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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Phase Portraits Given Hamiltonian

Given a Hamiltonian say $$ H = 5p^2 $$ What is the correct procedure for producing a phase portrait. My initial thoughts were to solve the system of equations $\frac{dq}{dt} = 0$ and $\frac{dp}{dt} ...
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Liouville's Theorem: Prove the relationship between $\nabla\cdot\underline{v}=0$ and Hamiltonian Conservation

Let: $\underline{r} = (q_{1}, q_{2},...., p_{1}, p_{2}..)^{T}$ where $p$ and $q$ are the normal canonical co-ordinates. Then for $\underline{v} = \dot{\underline{r}}$, it can be shown $\nabla \cdot ...
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Counting number of degrees of freedom in constrained system

Following Counting degrees of freedom in presence of constraints, we know that there would be N-2M-S dofs if we have M 1st-class constraints and S 2nd-class constraints in N-dim phase space. I don't ...
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Canonical Transformation [duplicate]

How can I prove that the following transformation is canonical: $\begin{cases}\overline{q}_i=\dfrac{q_i}{Q} \\ \overline{p}_i=Qp_i-2Pq_i \end{cases},\ i\in\overline{1,n}$ where $Q=\sum_{i=1}^n ...
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Recovering a symmetry transformation from a conserved charge

I'm going through some notes on how to apply the Hamiltonian formalism to systems with gauge invariance and I found a derivation of Noether's theorem I had never seen before. The idea is roughly that ...
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Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), ...
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A physical system is described by the following Lagrangian: $ L = \frac{m}{2} (\dot{\rho}² + \rho ² \dot{\phi} ² + \dot{z} ²) + a \rho² \dot{\phi}$ [closed]

Where $a$ is a constant and $(\rho,\phi,z)$ are cylindrical coordinates. I found the following Hamiltonian $ H =\frac{m}{2}(\dot{\rho}² + \dot{z}² + \rho²\dot{\phi}²)$. The problem asked me to find ...
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Energy conservation Hamiltonian dependency

Suppose the a system has a Hamiltonian $H = H(q,p)$, and suppose $H$ does not depend explicitly on time. If $H\neq E$ the total energy of the system, does this necessarily say that $E$ is not ...
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Can one write down a Hamiltonian in the absence of a Lagrangian?

How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to ...
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Hamilton's Equations

The last step of this derivation of Hamilton's Equations is what's making me doubt it. It is as follows: Assuming the existence of a smooth function $\mathcal{H}(q_i,p_i)$ in $(q_i(t), \,p_i(t))$ ...
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Hamiltonian form: inverting the general momenta

I am studying analytical mechanics and am confused with the part about Hamiltonian form. My textbook develops the whole theory of Legendre transforms, in order to invert the formula ...
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Find the error: If $L_x$ and $L_y$ are zero, then $L_z$ is conserved

From Goldstein's Classical Mechanics (2nd ed.), problem 38 of chapter 9 basically says the following: It's been shown that the Poisson bracket of two constants of the motion is also a constant of ...
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35 views

Calculate lapse function from the metric

I have a technical question about the lapse function: Assume I have some given (Lorentzian) metric $g$. I have seen the following definition of the lapse function $\alpha^{-2}=-g(\nabla f, \nabla ...
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2answers
54 views

Deriving Hamilton's equation of motion

I am trying to derive Hamilton's equations of motion without using Lagrange's method but am left with an additional factor of $1/2$. Where am I going wrong? Please note this in not a homework ...
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Are there fifth-kind and sixth-kind generating functions?

In Goldstein's Classical Mechanics (2nd ed.), Section 9-1, pgs. 382-385, the generating functions (hereafter denoted $F$) for canonical transformations are introduced. From here on out, I'll refer to ...
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Independent canonical coordinate variables?

In Goldstein's Classical Mechanics (2nd ed.) on section 9-1 page 382, there is a discussion about finding a canonical transformation $(q_i,p_i)\rightarrow (Q_j(q_i,p_i,t),P_j(q_i,p_i,t))$ from a given ...
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Help using Hamiltonian mechanics [closed]

I didn´t understand how to use Hamiltonian for some mechanical problems, in particular in a two-body $(m_1, m_2)$ attached by a string $(k,l). First, calculating The lagrangian: ...
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Question about CSCO for identical 1/2 spin particles?

I have two, identical 1/2 spin particles which interact through the hamiltonian $$H = \frac{p_1^2}{2m} + \frac{p_2^2}{2m} + V(|\vec{r_1}-\vec{r_2}|) $$ and I have to determine the observables that ...
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How to describe time-shifts in Noether's theorem in Hamiltonian formalism

As was described in, for example, this post, one can formulate Noether's Theorem also in Hamiltonian Mechanics. Symmetries are then represented by vector fields generated by observables whose Poisson ...
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How to show period is defined by $T=dS/dE$ (V.I. Arnold Mathemtical Physics)

I'm looking at a book by VI Arnold on mathematical physics and I've hit a roadblock pretty early on. I'll quote the question: "Let $S(E)$ be the area enclosed by the closed phase curve ...
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Relation between interaction Lagrangian and interaction Hamiltonian

I work with this interaction Lagrangian density $$\mathcal{L}_{int} = ia\bar{\Psi}\gamma^\mu\Psi Z_\mu +ib(\phi^\dagger\partial_\mu \phi - \partial_\mu\phi^\dagger \phi)Z^\mu,$$ where $Z^\mu$ is an ...
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Derive statistical entropy from information entropy

I'm trying to compute the entropy of an Hamiltonian system with a fixed energy from the information entropy. Consider a random variable $X$ (state) to be uniformly distributed in a domain $\Omega$ ...
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Birkhoff Method for Harmonic Oscillator Perturbation

Problem: Given Hamiltonian $$H = \frac12 (p^{2}+q^{2})+q^{3}-3qp^{2}$$ make a perturbative canonical transformation $(q,p) \rightarrow (Q,P)$ such that the new Hamiltonian, apart from terms of degree ...
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Constructing a Hamiltonian from a mass matrix?

I was solving some questions regarding the Hamiltonian, which required a lot of algebra, but as I finished and looked professor answer I saw that he constructed a matrix from the kinetic energy and ...
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Is time-1 map of a Hamiltonian vector field on a cylinder always twist?

I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are closed ...
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Canonical transformation question

Let $(\vec{r},\vec{p})$ denote set of canonical variables. Assume a system is described by the following Hamiltonian $$H(r,p) = \frac{1}{2m}(p_1^2 + (p_2 - \beta*x_1)^2 + p_3^2),$$ where $\beta$ ...
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Harmonic oscillator and cyclic coordinates

I am reading goldstein there is some comment I don't understand. Consider the following hamiltonian $$H = \frac{p^2}{2m} + \frac{kq^2}{2}$$, which can be rewritten as follows $$H = \frac{1}{2m}(p^2 ...
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Field theory: equivalence between Hamiltonian and Lagrangian formulation

Let $\mathscr{B}$ be a space of physics we have and $\mathscr{T}$ be the duration. Let $\mathscr{L}$ be a lagrangian density of the field such that the action is a functional of ...
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How can we get the interaction hamilton $H_\text{int}$ from the Lagrange $L$?

After we quantize the free field we continue on determining the form of $H$. We can impose, by example: $$H=H_0+\lambda V_\text{int}$$ My question is, can we determine $H_\text{int}$ by the ...
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How to impose canonical commutation relations when quantising a field

I believe this is a simple question, however I cannot find it explained anywhere what the term: "Impose canonical commutation relations" means. If I have a classical equation, and I wish to quantise ...
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Volume as a choice of measure in phase space

For equilibrium systems, we expect the Liouville theorem to hold. This theorem states that the density function of the states of the system is a constant of motion, which in turn can be translated ...
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How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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Hamiltonian of a 1D Linear Harmonic Oscillator [closed]

Show that for the one-dimensional linear harmonic oscillator the Hamiltonian is: $$H = \frac{1}{2}[P^2+\omega ^2 X^2]-\frac{1}{2}\omega \hbar$$ $$=\frac{1}{2}[P+i\omega X][P-i\omega X]+\frac{1}{2} ...
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Lagrangian vs Hamiltonian and symmetry of a theory

It is said that since the path-integral formulation of quantum mechanics/or quantum field theory uses the Lagrangian rather than the Hamiltonian, as the fundamental quantity, it preserves all the ...
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Generating function as a prove of a canonical transformation [closed]

Does the existence of a generating function $ F $ prove that a given transformations is canonical? How? The thing is: Show that the transformation is canonical $$ Q=p+iaq $$ and $$ ...
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Generallized Canonical Ensemble - Isobaric Ensemble

I am trying to understand the way generalized canonical ensembles like the pressure ensemble are derived from the standard canonical ensemble. In the derivation for the standard form, one defines a ...
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Sinusoidal variation of a generalised coordinate with its conjugate momentum

The hamiltonian of a system with 2 degrees of freedom is given by: H= 0.5(p12q14+p22q12-2aq1) where 'a' is a constant. I am to show that q1 varies sinusoidally with p1. My attempt was to write down ...
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What are the implications of integrating the Poisson bracket? [closed]

Reading Ref. 1 I admit I am a little lost in some places. I was hoping that someone in this area could explain the basic premise of integrating the Lie bracket and further is it connected to nlab's ...
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Is the Hamiltonian conserved or not?

The question is the very last sentence at the end of this post. In this post, I'll first show that the Hamiltonian is conserved since it does not have explicit dependence on time and then show that ...
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Quasilocal stress tensor

I have been reading through the paper hep-th/9902121 and have a few questions about the first five lines of the introduction: 1) "In a generally covariant theory, it is unnatural to assign a local ...
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how to reduce the order of the hamiltonian equation for electrical problem given below?

I am having my FYP in this Hamiltonian project to analysis the integrator for Hamiltonian system. Can anyone please guide me how to reduce the equations to first order using substitution method?
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How to reduce the order of Hamiltonian equation for electrical problem [duplicate]

I want to reduce the order of this Hamiltonian but I don't know how to proceed. The equations are given below: $$H(p,q) = \frac{1}{2} (kq^2) + \frac{p^2}{2m} $$ This is the Hamiltonian for a simple ...
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Full time-derivative, Poisson brackets and Hamilton's equations (classical mechanics)

While studying Poisson brackets in classical mechanics and the derivation of $\dot{q_j}=\{q_j,H\}$ and $\dot{p_j}=\{p_j,H\}$ form of Hamilton's equations I encountered a surpsing identity, which led ...
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Canonical Commutation Relations in arbitrary Canonical Coordinates

If one were to formulate quantum mechanics in an arbitrary canonical coordinate system, does he impose canonical commutation relations using Dirac's recipe? ...
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Dirac bracket for a constrained particle

I am trying to work through a simple example of how to use the Dirac bracket from the following paper. In particular section 4 where the authors consider a constrained particle with the following ...
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Conservation of angular momentum poisson brackets vs newtonian mechanics [closed]

So I have the following system. For a mass falling due to gravity the given Hamiltonian is $$ H = \frac{1}{2m}\left( P^{2}_{x} + P^{2}_{y} \right) + mgy $$ In Cartesian coordinates then, $$ x = v_o ...
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Diagonalizing a linearized optomechanical Hamiltonian

in the paper Nonlinear Interaction Effects in a Strongly Driven Optomechanical Cavity, the authors diagonalize the Hamiltonian (equation (2) in the paper) $$H_1=−Δd^†d+ω_Mb^†b+G(d+d^†)(b+b^†)$$ in ...
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How general are Noether's theorem in classical mechanics?

I'm going through the derivations of Noether's theorems and I have several criticisms as to how they are presented in popular sources (note that I'm only referring to classical mechanics here and not ...
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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: ...
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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 ...