The hamiltonian-formalism tag has no wiki summary.
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Lorentz invariance of the 3 + 1 decomposition of spacetime
Why is allowed decompose the spacetime metric into a spatial part + temporal part like this for example
$$ds^2 ~=~ (-N^2 + N_aN^a)dt^2 + 2N_adtdx^a + q_{ab}dx^adx^b$$
($N$ is called lapse, $N_a$ is ...
3
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161 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
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 ...
3
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3answers
111 views
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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3answers
364 views
Type of stationary point in Hamilton's principle
In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense - we all ...
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1answer
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Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?
I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude:
$$F(r, t) = \frac{k}{r^2}e^{-at}$$
...
3
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2answers
154 views
Ordering Ambiguity in Quantum Hamiltonian
While dealing with General Sigma models (See e.g. Ref. 1)
$$\tag{10.67} S ~=~ \frac{1}{2}\int \! dt ~g_{ij}(X) \dot{X^i} \dot{X^j}, $$
where the Riemann metric can be expanded as,
$$\tag{10.68} ...
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2answers
155 views
Elimination of velocities from momenta equations for singular Lagrangian
this doubt is related to Generalized Hamiltonian Dynamics paper by Dirac.
Consider the set of $n$ equations : $p_i$ = $∂L/∂v_i$,
(where $v_i$ is $q_i$(dot) = $dq_i/dt$, or time derivative of ...
3
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2answers
237 views
Hamilton's equations in terms of initial conditions
I'm trying to understand the way that Hamilton's equations have been written in this paper. It looks very similar to the usual vector/matrix form of Hamilton's equations, but there is a difference.
...
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
...
3
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3answers
230 views
When Hamiltonian and the total energy are the same
In which condition, the Hamiltonian is the same as the total energy of the system, or say $H=T+V$?
3
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1answer
140 views
Abstract, generic derivations of energy
How generic can be derivation of energy?
In a system with gravity and masses – it is potential energy and kinetic energy.
What if a constraint would be specified that no mass and velocity should be ...
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2answers
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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 ...
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,$$
and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$.
To answer to the first ...
2
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2answers
209 views
Why is it important that Hamilton's equations have the four symplectic properties and what do they mean?
The symplectic properties are:
time invariance
conservation of energy
the element of phase space volume is invariant to coordinate transformations
the volume the phase space element is invariant ...
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4answers
302 views
Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations
Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression:
$\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$
where, ...
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1answer
185 views
How to explain the different forms of the Hamilton-Jacobi equation?
In Arnold's Mathematical Methods of Classical Mechanics, he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get
$$
H\left(\frac{\partial S_1(Q, q)}{\partial q}, ...
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1answer
209 views
ADM Hamiltonian formalism and Quantum gravity
is there a Hamiltonian reformultion of gravity ?=? if so if we use the usual Quantization scheme we can not we quantizy the gravity ??
in terms of a Gauge Theory with the potential $ A_{\mu}^{i} $ ...
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1answer
174 views
Expectation of a commutation relation
Is there any significance to: $\langle[H,\hat{O}]\rangle =0$ (which can easily be shown) where $H$ is the Hamiltonian, $\hat{O}$ is an arbitrary operator? Thanks.
2
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1answer
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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 ...
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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3answers
315 views
Dirac equation as Hamiltonian system
Let us consider Dirac equation
$$(i\gamma^\mu\partial_\mu -m)\psi =0$$
as a classical field equation. Is it possible to introduce Poisson bracket on the space of spinors $\psi$ in such a way that ...
2
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1answer
163 views
A good example of a nonlinear symplectomorphism?
What is a good example of a simple, physically useful nonlinear symplectomorphism $\kappa: \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$? I'm not much of a physicist, and all the examples I've worked ...
2
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1answer
318 views
Origins of the principle of least time in classical mechanics
Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least ...
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2answers
305 views
Correct application of Laplacian Operator
Not a physicist, and I'm having trouble understanding how to apply the Laplacian-like operator described in this paper and the original. We let:
$$ \hat{f}(x) = f(x) + \frac{\int H(x,y)\psi(y) ...
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2answers
84 views
Heisenberg evolution equation for $\hat{\phi}$
Consider quantum Hamiltonian of free massive scalar particle:
$$\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, ...
2
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1answer
93 views
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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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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1answer
95 views
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?
2
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1answer
161 views
Degeneracy and the Hamiltonian
How many linearly independent eigenfunctions can be associated with one degenerate eigenvalue of the Hamiltonian operator? (Is there a limit since it contains a 2nd order differential operator?) ...
2
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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 ...
2
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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 ...
2
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1answer
132 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
...
2
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1answer
146 views
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 $\mathbb R^{2n}$. ...
2
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3answers
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Quantizing first-class constraints for open algebras: can Hermiticity and noncommutativity coexist?
An open algebra for a collection of first-class constraints, $G_a$, $a=1,\cdots, r$, is given by the Poisson bracket $\{ G_a, G_b \} = {f_{ab}}^c[\phi] G_c$ classically, where the structure constants ...
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1answer
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Equivalence of classical and quantized equation of motion for a free field
Suppose a classical free field $\phi$ has a dynamic given in Poisson bracket form by $\partial_o\phi=\{H, \phi\}$. If we promote this field to an operator field, the dynamic after canonical ...
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2answers
84 views
Commutation for constraints
Suppose from the Hamiltonian I got the Primary constraints $$(\Phi_m,\Phi)$$ And $\dot \Phi_m$ , $\dot \Phi$ leads to secondary constraints $$(\gamma_m,\gamma)$$ respectively. Now if the commutation ...
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2answers
213 views
Counting degrees of freedom in presence of constraints
In a $N$ dimensional phase space if I have $M$ 1st class and $S$ 2nd class constraints, then I have $N-2M-S$ degrees of freedom in phase space. How can I calculate the degrees of freedom in ...
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1answer
170 views
A question regarding particle trajectories in the symplectic manifold formalism
How to solve a free particle on a 2-sphere using symplectic manifold formalism of classical mechanics ?
Is there a way to get coriolis effect directly, without going into Newton mechanics?
And is ...
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0answers
300 views
Calculation of the non-Gaussity parameter for primordial cosmological perturbations by the ADM Formalism
Maldacena has used the ADM Formalism in one of his papers (http://arxiv.org/abs/astro-ph/0210603) in computing the the three point correlation function (i.e the non-Gaussianity) parameter for ...
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3answers
380 views
Noether's theorem and “translations” of the Hamiltonian function
In a nutshell, Noether's theorem states that for every continuous symmetry a corresponding conserved quantity exists.
Now, the Hamiltonian equations of motion (let's talk about a classical system ...
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1answer
199 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
140 views
Hamiltonian Flow Map
I'm reading this article and am struggling with some of the terminology. What is the flow map for a Hamiltonian system? I'm looking for a rigorous definition really!
Many thanks in advance.
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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' ...
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3answers
189 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:
$$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 ...
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1answer
116 views
Canonical transformation and Hamilton's equations
I was trying to prove, that for a transformation to be Canonical, one must have a relationship:
$$
\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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1answer
187 views
interpretation of $\{H,L^2\}$
In Hamiltonian mechanics, we show $\{H,L_z\}=0$, which can be interpreted as the conservation of angular momentum around $Oz$. Following the same idea, how can we interprete $\{H,L^2\}$?
Is the ...
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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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2answers
386 views
Canonical transformations and conservation of energy
I have an important doubt about the nature of canonical transformations in hamiltonian mechanics.
Suppose I have a one-degree-of-freedom lagrangian system, whose hamiltonian depends explicitly on ...
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1answer
372 views
Conjugate Variables and Fourier Transforms in Classical Physics
Let q be a generalized coordinate with a conjugate momentum p and a potential resulting in a periodic motion of q. What is the meaning of the Fourier transform of q(t) over its period? Can this be ...
