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21

Notice first that the phase space of any theory is nothing but the space of all its classical solutions. The traditional presentation of phase spaces by fields and their canonical momenta on a Cauchy surface is just a way of parameterizing all solutions by initial value data -- if possible. This is often possible, but comes with all the disadvantages that a ...


9

Your interpretation is not quite right. One sharp interpretation one can give to this "cutting" of phase space into cubes of size $h^{2N}$ (here $N$ is the dimension of the system's configuration space), is that it allows one to use classical phase space to count the number of energy eigenstates of the corresponding quantum hamiltonian. Instead of trying ...


8

No, it's not a problem. The reason is that, in order for expressions like $$\mu=-T\left(\tfrac{\partial S}{\partial N}\right)_{E,V}.$$ to be meaningful, you have to be using the grand canonical ensemble (or a generalisation thereof), in which particles are able to enter and leave the system. Consequently, $N$ stands not for an integer number of particles, ...


6

The phase-space represents the "number" of allowed final states (think of a discrete quantum system with degenerate final states, except that here we have a continuum). More final states makes the transition more likely to happen and thus gives it a shorter lifetime. Each of the reactions that you show has a three body final state and a single body ...


6

A metric structure $g$ and a symplectic structure $\omega$ are two very different structures, although sometimes they can co-exist in a compatible way. Unlike a symplectic structure, there are no Jacobi-like identity and no Darboux-like theorem for a metric structure. There exists a unique torsionfree metric connection $\nabla$ on a pseudo-Riemannian ...


5

See this article on the history of phase space. Assuming the article is to be trusted, Boltzmann noted that in a 2-D system the trajectories looked like Lissajous figures, and the shape of the Lissajous figure is determined by the relative phase of the two input signals. He then used the work phase to refer to that part of the configuration that was ...


5

Consider a non-relativistic massless particle with charge $q$ on a 2D torus $$\tag{1} x ~\sim~ x + L_x , \qquad y ~\sim~ y + L_y, $$ in a constant non-zero magnetic field $B$ along the $z$-axis. Locally, we can choose a magnetic vector potential $$\tag{2} A_x ~=~ \partial_x\Lambda, \qquad A_y ~=~ Bx +\partial_y\Lambda, $$ where $\Lambda(x,y)$ is ...


5

There are several different notions of microstates or distinguishability that might be relevant to your question. Coarse-graining of phase space into Planck cells. Consider two classical variables $x$ and $p$ with $x \sim x+x_0$ and $p \sim p+p_0$. You can think of this system as describing a particle that lives on a circle of radius $x_0$ and where ...


5

Let there be given a $2n$-dimenional real symplectic manifold $(M,\omega)$ with a globally defined real function $H:M\times[t_i,t_f] \to \mathbb{R}$, which we will call the Hamiltonian. The time evolution is governed by Hamilton's (or equivalently Liouville's) equations of motion. Here $t\in[t_i,t_f]$ is time. On one hand, there is the notion of complete ...


4

Complete integrability is far stronger than solvability of the initial value problem. Complete integrability implies the absence of chaotic orbits. More precisely, all bounded orbits are quasiperiodic, lying on invariant tori. Perturbations of a completely integrable system preserve only some of these tori; this is the KAM theorem. ...


4

It is misleading to write $\rho_i$ for the components of $\psi$, and as they are complex numbers, you cannot use these in the formula for entropy. The space of wave functions is (not $[0,1]\times S^1$ but) the Poincare sphere (or Bloch sphere) S^2, parameterized by quaternions (corresponding to points on the complexified circle). ...


4

$U(1)$ Chern-Simons theory with (physical) space a 2-torus is such an example. Its phase space is the gauge equivalence classes of flat connections on the 2-torus. These are specified by the holonomies around two 1-cycles forming a basis of $H_1(T^2)$. This is of course a 2-torus $U(1) \times U(1)$. Because of the form of the Chern-Simons action, these ...


3

What is meant is that $$\frac{d\rho(q(t), p(t), t)}{dt} = 0$$ when $q,p$ are solutions to Hamilton's equations. While it is notationally convenient and space-saving to not write everything out in this detail, it is as you noted confusing. This particular confusion actually has a name -- it's the first fundamental confusion of calculus. (There's a second ...


3

The formula you write down is one of thermodynamics. In the statistical mechanics version it is valid in the grand canonical ensemble only if you interpret the extensive variables as expectation values. (See, e.g., Chapter 9 of my online book Classical and Quantum Mechanics via Lie algebras, arXiv:0810.1019.) But expectation values are continuous even when ...


3

Generally, coadjoint orbits of a Lie group provide important examples of global symplectic manifolds. In general such systems are obtained by symplectic reduction from a more fundasmental description. For example, the spinning top is modelled for constant $J^2$ on a symplectic manifold $S^2$ that is a coadjoint orbit of the rotation group $SO(3)$. It is ...


3

1) On a symplectic manifold $(M,\omega)$, Liouville's theorem is often stated as that every Hamiltonian vector field $X_f=\{f,\cdot\}$ is divergence-free $$ {\rm div}_{\rho} X_f~=~0 ,$$ where the volume density $\rho$ comes from the canonical volume form $$\Omega~=~\rho dx^1 \wedge \ldots \wedge dx^{2n}.$$ Here the canonical volume form ...


3

If you are looking for a detailed answer as to why in general Lagrangians depend only on first derivatives, then you should read the answer in this question, as Qmechanic rightfully said. However, I suspect that you are asking something else: Given that the equations of motion depend only on first derivatives (like Newton's law), why aren't second ...


3

There are systems which are not integrable (in Poincaré sense) because interactions destroy the invariants. Consider a Hamiltonian $H = H_0 + \lambda V$, where $H_0$ is the unperturbed Hamiltonian and $\lambda$ the coupling constant. If you start with the interactions turned off you can find invariants of motion $\Phi^0$ by the usual Poisson bracket ...


3

Gibbs' thought on this was (Elementary principles, page 204 footnote) "Strictly speaking, $\psi_{\rm gen}$ is not determined as function of $\nu_1,\ldots\nu_h$, except for integral values of these variables. Yet we may suppose it to be determined as a continuous function by any suitable process of interpolation." Here $\psi_{\rm gen}$ is the free energy of ...


3

The problem with the phase space flow in Hamiltonian mechanics is that the flow itself is non-dynamical, that is, the flow is immediately defined for a given Hamiltonian, so there is no independent equation governing its evolution. Thus, Liouville equation is simply a transport of a scalar variable in a given flow. So, dimensional analysis of the flow ...


2

The problem you are having is that there are two different uses of the word "phase". One is, as you point out, the argument of the $sin$ function. The other use of the word is the ordered pair $(x, p)$, that is, coordinate and momentum. $(x,p)$ specifies a point in a two-dimensional "phase space" where one axis is $x$ and the other $p$. The state of a ...


2

In the case of a particle decaying to two identical particles, this is true. There is no angle-dependence in the scattering amplitude, so the integration is indeed trivial. When calculating scattering cross sections, this is in general not true: the scattering amplitude can depend on the phase space angle. It is also not true for the general $n$-particle ...


2

We are not living in the $q$ part of phase space : we indeed live in the full phase space since we're definitely not fixed-momentum objects. However we give more importance to our position than to our velocity/momentum; then the question gets out of physics into psychology. In my opinion, part of it may be because of the way we gather knowledge : we fix it ...


2

The key inside to OP's question has already been provided by Ikiperu in above comments. Here we just want to show that the problem becomes very simple to study in the corresponding Lagrangian formalism. The Hamiltonian reads $$\tag{1} H(p,q) ~:=~ \frac{p^2}{2m} + \lambda pq + \frac{m\lambda^2}{2}\frac{q^6}{q^4+\alpha^4}. $$ Since there is no explicit time ...


2

Conservation of Ă©tendue is essentially the same thing as Liouville's theorem applied to the space of rays of light in geometric optics. This is central to non-imaging optics, for example in the design of car headlamps or in concentrating sunlight in photovoltaic cells.


2

Phase spaces which are not cotangent bundles can be realized in mechanical systems with phase space constraints . The phase space given by Arnold: the two sphere $S^2$ can be mechanically realized as the reduced dynamics of an energy hypersurface of a two dimensional isotropic harmonic oscillator: $ |p_1^2|+|p_2^2|+|q_1^2|+|q_2^2| = E$ We observe that the ...


2

''similarites/differences between phase space and Hilbert space?'' There are no similarities except that they are both infinite-dimensional vector spaces. How to use the two is completely different. Thus there is no book that can answer that. But if you mean the differences and similarities of classical and quantum physics, my book Classical and Quantum ...


2

At a given point in space, there are several values of $f_i$, one for each discrete velocity $v_i$. So you have to apply the lattice-Boltzmann equation as many times. This is exactly the point of the lattice-Boltzmann method: each "element" is actually defined by its discrete value of the velocity. You apply the equation for each of these elements which have ...


2

In solid state physics, the bulk of a crystal is usually given periodic boundary conditions to avoid the sticky problem of what to do at the termination of the crystal. So the crystal is all bulk, no surface. This turns out to be a very good approximation to the bulk of a real crystal. It also gives the solid the topology of a 3-torus.


1

So, in classical mechanics, we know nothing of this strange "spacetime" and its metric. We know only that systems are described by $n$ continuous generalized coordinates $q^i$ with a certain range, and we take the manifold $\mathcal{M}$ consisting of all possible different $\vec q$ as our starting point. Note that there is no metric, no form, nothing on ...



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