canonical momenta vanishes. This ...
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Invertibility of the Legendre Transformation

The above image shows the Legendre Transformation in the context of an introduction to the Hamiltonian formalism. My question is in 4.6, wherein $u(x, y)$ has been defined; what is the guarantee ...
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Why do we even need Hamiltonian formalism if Lagrangian formalism is basically the same at the basic level? [duplicate]

I have just started studying Classical Mechanics and I just can't wrap my head around why we legendre transform lagrangian to hamiltonian if it just reverses velocity and conjugate momentum.
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Contradiction in canonical transformation

The problem I'm supposed to solve is finding $Q$, such that $(p,q)\rightarrow(P,Q)$ is a canonical transformation. In this case $\mathcal{H}=\frac{p^{2}+q^{2}}{2}$ and the new hamiltonian $\mathcal{K}$...
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Does a Lagrangian always generate a unique Hamiltonian?

Hamiltonian is related to Lagrangian with the equation: $$H= p\dot{q}- L(q,\dot{q},t) $$ Now, $H$ is function of $p,q,t$ so the Hamiltonian to be unique, $\dot{q}$ must be expressed uniquely using $...
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Hamiltonian matrix elements involving ladder operators for spin-1 state

I am reading the Doctorate thesis 'Zero-Field Anisotropic Spin Hamiltonians in First-Row Transition Metal Complexes: Theory, Models and Applications' (link: https://tel.archives-ouvertes.fr/tel-...
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Expanding universe and Time-translation invariance

My understanding is that there are 5 implicit assumptions when introducing the Lagrangian formulation. Assumptions: We appear to know that the principle of stationary action is true for the universe ...

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