basic_tools:phase_space
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| ====== Phase Space ====== | ====== Phase Space ====== | ||
| + | //see also [[basic_tools:configuration_space|Configuration Space]] and [[basic_tools:hilbert_space]]// | ||
| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | A phase space is a mathematical tool that allows to grasp important aspects of complicated systems. | + | A phase space is a mathematical tool that allows us to grasp important aspects of complicated systems. |
| Each point of the phase space represents one specific configuration a given system can be in. | Each point of the phase space represents one specific configuration a given system can be in. | ||
| Line 16: | Line 17: | ||
| Try to imagine a | Try to imagine a | ||
| 'space' of a large number of dimensions, one dimension for each of the | '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). | 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 | 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 | 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 | 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 | 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 | molecules, the number of phase-space dimensions might be something like | ||
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| It is conventional to say that the time evolution | It is conventional to say that the time evolution | ||
| - | is governed by a flow in phase space. | + | 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 |}} | ||
| + | <-- | ||
| Line 59: | Line 75: | ||
| 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 $(\omega)$. | 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. Functions on the manifold are smooth maps $f \ : \ M \longarrow R$. 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. | + | 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. | 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> | ||
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| <blockquote> | <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> | <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> | ||
basic_tools/phase_space.1523179268.txt.gz · Last modified: 2018/04/08 09:21 (external edit)
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