The hamiltonian-formalism tag has no wiki summary.
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How to understand the matrix behind a Hamiltonian?
thanks to the answers I received to my previous questions, I could derive correctly an elegant partition function for my problem which resembles a second quantized model taking the particles to be ...
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1answer
58 views
Is this a valid derivation of the Legendre transformation from the Euler-Lagrange condition
E-L condition:
$$\frac{d p}{dt}=\frac{\partial L}{\partial q}$$
Where $p=\frac{\partial L}{\partial \dot{q}}$
Are the following steps valid:
$$\frac{\partial q}{dt} dp=\partial L$$
$$\dot{q} \: ...
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3answers
101 views
How is a Hamiltonian constructed from a Lagrangian with a Legendre transform
many textbooks tell me that Hamiltonians are constructed from Lagrangians like
$$L=L(q,\dot{q})$$
with a Legendre transformation to obtain the Hamiltonian as
$$H=\dot{q}\frac{\partial L}{\partial ...
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0answers
90 views
The Hamiltonian for clocks?
I am rather a theoretician and looking for a formalism to represent biological clocks by Hermitian operators.
The simplest thought model I am looking for is a formal representation of a clock (for ...
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1answer
85 views
Why do Lagrangians and Hamiltonians give the equations of motion? [duplicate]
I remember asking my second year Mechanics teacher about why do the Lagrangians give the equations of motion. His answer was that there is no answer to that, it is an empirical fact, and that asking ...
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24 views
Hamiltonian system: match transformations and constants of motion
I have a problem about the interpretation of an exercise.
Given the following Hamiltonian
$$H=\frac{\mathbf{p_0}^2}{2m}+\frac{\mathbf{p_1}^2}{2m}+\frac{\mathbf{p_2}^2}{2m}-2V(\mathbf{r_1}- ...
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58 views
A quicker way to verify that a function is a constant of motion?
I have three particles that we can indicate with $\alpha$ ($\alpha$=0,1,2), they are identified by the $r^i_\alpha$ coordinates and $p^\beta_j$ conjugata momenta ($\beta=0,1,2$ and $i,j=1,2,3$).
I ...
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1answer
56 views
How to find constant of motion for Hamiltonian system?
I have to find a constant of motion associated to this Hamiltonian but I don't know how to proceed.
$$H=\frac{\mathbf{p_0}^2}{2m}+\frac{\mathbf{p_1}^2}{2m}+\frac{\mathbf{p_2}^2}{2m}-2V(\mathbf{r_1}- ...
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2answers
58 views
Hamiltonian equations: can I divide a solution of motion for a constant?
I'm solving an exercise about Hamiltonian equations. I have followed the proceeding below. The results given by the book are different to mine because its first result is the half of mine (and the ...
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0answers
19 views
Delivered/Reflected Power by Drive on a Hamiltonian System
Imagine a SHO with a drive F(t). (or in general a Hamiltonian system)
What is the power delivered to the system and can we talk about the power reflected? is i am imagining say a MW oscillator ...
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1answer
57 views
Hamiltonian reduction having constant of the motion
I have this $2^n*2^n$ matrix that represent the evolution of a system of $n$ spin.
I know that I can have only one excited spin in my configuration a time.
(eg: 0110 nor 0101 ar not permitted, but ...
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27 views
Proving conservation of angular momentum in an elliptic billiard problem
This is for a course focusing on the connections between Newtonian, Lagrangian and Hamiltonian formalisms.
We're given an elliptic billiard table with foci 1 and 2, where $L_1$ and $L_2$ are the ...
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1answer
39 views
Is a particle subject to dissipation proportional to its velocity a hamiltonian system?
Why or why not? Im pretty sure that this isn't a hamiltonian system because it involves a dissipation term, but using the hamiltonian flow it gives me that the system is hamiltonian.
Thank you very ...
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0answers
56 views
Gradient involved commutator in $\phi^4$ theory
In a phi fourth theory, the Hamiltonian density is:
$$\mathcal{H}=\frac{1}{2}\pi^2+\frac{1}{2}(\nabla \phi)^2+\frac{1}{2}m^2\phi^2+\frac{\lambda}{4!}\phi^4$$
Now I impose the usual equal time ...
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1answer
43 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 $$\frac{\delta f[u]}{\delta u(x)}\equiv \frac{\partial f}{\partial ...
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1answer
69 views
Peculiar Hamiltonian Phase space
I was solving an exercise of classical mechanics :
Consider the following hamiltonian
$H(p,q,t) = \frac{p^2}{2m} + \lambda pq + \frac{1}{2}m\lambda^2\frac{q^6}{q^4+\alpha^4}$
Where ...
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2answers
102 views
What would happen if energy was conserved but phase space volume wasn't? (and vice-versa)
I'm trying to understand the relationship between the two conservation laws. As I understand, Liouville's result is a weaker condition: it relies merely on the particular form assumed by Hamilton's ...
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3answers
161 views
Physical interpretation of Poisson bracket properties
In classical Hamiltonian mechanics evolution of any observable (scalar function on a manifold in hand) is given as
$$\frac{dA}{dt} = [A,H]+\frac{\partial A}{\partial t}$$
So Poisson bracket is a ...
3
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1answer
140 views
Phase space in quantum mechanics and Heisenberg uncertainty principle
In my book about quantum mechanics they give a derivation that for one particle an area of $h$ in $2D$ phase space contains exactly one quantum mechanical state.
In my book about statistical physics ...
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1answer
51 views
Is symplectic form in Hamiltonian mechanics a physical quantity?
Is symplectic form $dp_i \wedge dq_i$ in Hamiltonian mechanics a physical quantity? It feels to me to be something different than say energy, momentum or mass. Like just certain structure.
The real ...
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1answer
72 views
Quantum mechanical analogue of conjugate momentum
In classical mechanics, we define the concept of canonical momentum conjugate to a given generalised position coordinate. This quantity is the partial derivative of the Lagrangian of the system, with ...
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2answers
211 views
primary constraints for constrained Hamiltonian systems
I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum
dynamics of constrained Hamiltonian ...
2
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1answer
110 views
rate of change of spring potential energy $\frac{dU}{dt}$
Suppose we have a setup like this. In orange are two wooden sticks sort of things, and they are attached to the block of mass $m$(as usual) at a joint which is hinge type something. A similar ...
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2answers
95 views
Heisenberg evolution equation for $\hat{\phi}$
Consider quantum Hamiltonian of free massive scalar particle:
$$\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
141 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
155 views
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
60 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
56 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
198 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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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
225 views
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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64 views
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 ...
4
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2answers
209 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
100 views
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 ...
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1answer
88 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
99 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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2answers
112 views
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
92 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
95 views
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$$
...
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2answers
83 views
Hamiltonian constraint in spherical Friedmann cosmology
I'm taking a GR course, in which the instructor discussed the 'Hamiltonian constraint' of spherical Friedmann cosmology action. I'm not quite clear about the definition of 'Hamiltonian constraint' ...
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1answer
61 views
Solution of motion in hamiltonian formalism
I have these canonical equations:
$$\dot p = - \alpha pq$$
$$ \dot q =\frac{1}{2} \alpha q^2$$
I have to find $q(t)$ and p$(t)$, considering initial conditions $p_0$ and $q_0$.
I thought to simply ...
2
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1answer
164 views
Find the Hamiltonian given $\dot p$ and $\dot q$
I have these equations:
$$\dot p=ap+bq,$$
$$\dot q=cp+dq,$$
and I have to find the conditions such as the equations are canonical. Then, I have to find the Hamiltonian $H$.
To answer to the first ...
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1answer
152 views
Graphical representation of Hamilton's equation of motion [closed]
Position time graph for the Hamilton's equations motion for a simple pendulum.
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3answers
196 views
Factors of $c$ in the Hamiltonian for a charged particle in electromagnetic field
I've been looking for the Hamiltonian of a charged particle in an electromagnetic field, and I've found two slightly different expressions, which are as follows:
$$H=\frac{1}{2m}(\vec{p}-q \vec{A})^2 ...
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1answer
329 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:
$$\dot ...
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2answers
267 views
Hamiltonian or not?
Is there a way to know if a system described by a known equation of motion admits a Hamiltonian function? Take for example
$$ \dot \vartheta_i = \omega_i + J\sum_j \sin(\vartheta_j-\vartheta_i)$$
...
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2answers
150 views
Hamiltonian and non conservative force
I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$.
I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is ...
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1answer
237 views
Cyclic Coordinates in Hamiltonian Mechanics
I was reading up on Hamiltonian Mechanics and came across the following:
If a generalized coordinate $q_j$ doesn't explicitly occur in the
Hamiltonian, then $p_j$ is a constant of motion ...
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1answer
122 views
Canonical transformation and Hamilton's equations
I was trying to prove, that for a transformation to be Canonical, one must have a relationship:
$$
\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
165 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 ...
