What is Poincare: Definition and 154 Discussions

Poincaré is a French surname. Notable people with the surname include:

Henri Poincaré (1854–1912), physicist, mathematician and philosopher of science
Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
Lucien Poincaré (1862–1920), physicist, brother of Raymond and cousin of Henri
Raymond Poincaré (1860–1934), French Prime Minister or President inter alia from 1913 to 1920, cousin of Henri

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  1. R

    Were Poincaré's leaps in proofs ever proven to be false?

    According to the experts, Poincaré made huge leaps in his proofs, often leaving lesser mathematicians scratching their heads. I'm wondering if some of those leaps later turned out to be false and if so how often.
  2. O

    Finding local flat space of the Poincare half-disk

    ΩHomework Statement Given the metric ds^2 = \frac{dx^2 + dy^2}{y^2} find a set of coordinates which yield local flat space. i.e. ( g_{\mu\nu} = \delta_{\mu\nu} + second order terms). My text outlined a process to go through to find the flat space coordinates, but the actual execution is...
  3. M

    Proving the Last Term in the Poincaré Group Lie Algebra Identity

    Homework Statement The problem statement is to prove the following identity (the following is the solution provided on the worksheet): Homework Equations The definitions of L_{\mu \nu} and P_{\rho} are apparent from the first line of the solution. The Attempt at a Solution I get to the...
  4. B

    Using Poincare Bendixson's Theorem to prove a periodic Orbit

    Homework Statement The System is x˙ = 2x -y - x(x^2+y^2) y˙ = 5x - 2y(x^2+y^2) Using a trapping region to show there is a periodic orbit Homework Equations Use Poincare Bendixson's TheroemThe Attempt at a Solution I tried constructing 2 Lyapunov type functions to show that DV/dt>0 and...
  5. A

    Can we do other than Lorentz (or Poincare) Transformation in SR?

    I want to discuss this because I afraid that the answer is no. In SR we stuck with the transformations from Poincare Group because this transformations leave invariant the exact form of the Lorentz Metric tensor. Any other transformation will change the components of the Lorentz Metric Tensor...
  6. S

    Poincare Transformation: Understanding its Properties and Group Structure

    Dear all, Poincare transformation construct a group, better to say noncompact Lie group. I want to prove this fact but I don't know how...; I mean the general characteristics- associativity, closure, identity element and inversion element. I would appreciate it if anyone could help me or...
  7. F

    Diffeomorphic Invariance implies Poincare Invariance?

    I have been quite puzzled for some time with the concept of Diffeomorphic Invariance. Here is what I think about it, 1) Diffeomorphic Invariance is the invariance of the theory under general coordinate transformations. For instance the Einstein-Hilbert action is diffeomorphic invariant...
  8. C

    Poincaré recurrence and maximum entropy

    Fluctuation theorem says that there will be fluctuations in microscopic scale that results local decreases in entropy even in isolated systems. According to the Poincaré recurrence theorem, after sufficiently long time, any finite system can turn into a state which is very close to its initial...
  9. J

    Is Poincare wrong about no preferred geometry?

    I heard that some physicists are trying to determine the spacial/geometric curvature of the universe by measuring the angles of distant stars (a very large triangle). Is this possible? Or is Poincare correct when he said that there is no preferred geometry and that there is no experiment...
  10. Y

    Polarization and Poincare circle.

    What is the theory behind mapping of the latitude and longitude of the sphere in the Poincare Circle to the polarization of the TEM wave? That is, why: 1) Linear polarization when ε=0 deg? 2) Circular polarization when ε=+/- 45 deg? 3) Elliptical when ε is not 0 or +/- 45 deg? 4) RH rotation...
  11. T

    Describing polarization of a coheret state parametrizing on poincare sphere

    The question is here: http://physics.stackexchange.com/questions/45341/representing-a-polarization-vector-for-light-as-a-manifold-of-two-state I'd appreciante any help
  12. mnb96

    Poincaré disk vs Riemann sphere

    Hello, it is well-known that with stereographic projection we can obtain a 1-1 correspondence between the points of the 2d Cartesian plane (plus the point at infinity), and the points on the Riemann sphere. What is the geometrical construction that corresponds to a 1-1 mapping between the...
  13. tom.stoer

    Photon helicity: Wigner's unitary rep. of Poincare group and gauge symmetry

    1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |kμ, h> with kμkμ = 0 and h = ±1 and we have two d.o.f. 2) For gauge theories with...
  14. A

    Poincaré group is semi-product of translations and Lorentz-group

    Hi everybody, Can somebody help me with the following proof. Show that for the Poincaré group P=T\odot L Where T is the group of translations and L is the Lorentz group and P is the semi-direct product of the two subgroups I know the axioms for a semi-direct product in this case are...
  15. jfy4

    Poincare Algebra from Poisson Bracket with KG Action

    Homework Statement Consider the Klein-Gordan action. Show that the Noether charges of the Poincare Group generate the Poincare Algebra in the Poisson brackets. There will be 10 generators.Homework Equations \{ A,B \}=\frac{\delta A}{\delta \phi}\frac{\delta B}{\delta \pi}-\frac{\delta...
  16. C

    The Poincare Group: A Study of Second Part of 3.26 and 3.27

    Hi all! I'm trying to study the Poincare group and I have one problem. I'm reading a book: Gross D. Lectures on Quantum Field Theory (there is section about it). So I do not understand how the second part of (3.26 and 3.27) folows from the first part i.e I do not understand how was obtained...
  17. J

    Poincare Recurrence and the Klein-Gordon Equation

    There exists Green's Functions such that the solutions appear to be retro-causal. The Klein-Gordon equation allows for antiparticles to propagate backwards in time. Does this mean the future can influence the past and present? Then again The Poincare Recurrence Theorem states that over a...
  18. O

    Computing arc length in Poincare disk model of hyperbolic space

    I am reading Thurston's book on the Geometry and Topology of 3-manifolds, and he describes the metric in the Poincare disk model of hyperbolic space as follows: ... the following formula for the hyperbolic metric ds^2 as a function of the Euclidean metric x^2: ds^2 = \frac{4}{(1-r^2)^2} dx^2...
  19. S

    Poincaré disk: metric and isometric action

    Hi! I'm trying to give a few examples of symmetric manifolds. In the article "Introduction to Symmetric Spaces and Their Compactification" Lizhen Ji mentions the Poincaré disk as a symmetric space in the following way: D = \{z \in \mathbb C | |z| < 1\} with metric ds^2 =...
  20. P

    Poincare Conjecture: Explaining Lower Dimension Equivalent

    I've been doing a project on Henri Poincare and I am attempting to explain his conjecture to my Calculus class so I am using the common lower dimensional equivalent to do so. If a rubber band is wrapped around an object and becomes smaller and smaller until it is a point than that object is...
  21. G

    Poincare disk model. Circle question.

    Homework Statement Use the Poincare (disk) model to show that in the hyperbolic plane, there exists two points A, B lying on the same side S of a line l such that no circle through A and B lies entirely within S. Homework Equations The hint was to use this proposition: A P-circle is a...
  22. C

    Poincaré recurrence applicability condition?

    This is how Wikipedia summarizes the Poincaré Recurrence Theorem: This is wrong, isn't it? Don't you need to ensure the phase space is bounded, and isn't conservation of energy an insufficient justification for that? Like, imagine throwing two baseballs away from each other into infinite...
  23. R

    Proving periodic orbits using Poincare Bendixson's Theorem and a trapping region

    Homework Statement System in polar coordinates \dot{r} = 2r - r3(2 + sin(\theta)), \dot{\theta} = 3 - r2 Use a trapping region to show there is at least one periodic orbit? Homework Equations By using Poincare Bendixson's Theorem The Attempt at a Solution I am struggling to...
  24. A

    Tensor analog of Poincare Lemma

    Hi everyone, I know that there is a result that corresponding to a closed p-form α, I can find a p-1 form β, such that dβ = α. I wanted to ask, what the tensorial analog of this would be. I mean would it be right to say that on a manifold with a lorentzian metric, If I have a vector...
  25. F

    Mathematica (Mathematica) Poincarè map of restricted three body problem

    Hi everybody, sorry for the inconvenience. I try to plot the poincarè map of the restricted three body problem. I find in this forum the follow script that do this for the Lorenz system: mysol = NDSolve[{x'[t] == -3 (x[t] - y[t]), y'[t] == -x[t] z[t] + 26.5 x[t] - y[t], z'[t] ==...
  26. F

    Drawing Phase planes, and computing the Poincare index

    Here is what I've done so far How do I draw the Phase portrait for this system? Have I done everything correct so far? Thanks
  27. D

    Circumference of a circle in Poincare Half Plane

    i am trying to figure out how to calculate the circumference of a circle in the Poincare Half Plane. I know that vertical lines are geodesics so using the arclength formula, the distance between 2 points (x_0, y_0) and (x_1, y_1) on a vertical line is ln(y_1/y_0) . Thus, if i have a circle...
  28. T

    Poincaré Stability: How to Show Paths Remain in Circle?

    Hey Guys! I'm having MAJOR difficulty with a problem regarding stability of a DE. The problem goes as following: Find the equation of the phase paths of x˙=1+x^2, y˙=−2xy. It is obvious from the phase diagram that y=0 is Poincaré stable. Show that for the path y=0, all paths which start in...
  29. W

    Poincare vs Lorentz Group

    The words Poincare and Lorentz sound pretty elegant. I think they are French words like Loreal Or Laurent. I know Poincare has to do with spacetime translation and Lorentz with rotations symmetry. But how come one commonly heard about Lorentz symmetry in Special Relativity and General...
  30. M

    Henri poincare got a 35 on an iq test?

    I read here http://www.eoht.info/page/Genius that henri poincare got a 35 on an iq test. I think that is really interesting, if its true! I've heard that people with mental deficiency score in the 60's, so 35 would be really low! I used to think iq meant something years ago, but I dropped...
  31. S

    Inhomogeneous (poincare) lorentz transormation

    I'm reading a physics book and in the section on relativity they are using the Einstein summation convention, with 4vectors and matrices. They say that the transformations take the form: x^{\prime\mu}=x^{\nu}\Lambda^{\mu}_{\nu}+C^{\mu} where it is required that \Lambda^{\mu}_{\nu} satisfy the...
  32. E

    How do you find the generator corresponding to a parameter of a Lie group?

    Alright, so I was reading Ryder and he defines the generator corresponding to a^{\alpha} as the following X_{\alpha}=\frac{\partial x'^{\mu}}{\partial a^{\alpha}}\frac{\partial}{\partial x^{\mu}} (\alpha =1,...r) for r-parameter group of transformations Now this makes sense for...
  33. Demon117

    The Poincaré Group and Geodesics in Minkowski Spacetime

    The poincare' group is the group of isometries of Minkowski spacetime, in a nutshell. In terms of an actual physical definition it is the group of all distance preserving maps between metric-spaces in Minkowski spacetime. What is the difference between this and geodesics?
  34. jfy4

    Diffeomorphisms in Flat-Space: Are All Metric Preserving?

    Hi, I'm sorry to have to ask this, but I can't seem to reason this one out by myself at the moment. Given the metric is the Minkowski spacetime, is the group of diffeomorphisms the poincare group, or are there diffeomorphisms for flat-space that are not metric preserving? I would really...
  35. B

    Lie algebra of the Poincare group

    Given [M^{\mu \nu},M^{\rho\sigma}] = -i(\eta^{\mu\rho}M^{\nu\sigma}+\eta^{\nu\sigma}M^{\mu\rho}-\eta^{\mu\sigma}M^{\nu\rho}-\eta^{\nu\rho}M^{\mu\sigma}) and [ P^{\mu},P^{\nu}]=0 I need to show that [M^{\mu\nu},P^{\mu}] = i\eta^{\mu\rho}P^{\nu} - i\eta^{\nu\rho}P^{\mu}We've been given the...
  36. K

    How to calculate the Poincare dual of a ray on R2-{0}?

    S={(x,0)|x>0} on R^2-{0},I need to calculate the closed Poincare dual of S. Assume \omega=f(x,y)dx+g(x,y)dy on R^2-{0} have compact support.Then we need to find a form \eta in H^1 (R^2 - {0} ) satisfying \int\limits_S {i^* \omega = \int\limits_M {\omega \wedge \eta } } , The book...
  37. A

    Question on the representation of Poincare algebra generators on fields

    Hi, I am working through Maggiore's QFT book and have a small problem that is really bothering me. It involves finding the representation of the Poincare algebra generators on fields. I always end up with a minus sign for my representation of a translation on fields compared to Maggiore...
  38. C

    How Does the Poincare Group Relate to Special Relativity and Reference Frames?

    I've read that it is also a Lie group. But what does that have anything to do with special relativity or different reference frames? The wiki definition is "a 10-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz group is a...
  39. H

    Galilei, Poincare and conformal symmetry

    Reading this article: http://arxiv.org/abs/math-ph/0102011 made me wonder: 1.) So, it appears that Galilean transformations are not the most general symmetry transformations of nonrelativistic mechanics. Fine. 2.) The article states that the two additional symmetries are the nonrelativistic...
  40. S

    What are the representations and generators of Lorentz and Poincare Groups?

    I'm new here and I have checked the FAQs. I'm not sure if this question has been posted before. This may actually be a silly question. Why do we study Lorentz and Poincare Groups? I have studied a bit of the theory but was wondering what exactly are we talking about when we study the...
  41. C

    Does the poincare conjecture disprove the shape of strings in string theory

    Considering that the smallest particles in nature are supposed to be strings, which are donut and line shapes. And the poincare conjecture says the simplest shape in nature is a sphere. wouldn't it make sense that the true fundamental particles are sphere shaped and that if they combine to form...
  42. M

    Local Poincare Transformation

    Can any continuous coordinate transformation on a differential manifold be viewed as a poincare transformation locally in every tangent space of this manifold? Thx!
  43. P

    Minkowski vacuum: Poincare invariant, quasi-free state

    Minkowski vacuum is Poincare invariant and quasi-free state. I wonder if these two conditions fully define it or there are more states which fulfill these conditions (or maybe Poincare invariance alone is sufficinet). Thanks for answers.
  44. R

    Is Minkowski space the only Poincare invariant space?

    Hi everyone, I was wondering: if a space is invariant under Poincare transformations, does that mean it has to be Minkowski space? Or could it have some further isometries? By the same token, if a space is invariant under the orthogonal transformations, does it have to be Euclidean? I...
  45. M

    Show Poincare Disk is incidence geometry

    Homework Statement I have to show that the Poincare disk satisfies the incidence axiom that any line contains at least two points. Homework Equations There are two kinds of lines on the Poincare disk. I've found 2 points for the first kind, which are straight lines going through the...
  46. P

    Conformal group -> Poincaré group

    Hi. I want to talk about the derivation of the form of space-time transformations T between inertial references. As it's known, this is the Poincaré group, defined as the group that leaves invariant the (+,-,-,-) metric. This derives from two things: c is constant and space-time is homogeneous...
  47. K

    Mathematica Mathematica for Poincaré sections - or maybe a different tool?

    Hi, I have a system of ODE's for which I want to compute points for a Poincaré section. I used NDSolve and EventHandler in Mathematica but EventHandler appears rather messy/limited for my condition: Suppose I want to save 4 iterations of a Poincaré map and then abort integration imediately...
  48. haushofer

    Gauging Poincare to obtain Einstein gravity

    Hi, I'm rather confused about the procedure in which people obtain gravity from gauged (super)-Poincare algebras. Let me outline what this procedure is. *First you gauge the Poincare algebra with generators P and M *You obtain two gauge fields: the vielbein (associated with P) and the...
  49. A

    Circular Polarization in Electrical Engineering: Examining the Poincaré Sphere

    *I am using the conventions of circular polarisation according to electrical engineering , not the one used in optics* Let us take a uniform plane TEM wave traveling in +z direction which is composed of two linearly polarised TEM waves , one whose electric field lies in X direction , the other...
  50. Fredrik

    How Does Changing Velocity Affect Inertial Frame Transformations?

    Suppose that we write a function that changes coordinates from one inertial frame to another in the form x\mapsto\Lambda x+a where \Lambda is linear, with components \Lambda=\gamma\begin{pmatrix}1 & \alpha\\ -v & \beta\end{pmatrix} in the standard basis. (This is the most general...
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