Questions tagged [phase-space]
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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Why do “points” in phase space correspond to possible physical states of a system, rather than “lines”?
Why aren't “lines” the physical states of system, instead of “points”? In a simple harmonic oscillator, the motion of particles changes back and forth between kinetic and potential energy, but it ...
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Poincaré recurrence theorem in an infinite phase space
Regarding the Poincaré recurrence theorem there where already a few questions asked about boundness.
However, I was wondering whether the theorem could still, in some form, hold within a Hamiltonian ...
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Alternate statement for Liouville's theorem: Can you say that the relative velocity between any two representative points is zero?
So Liouville's theorem basically says the local density of representative points stays constant or that the flow of representative points resembles that of an incompressible fluid. Can you then say ...
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Regarding movement of representative points in phase space (Liouville's theorem)
Okay, so I'm studying statmech from Pathria.
Liouville's theorem talks about the flow of representative points in phase space. These representative points represent each and every microstate ...
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Interpreting one-dimensional Newtonian mechanics using complex numbers? (via Hamiltonian mechanics)
Let the configuration space of a single "point particle" be the one-dimensional affine space $\mathbb{A}^1 \cong \mathbb{R}$, with a chosen linear coordinate chart identifying some ...
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Sufficiency and necessity of symplectic conditions for time dependent canonical transformation
I'm studying Goldstein's classical mechanics book. I'm currently reading section 9.4, in particular, reguarding symplectic formalism, the author first proves for restricted canonical transformations, ...
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From phase portrait to equation
Given a certain phase portrait/phase space, what is the right approach in order to find an equation $\dot{x}=f(x)$ (or a set of equations $\dot{x_n}$) with a flow consistent with that portrait?
More ...
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Infinitesimal canonical transformations and Lie algebras
I'm currently studying classical mechanics, partly from Goldstein's book. I'm reading the part about infinitesimal canonical transformations (ICT) in the Poisson bracket formulation (section 9.6). ...
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Relation between the continuity equation of $\rho$ and Liouville's theorem
I have a question about a proof about the Liouville theorem and incompressibility of the phase space fluid.
It is a proof that $\nabla \cdot v = 0$ from the Liouville equation, starting from the ...
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Boltzmann correction factor for free particles but not for harmonic oscillators?
A) Suppose the microcanonical statistical mechanics of $N$ one-dimensional identical free particles of mass $m=1/2$, in an interval of size $L$. We must integrate over $2N$-dimensional phase space in ...
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What the Liouville equations for the one-dimensional Classical Caldirola-Kanai oscillator?
The Liouville equation is valid for a conservative system where the Jacobian equal one. So, the volume of the ensemble in the classical phase space in canonical coordinates (q,p) does not change. In ...
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How many independent variables for the distribution function in Liouville theorem?
I have a very simple question about Liouville theorem:
$$\frac{\partial f_N}{\partial t} + \sum_{i=1}^n \mathbf{\dot{q}_i} \frac{\partial f_N}{\partial \mathbf{q}_i} + \sum_{i=1}^n \mathbf{\dot{p}}_i \...
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Why is the Poisson bracket $\{q_i,p_j\}=\delta_{ij}$? [closed]
Why is $\{q_i,p_j\}=\delta_{ij}$?
Here the Poisson bracket $$\{F,H\}=\sum_{i=1}^f\frac{\partial F}{\partial q_i}\frac{\partial H}{\partial p_i}-\frac{\partial F}{\partial p_i}
\frac{\partial H}{\...
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Phase line and its configuration space
If I consider a system represented by a phase line or a one dimensional phase space, which is the configuration space? Is the dimension of the configuration space zero? Can I consider this phase line ...
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From the Poisson bracket equation to the Euler equations in fluid dynamics
In the book Introduction to Mechanics and Symmetry (Marsden, 1998, pp. 20–21), it is stated that the Euler equations for an incompressible, ideal fluid:
$$
\frac{\partial \mathbf{v}}{\partial t} + (\...
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