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--- basic_tools:phase_space [2018/04/08 09:35]
jakobadmin [Abstract]
+++ basic_tools:phase_space [2020/05/12 06:25] (current)
173.245.129.4 grammar fix
@@ Line 1: / Line 1: @@
 ====== Phase Space ======
+//see also [[basic_tools:configuration_space|Configuration Space]] and [[basic_tools:hilbert_space]]//
 <tabbox Intuitive>
-{{ :basic_tools:phasespace.png?nolink&350|}}
+A phase space is a mathematical tool that allows us to grasp important aspects of complicated systems.
-A phase space is a mathematical tool that allows to grasp important aspects of complicated systems.
 Each point of the phase space represents one specific configuration a given system can be in.
@@ Line 12: / Line 12: @@
 The time evolution of a system can then be represented as a path in phase space.
-<tabbox Concrete>
 <blockquote>
-The form of the Hamiltonian equations allows us to 'visualize' the evolution
+Try to imagine a
-of a classical system in a very powerful and general way. Try to imagine a
 'space' of a large number of dimensions, one dimension for each of the
-coordinates $x_1, x_2,\ldots$, $p_1, p_2,\ldots$ (Mathematical spaces often have many;
+coordinates $x_1, x_2,\ldots$, $p_1, p_2,\ldots$ (Mathematical spaces often have many
 more than three dimensions.) This space is called phase space (see Fig. 5.10).
-For n unconstrained particles, this be a space of 6n dimensions (three,
+For n unconstrained particles, this is a space of 6n dimensions (three
 position coordinates and three momentum coordinates for each particle). The
 reader may well worry that even for a single particle this is already twice as
 many dimensions as she or he would normally be used to visualizing! The
 secret is not to be put off by this. Whereas six dimensions are, indeed, more
-dimensions than can be readily pictured, it would actually not be of much
+dimensions than can be readily pictured, it would actually not be of much use
-to us if we were in fact able to picture it. For just a room full of
+to us if we were in fact able to imagine it. For just a room full of
 molecules, the number of phase-space dimensions might be something like
@@ Line 36: / Line 34: @@
 {{ :basic_notions:phasespacepenroseemprerornewmind.png?nolink&600 |}}
-Now, how are we to visualize Hamilton's equations in terms of phase
-Space? First, we should bear in mind what a single point Q of phase space
-actually represents. It corresponds to a particular set of values for all the
-position coordinates $x_1,x_2,....$ and for all the momentum coordinates $p_1,p_2, ...
-$ That is to say, Q represents our entire physical system, with a particular
-state  of motion specified for every single one of its constituent particles.
-HamIlton's equations tell us what the rates of cbange of all these coordinates
-are,.wben we know their present values; i.e. it governs how all the individual
-particles are to move. Translated into phase-space language, the equations
-are tellingus how a single point Q in phase space must move, given the
-present location of Q in phase space. Thus, at each point of phase space, we
-have a little arrow - more correctly a vector- which tells us the way that Q is
-moving, in order to describe the evolution of our entire system in time. The
-whole arrangement of arrows constitutes what is known as a vector field (Fig.
-.11). Hamilton's equations thus define a vector field on phase space.
-Let us see how physical determinism is to be interpreted in tel filS of phase
-space. For initial data at time t = 0, we would have a particular set of values
-~specified for all the position and momentum coordinates; that is to say, we
-have a particular choice of point Q in phase space. To find the evolution of
-the system in time. we simply follow the arrows. Thus the entire evolution of our system with time - no matter how complicated that system might be - is described in phase space as just a single point moving along following the particular arrows that it encounters. We can think of the arrows as indicating the "velocity" of our point Q in phase space. For a "long" arrow, Q moves along swiftly, but if the arrow is "short", Q's motion will be sluggish. To see what our physical system is doing at time t, we simply look to see where Q has
-moved to, by that time, by following arrows in this way. Clearly this is a deterministic procedure. The way that Q moves is completely determined by,
-the Hamiltonian vector field.
-{{ :basic_notions:vectorfieldphasespacepenroseemprerornewmind.png?nolink&600 |}}
+<cite>page 174ff in "The Emperors new Mind" by R. Penrose</cite></blockquote>
+<tabbox Concrete>
+Each point of the phase space corresponds to one particular state of the system.
+$\{q_i \}$ defines a point in n-dimensional
+[[basic_tools:configuration_space|configuration space]] $C$. Time evolution is a path in $C$. However, the complete __state__ of the system
+is defined by $\{q_i \}$ and $\{p_i \}$. Only with this information, we are able to determine
+the state at all times in the future. The pair $\{q_i, p_i \}$ defines a point in a $2n$-dimensional
+space we call phase space.
+Since each point in phase space is sufficient
+to determine the time
+evolution of the system, paths in phase space can never cross. Otherwise, the time-evolution would not be unique.
+It is conventional to say that the time evolution
+is governed by a **flow** in phase space.
+To understand this imagine that you follow one individual trajectory in phase space. Say, you take a pencil, put it down at one point in phase space and then start to draw the correct trajectory for the system which follows from [[equations:hamiltons_equations|Hamilton's equations]]. Now, this trajectory is only one thing that can happen in our system. When we are interested in the bigger picture, we need to follow all trajectories that are possible. This means, that we take an infinite number of pencils, put them down and the draw the trajectories. Each trajectory is calculated with Hamilton's equations. The set of all possible trajectories defines a flow in phase space.
+----
+**Examples**
+-->Phase space of a harmonic oscillator#
+{{ :basic_tools:phasespaceoscillator.png?nolink&600 |}}
+<--
+-->Phase space of a pendlum#
+{{ :basic_tools:phasespacependulum2.png?nolink&600 |}}
+<--
-<cite>page 174ff in "The Emperors new Mind" by R. Penrose</cite></blockquote>
 <tabbox Abstract>
 The space of states in classical mechanics is modeled as a manifold $M$ equipped with a symplectic structure: $(M,ω)$. This manifold is what we usually call phase space. The phase space is a symplectic manifold which simply means that is a manifold equipped with a symplectic structure. A symplectic structure is a distinguished 2-form $(\omega)$.
-Such a 2-form is an object that eats two vector fields on our manifold and returns another function on the manifold.
+Such a 2-form is an object that eats two vector fields on our manifold and returns another function on the manifold. Functions on the manifold are smooth maps $f \ : \ M \rightarrow R$. These functions are what we call “the observables of our classical system”. So in words, this means that the observables of our classical system map each state to a real number.
+One of the most important function on our phase space manifold is the Hamiltonian function. This function represents the energy of the system and describes the time-evolution of phase space points.
+----
+  * https://mathoverflow.net/questions/16460/how-to-see-the-phase-space-of-a-physical-system-as-the-cotangent-bundle
+----
+<blockquote>In classical mechanics, the phase space is the space of all possible states of a physical system; by “state” we do not simply mean the positions q of all the objects in the system (which would occupy physical space or configuration space), but also their velocities or momenta p (which would occupy momentum space). One needs both the position and momentum of system in order to determine the future behavior of that system. Mathematically, the configuration space might be defined by a manifold M (either finite1 or infinite dimensional), and for each position q ∈ M in that space, the momentum p of the system would take values in the cotangent2 space Tq∗M of that space. Thus phase space is naturally represented here by the cotangent bundle T∗M := {(q,p) : q ∈ M,p ∈ Tq∗M}, which comes with a canonical symplectic form ω := dp ∧ dq.
+This may seem surprising; since velocity q ̇ naturally lives in the tangent space TqM, one would expect momentum to also. However, from Lagrangian mechanics, in which the system
+R
+evolves by finding formal critical points of a Lagrangian
+defined as p := ∂L , which lives most naturally in the cotangent space. Dually, the Hamiltonian
+∂ q ̇
+links momentum to velocity by Hamilton’s equation q ̇ = ∂H . ∂p
+<cite>http://www.math.ucla.edu/~tao/preprints/phase_space.pdf</cite></blockquote>
@@ Line 78: / Line 101: @@
 <blockquote>
-[[frameworks:hamiltonian_formalism|Hamiltonian Mechanics]] is geometry in phase space. [...]
+[[formalisms:hamiltonian_formalism|Hamiltonian Mechanics]] is geometry in phase space. [...]
 <cite>Mathematical Methods of Classical Mechanics  [[https://books.google.de/books?id=5OQlBQAAQBAJ&lpg=PA160&ots=u7Qs-TMaNb&dq=%22Hamiltonian%20Mechanics%20is%20geometry%20in%20phase%20space.%22&hl=de&pg=PA160#v=onepage&q&f=false|Vladimir Arnold]]</cite>

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