# 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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### Transformation between a dynamical system to a Hamiltonian system [duplicate]

Consider a dynamical system characterized by these equations $$\dot{x}=x-xy \\ \dot{y}=-y+xy$$ If we transform $\ln(y)=q$ and $\ln(x)=p$, the system can be changed into a Hamiltonian system with $q$ ...
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### An “upper ceiling” for thermodynamics?

Roger Penrose said in "A Road to Reality" (p.701): “There is a common view that the entropy increase in the second law is somehow just a necessary consequence of the expansion of the universe. ...
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### Hamilton-Jacobi Equation: Why does any $F(q,Q,0)=f(q,Q)$ lead to a solution?

I) Given the Hamilton-Jacobi equation,$$\frac{\partial F(q,Q,t)}{\partial t}+H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)=0$$ it is stated that any function $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ ...
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### Time Dependence on Landau & Lifshitz's Proof of Poisson's Bracket Canonical Invariance

I'm reading Landau & Lifshitz's Mechanics and, at a certain point when discussing canonical transformations, they prove that Poisson brackets are canonical invariants. The proof starts with ...
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### How to check if a generating function produces an identity transformation without substituting the CT equations in the Hamiltonian?

In chapter 9, Goldstein ($3^{rd}$ ed.) includes a discussion and a few "trivial special cases" of Canonical Transformation which keeps the form of the Hamiltonian unchanged and named it Identity ...
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### A proof of Liouville’s theorem

I have found a proof of Liouville's theorem on the internet, which fits me very well except one step I don't understand, the derivation is as follows: In the derivative, it must have used the ...
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### The complex form of Hamilton canonical equations

I found an excerpt on page 171 of "The variational principles of mechanics" written by Cornelius Lanczos stated that If, however, the conjugate variables $q_k$, $p_k$ are replaced by the complex ...
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### Relation between Wigner flows and entropy

After reading this question What's the intuitive reason that phase space flow is incompressible in Classical Mechanics but compressible in Quantum Mechanics? and some papers of Steuernagel’s group,...
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### Legendre transform and coordinate system independence

I'm self-learning analytical mechanics. Consider a classical mechanical system. Even if it's clear to me that via (the usual) Legendre transform we can get a unique Hamiltonian function from a ...
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### How is it possible to have four types of generating functions?

Since the Hamilton's equations of motion remain unchanged in form under a canonical transformation $(q,p)\to (Q,P)$, the Lagrangians must differ by a total time derivative of a function of $q,t$. In ...
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### Density Of states derivation

In the aspect of density of state derivation or simply assuming the frequency of a solid as a continuous distribution we have to come up with an equation expressing the density of states. Its derived ...
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### Particle in the 1D infinite square well

I'm trying to draw the phase space for a particle moving freely between 0 and $L$. I guess $H=E$(total energy, constant)$=\frac{p_{x}^2}{2m}$ so $p_{x}=\pm\sqrt{2mE}$ for every $x$ between $0$ and $L$,...
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### Why are 2 dimensions needed for every 1 dimension of space in order to determine the motion of a physical system?

In classical mechanics, the phase space of a mechanical system has twice the number of dimensions of "actual" space (i.e. position space). That is, in phase space, each particle has both a position ...
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### Thermodynamical conjugate variables

In thermodynamics the potentials are typically only a function of 2 variables, say $$U=U(S,V)$$ with entropy $S$ and volume $V$. I see that conjugate pairs $S,T$ or $p,V$ always have the unit of ...
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### Why is a single function sufficient to specify a canonical transformation?

Spivak argues at page 577 in his book Physics for Mathematicians: What are the $2n$ relations he is talking about?
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### Star-shaped phase space

I am asked to classify the following phase spaces. The phase spaces 2 and 3 are fairly simple (harmonic oscillator and a elastically reflected particle). However, I fail to classify the phase space ...
In Classical Mechanics, a gauge transformation is of the form $$L \to L' = L + \frac{dF(q,t)}{dt} \, .$$ Any transformation of this kind leaves the Euler-Lagrange equation ...
Consider a $2n$ real symplectic space - the usual $\mathbb R^{2n}$. Suppose that the same space could be endowed too with an Euclidean structure, by which the vectors of the symplectic basis are ...