A notional even-dimensional space representing all relevant states of a dynamical system; it normally consists of all components of position and momentum/velocity involved in that unique specification. Use for both classical and quantum physics.

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Quantum versus classical computation of the density of states

If I consider for instance N non interacting particles in a box, I can compute the energy spectrum quantum mechanically, and thus the number of (quantum) microstates corresponding to a total energy ...
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78 views

Action variables in canonical transformations

Let's suppose we have a Hamiltonian $H(p_k, q_k)$ and we want to transform it via a canonical transformation to one Hamiltonian which doesn't depend on the new coordinates $w_k$, but only in the ...
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71 views

Fixing time in Feynman phase space path integral

The phase space version of Feynman's path integral expression for the free particle propagator involves a (formal) sum over paths in phase space with fixed $q$ endpoints and (as far as I'm aware) ...
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68 views

Differential cross section $d\sigma/dp^{\gamma}_{T}$?

Why we care about $d\sigma/dp^{\gamma}_{T}$? What the physical meaning of it? Why not plot $\sigma$ follow $p^{\gamma}_{T}$?. As in this picture.
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Motivation for covariant phase space

The covariant phase space idea, in one sentence, is that there is a natural symplectic structure on the space of the classical trajectories of a system and that the usual $(q,p)$ coordinates just ...
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75 views

Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = \...
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On the Liouville-Arnold theorem

A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to ...
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47 views

Non-canonical transformation

I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
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189 views

A question on Lagrangian dynamics an the velocity phase space

I've struggled in the past with understanding why we can treat position and velocity as independent variables in the Lagrangian, but I think I may have finally become a bit more enlightened on the ...
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Group of translations in two dimensions - A weird treatment

Again, as usual Schwinger leaves me startled as he writes, the Hermitian displacement operator in 2D is $$ G = p_1\delta x_1 +p_2 \delta x_2 $$ Now, we know clearly that this group is an Abelian ...
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38 views

Definition for a trajectory in phase-space

When we say "a trajectory in phase space", when the parameter is time, do we mean the set of points in phase-space corresponding to a continuous segment in time? Does it have to be continuous? Does it ...
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59 views

What is the phase space volume in terms of angular momentum?

Given a rigid rotor Hamiltonian, defined along the principle axes as $$ H = \sum_{i=1}^3 \frac{L_i^2}{2I_i} $$ say we would like to compute the classical partition function of this system. Is the ...
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55 views

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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Preservation of phase space volume: the extension from “small” times to generic times

Having a classical system whose evolution is described by \begin{equation} \dot{\phi_t}(x) = f(\phi_t (x))\\ \phi_0 (x) = x \end{equation} denoting with $\phi_t (x)$ the evolution for a time t of the ...
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68 views

Time evolution of the density of phase points for an ensemble

I want to calculate the time evolution of the density of phase points for an ensemble of N harmonic oscillators. However, I intended to do so without using the Liouville equation. Sure, I want to ...
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52 views

What are resonant tori?

What is the definition of a resonant/invariant torus (in the phase space of a Hamiltonian system)? Are there non-resonant tori?
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233 views

What does it mean for a phase space trajectory to be “long” and “stable”?

What does it mean for a phase space trajectory to be "long" and "stable"? I understand the concept of a trajectory in phase space but not how these adjectives can be applied to one. Thanks.
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12 views

Phase space - Diameter of attractor

For a dynamical system, with a phase space reconstructed from single scalar measurement, how do I calculate the diameter of the attractor (such as the one shown in figure below)?
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29 views

Interpretation of contourplot pendulum

I've made this plot of a function that evaluates the size of the angle on the x-axis, and the velocity of the angle for the pendulum on the y-axis. I'm having a hard time interpreting the meaning of ...
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After positron is produced after the target,what will the magnetic strength of the solenoid change phase space of the particle?

When positrons are generated from a particle beam hitting a dense target, solenoids are used to focus them, so what is the relationship between the magnetic fields of the axial direction(parallel to ...
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63 views

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 $\beta$ is ...
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243 views

Meaning of phase space density

I am trying to understand Liouville's theorem physically. It says that $\frac{\partial \rho}{\partial t} + \{\rho,H\} = 0$. Thus, we have $\frac{d \rho(q(t),p(t),t)}{dt}=0$. I would like to ...
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63 views

Can statistical mechanics be formulated generally in terms of phase space?

In many statistical mechanics books, notably Landau and Lifschitz' volume in the course on theoretical physics, the quantities central to statistical mechanics such as entropy are defined in terms of ...