The hamiltonian-formalism tag has no wiki summary.
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Hamiltonion in 2-dimensions?
I am trying to construct a Hamiltonian for a system in 2 dimensions using MATLAB.
I am not sure how this Hamiltonian will look like in matrix form.
If somebody can help me visualize this matrix that ...
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1answer
34 views
Finding Hamilton's equations given a Hamiltonian
I am trying to find Hamilton's equations for a general Hamiltonian given by $$H[u]=\int_\mathbf{R} \phi(u,u_x)dx$$
Suppose $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial ...
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1answer
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Peculiar Hamiltonian Phase space
I was solving an exercise of classical mechanics :
Consider the following hamiltonian
$H(p,q,t) = \frac{p^2}{2m} + \lambda pq + \frac{1}{2}m\lambda^2\frac{q^6}{q^4+\alpha^4}$
Where ...
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2answers
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What would happen if energy was conserved but phase space volume wasn't? (and vice-versa)
I'm trying to understand the relationship between the two conservation laws. As I understand, Liouville's result is a weaker condition: it relies merely on the particular form assumed by Hamilton's ...
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3answers
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Physical interpretation of Poisson bracket properties
In classical Hamiltonian mechanics evolution of any observable (scalar function on a manifold in hand) is given as
$$\frac{dA}{dt} = [A,H]+\frac{\partial A}{\partial t}$$
So Poisson bracket is a ...
3
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1answer
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Phase space in quantum mechanics and Heisenberg uncertainty principle
In my book about quantum mechanics they give a derivation that for one particle an area of $h$ in $2D$ phase space contains exactly one quantum mechanical state.
In my book about statistical physics ...
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1answer
49 views
Is symplectic form in Hamiltonian mechanics a physical quantity?
Is symplectic form $dp_i \wedge dq_i$ in Hamiltonian mechanics a physical quantity? It feels to me to be something different than say energy, momentum or mass. Like just certain structure.
The real ...
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1answer
57 views
Quantum mechanical analogue of conjugate momentum
In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
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2answers
208 views
primary constraints for constrained Hamiltonian systems
I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum
dynamics of constrained Hamiltonian ...
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1answer
103 views
rate of change of spring potential energy $\frac{dU}{dt}$
Suppose we have a setup like this. In orange are two wooden sticks sort of things, and they are attached to the block of mass $m$(as usual) at a joint which is hinge type something. A similar ...
2
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2answers
84 views
Heisenberg evolution equation for $\hat{\phi}$
Consider quantum Hamiltonian of free massive scalar particle:
$$\hat{H} = \int d^3x \left[\frac{1}{2} \hat{\pi}^2 (t, \vec{x}) + \frac{1}{2} \partial_i \hat{\phi}(t, \vec{x}) \partial_i \hat{\phi}(t, ...
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1answer
133 views
The relation between Hamiltonian and Energy
I know Hamiltonian can be energy and be a constant of motion if and only if:
Lagrangian be time-independent,
potential be independent of velocity,
coordinate be time independent.
Otherwise
...
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2answers
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From Lagrangian to Hamiltonian in Fermionic Model
While going from a given Lagrangian to Hamiltonian for a fermionic field, we use the following formula. $$ H = \Sigma_{i} \pi_i \dot{\phi_i} - L$$ where $\pi_i = \dfrac{\partial L}{\partial ...
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1answer
55 views
Deriving equations of motion of polymer chain with Hamilton's equations
This is related to a question about a simple model of a polymer chain that I have asked yesterday. I have a Hamiltonian that is given as:
$H = \sum\limits_{i=1}^N \frac{p_{\alpha_i}^2}{2m} + ...
2
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1answer
52 views
Hamiltonian of polymer chain
I'm reading up on classical mechanics. In my book there is an example of a simple classical polymer model, which consists of N point particles that are connected by nearest neighbor harmonic ...
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2answers
163 views
Weyl Ordering Rule
While studying Path Integrals in Quantum Mechanics I have found that [Srednicki: Eqn. no. 6.6] the quantum Hamiltonian $\hat{H}(\hat{P},\hat{Q})$ can be given in terms of the classical Hamiltonian ...
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0answers
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The consistency conditions of constrained Hamiltonian systems
I am studying the Hamiltonian description of a constrained system. There are some questions puzzled me for days, which I have been stuck on it. From the lagrangian, we can obtain the primary ...
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2answers
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Quantum Mechanics Notation for BRA KET
I've been given this homework problem, but I do not understand its notation.
Please perform the following where the wavefunctions are the normalized eigenfunctions of the harmonic oscillator ...
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0answers
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Second order action ADM formalism
I am trying to derive the second order action
$$S_{(2)}~=~\frac{m_{pl}^{2}}{8}\int a^{2}[(h_{ij}')^{2}-(\partial_{i}h_{ij})^{2}]d^{4}x, $$
used for tensor fluctuations derived from the ADM ...
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2answers
191 views
The string Poisson bracket
Where does the factor $\frac{1}{T}$ ($T$ is the string tension) in this Poisson bracket come from?
$$ \{X^{\mu}(\tau,\sigma),\dot{X}^{\nu}(\tau,\sigma')\} ~=~ ...
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3answers
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Other application of Liouville's theorem besides thermodynamics
Are there any other important practical and theoretical consequences of Liouville's theorem on the conservation of phase space volume besides the calculation of the microcanonical potential in ...
3
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1answer
80 views
Is Hamilton-Jacobi equation valid for only conserved systems?
From derivation of Hamilton-Jacobi (HJ) equation one can see that it is only applicable for conserved systems, but from some books and Wikipedia one reads the HJ equation as
...
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1answer
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Does a constant of motion always imply a Hamiltonian formulation?
If a continuous dynamical system has a constant of motion that is a function of all its variables, and is not already evidently Hamiltonian, is it always possible to use a change of variables and ...
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Are Poisson brackets of second-class constraints independent of the canonical coordinates?
Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\{\chi_N(q,p),\chi_M(q,p)\}_P$ . Is ...
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1answer
80 views
Eikonal approximation for wave optics. Why follow the unit vector parallel to the Pointing vector?
The description of the passage from wave optics to geometrical optics claims that light rays are the integral curves of a certain vector field (the Pointing vector direction, normalized to 1). Here ...
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1answer
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Find generating function $F_1$ for canonical trasformation
I'd like to know the steps to follow to find the generating function $F_1(q,Q)$ given a canonical transformation.
For example, considering the transformation
$$q=Q^{1/2}e^{-P}$$
$$p=Q^{1/2}e^P$$
...
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2answers
77 views
Hamiltonian constraint in spherical Friedmann cosmology
I'm taking a GR course, in which the instructor discussed the 'Hamiltonian constraint' of spherical Friedmann cosmology action. I'm not quite clear about the definition of 'Hamiltonian constraint' ...
2
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1answer
55 views
Solution of motion in hamiltonian formalism
I have these canonical equations:
$$\dot p = - \alpha pq$$
$$ \dot q =\frac{1}{2} \alpha q^2$$
I have to find $q(t)$ and p$(t)$, considering initial conditions $p_0$ and $q_0$.
I thought to simply ...
2
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1answer
157 views
Find the Hamiltonian given $\dot p$ and $\dot q$
I have these equations:
$$\dot p=ap+bq,$$
$$\dot q=cp+dq,$$
and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$.
To answer to the first ...
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1answer
126 views
Graphical representation of Hamilton's equation of motion [closed]
Position time graph for the Hamilton's equations motion for a simple pendulum.
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3answers
190 views
Factors of $c$ in the Hamiltonian for a charged particle in electromagnetic field
I've been looking for the Hamiltonian of a charged particle in an electromagnetic field, and I've found two slightly different expressions, which are as follows:
$$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 ...
0
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1answer
292 views
Hamilton's equations for a simple pendulum
I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with
$$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$
Now Hamilton's equations will be:
$$\dot ...
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2answers
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Hamiltonian or not?
Is there a way to know if a system described by a known equation of motion admits a Hamiltonian function? Take for example
$$ \dot \vartheta_i = \omega_i + J\sum_j \sin(\vartheta_j-\vartheta_i)$$
...
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2answers
132 views
Hamiltonian and non conservative force
I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$.
I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is ...
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1answer
201 views
Cyclic Coordinates in Hamiltonian Mechanics
I was reading up on Hamiltonian Mechanics and came across the following:
If a generalized coordinate $q_j$ doesn't explicitly occur in the
Hamiltonian, then $p_j$ is a constant of motion ...
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1answer
116 views
Canonical transformation and Hamilton's equations
I was trying to prove, that for a transformation to be Canonical, one must have a relationship:
$$
\left\{ Q_a,P_i \right\} = \delta_{ai}
$$
Where $Q_a = Q_a(p_i,q_i)$ and $P_a = P_a(p_i,q_i)$.
Now ...
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2answers
142 views
Good book for Analytical Mechanics
What is a good book for Analytical Mechanics?
To be more specific, I would prefer a book that:
Is written "for mathematicians", i.e. with high mathematics precision (for example, with less emphasis ...
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3answers
201 views
Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation
In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
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1answer
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Meaning of a canonical transformation “preserving” a differential form?
In Chapter 9 of Arnold's Mathematical Methods of Classical Mechanics, we find the following definition:
Let $g$ be a differentiable mapping of the phase space $\mathbb R^{2n}$ to $\mathbb R^{2n}$. ...
1
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1answer
81 views
Noise spectrum of two systems and interacting Hamiltonian
I've been discovering recently the concept of noise spectrum, defined as:
$$S_{xx}[\omega] = \int dt<x(t)x(0)>\text{e}^{-i\omega t}$$
Roughly the Fourrier transform of the two-point function.
...
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1answer
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Operator Ordering Ambiguities
I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity.
What does that mean?
I tried googling but to no avail.
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1answer
105 views
Potential Energy tends to infinity on the N-Body Problem
I need help to solve this problem related with the N-Body problem, i dont understand quite well what I need to define or to express in order to solve it.
We assume a particular solution to the N-Body ...
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5answers
272 views
Why don't we use the concept of force in quantum mechanics?
I'm a quarter of the way towards finishing a basic quantum mechanics course, and I see no mention of force, after having done the 1-D Schrodinger equation for a free particle, particle in an ...
2
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4answers
302 views
Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations
Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression:
$\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$
where, ...
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4answers
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What makes an equation an 'equation of motion'?
Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint.
For example, in the ...
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2answers
162 views
Relativistic Hamiltonian Formulations [duplicate]
Possible Duplicate:
Hamiltonian mechanics and special relativity?
The Hamiltonian formulation is beautifully symmetric. It's a shame that the explicit time derivatives in Hamilton's ...
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2answers
444 views
Poisson brackets: prove that they are canonical invariants
EDIT: I haven't forgotten to accept answer, the question is still open..
I need a clarification about Poisson brackets.
I'm studying on Goldstein's Classical Mechanics (1 ed.).
Goldstein proves ...
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3answers
504 views
What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)
What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)?
I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ...
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Non-Integrable systems
Integrable systems are systems which have $2n-1$ time-independent, functionally independent conserved quantities (n being the number of degrees of freedom), or n whose Poisson brackets with each other ...
2
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1answer
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Yang Mills Hamiltonian
Do you know why in the quantization of SU(2) Yang Mills Gauge Theory, it is always chosen the Weyl (temporal) gauge to derive the Hamiltonian?
Is it possible to fix another gauge?
