What is Symplectic: Definition and 53 Discussions

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.

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  1. A

    A Finding the Hermitian generator of a Symplectic transformation

    Consider a set of ##n## position operators and ##n## momentum operator such that $$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$ Lets now perform a linear symplectic transformation $$q'_{i} =A_{ij}q_{j}+B_{ij}p_{j},$$ $$p'_{i} =C_{ij}q_{j}+D_{ij}p_{j}.$$ such that the canonical commutation...
  2. cianfa72

    I Darboux theorem for symplectic manifold

    Hi, I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem. We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form...
  3. A

    A Parallel transport on a symplectic space

    Sorry if the question is not rigorously stated.Statement: Let ##(q,p)## be a set of local coordinates in 2-dimensional symplectic space. Let ##\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{n})## be a set of local coordinates of certain open set of a differentiable manifold ##\mathcal{M}.## For...
  4. M

    Conservation laws in Newtonian and Hamiltonian (symplectic) mechanics

    In Newtonian mechanics, conservation laws of momentum and angular momentum for an isolated system follow from Newton's laws plus the assumption that all forces are central. This picture tells nothing about symmetries. In contrast, in Hamiltonian mechanics, conservation laws are tightly...
  5. A

    A N-Body Simulation using symplectic integrators

    Hi, I hope I am in the right section of the forum. I was trying to understand the following algorithm: https://benchmarksgame-team.pages.debian.net/benchmarksgame/program/nbody-python3-1.html and particulary this part: def advance(dt, n, bodies=SYSTEM, pairs=PAIRS): for i in range(n)...
  6. m4r35n357

    I Symplectic integrator, non-separable Hamiltonian

    I have been attempting to modify a symplectic integrator that I wrote a while ago. It works very well for "separable" hamiltonians, but I want to use it to simulate a double pendulum. I am using the Stormer-Verlet equation (3) from this source. From the article "Even order 2 follows from its...
  7. gasar8

    A Lorentz Invariant Phase Space: Symplectic Geometry

    I have an assignment to show that specific intensity over frequency cubed \frac{I}{\nu^3}, is Lorentz invariant and one of the main topics there is to show that the phase space is Lorentz invariant. I did it by following J. Goodman paper, but my professor wants me to show this in another way...
  8. C

    A Symplectic geometry of phase space

    What is a symplectic manifold or symplectic geometry? (In intuitive terms please) I have a vague understanding that it involves some metric that assigns an area to a position and conjugate momentum that happens to be preserved. What is 'special' about Hamilton's formulation that makes it more...
  9. R

    [Symplectic geometry] Show that a submanifold is Lagrangian

    Homework Statement Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to...
  10. CassiopeiaA

    A Symplectic Condition For Canonical Transformation

    I am reading Chapter 9 of Classical Mech by Goldstein.The symplectic condition for a transformation to be canonical is given as MJM' = J, where M' is transpose of M. I understood the derivation given in the book. But my question is : isn't this condition true for any matrix M? That is it doesn't...
  11. L

    A Hyperhermitian condition

    Let ##V## be a quaternionic vector space with quaternionic structure ##\{I,J,K\}##. One can define a Riemannian metric ##G## and hyperkahler structure ##\{\Omega^{I},\Omega^{J}, \Omega^{K}\}##. Do this inner product $$\langle p,q \rangle :=...
  12. F

    A Symplectic Majorana Spinors in 5 Dimension

    I need to know if the Symplectic Majorana spinors in 5 dimension have any advantage with respect to the Dirac spinors in 5 dimension, since they have the same number of components. For example if the Symplectic Majorana spinors have a manifested symmetry that the Dirac spinors don't have, or if...
  13. F

    Topology Learn Differential Geometry: Books for Bachelor in Geometric Quantization

    I am taking my bachelor in geometric quantization but I have no real experience in differential geometry ( a part of my project is to learn that). So I find myself in need of some good books that cover that the basics and a bit more in depth about symplectic manifolds. If you have any...
  14. A

    Rovelli Quantum Gravity: Clarification on Symplectic Forms & Hamiltonian

    Please refer to p. 99 and 100 of Rovelli’s Quantum Gravity book (here). I wonder what is the signification of the “naturalness” of the definition of ##\theta_0=p_idq^i##? If I take ##\theta_0'=q^idp_i## inverting the roles of the canonical variables and have the symplectic 2-forms of the...
  15. M

    Simple Symplectic Reduction Example

    Homework Statement I'm struggling to perform a symplectic reduction and don't really understand the process in general. I have a fairly solid understanding of differential equations but am just starting to explore differential geometry. Hopefully somebody will be able to walk me through this...
  16. W

    Symplectic And Contact Manifolds

    Hi, we know that every contact manifold has a symplectic submanifold. Is it know whether every symplectic manifold has a contact submanifold? A contact manifold is a manifold that admits a (say global) contact form: a nowhere-integrable form/distribution (as in Frobenius' theorem) ## w## so...
  17. R

    MHB Harmonic oscillator and symplectic Euler method

    Given the equations for the harmonic oscillator $\frac{dy}{dz}=z, \frac{dz}{dt}= -y$if the system is approximated by the symplectic Euler method, then it gives$z_{n+1}= z_{n}-hy_{n}, \\ y_{n+1}= y_{n}+hz_{n+1}$which shows that the circle $y^2_{n} + z^2_{n} = 1$ is mapped into an ellipse...
  18. N

    Symplectic Structure of Thermodynamics and the Hamiltonian

    (inspired partially by this blog post: http://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/ To my understanding, the thermodynamic configuration space has a nice symplectic structure. For example, using the language of classical mechanics, starting...
  19. M

    Celestial mechanics with symplectic integrators

    Homework Statement The problem is that I just don't understand how the algorithm described here in section 2 hangs together... I have to present this on Thursday morning and that sensation of 'I'll never understand this soon enough' is growing ominously. Homework Equations Equations 2,6,7...
  20. B

    Finding energy as a function of Symplectic area?

    Homework Statement Find the energy E of the harmonic oscillator (H(x,p)=p2/2m+mω2x2) as a function of the system's symplectic area. Homework Equations Canonical equations and A=\int p dx (over one period) The Attempt at a Solution From Hamilton's equations I get ...
  21. A

    Understanding Symplectic Notation and its Equivalence to Traditional Methods

    I have some trouble understanding the attached section of my book. Basically I can't see why the marked equations are equivalent - that is the first two are contained in the last one. I can follow the derivation but when I do an example for myself where I just have two variables (q,p) being...
  22. A

    Symplectic Notation: Confused by Subscripts i & j?

    The attached is a section of the derivation of canonical transformation from the symplectic formulation. I tend to get very confused by the subscripts i and j. For me they both run from 1 to 2n and can be used interchangeably. But of course that is not the case since equation (9.53) on the...
  23. K

    Lagrangian subspaces of symplectic vector spaces

    Homework Statement If (V,\omega) is a symplectic vector space and Y is a linear subspace with \dim Y = \frac12 \dim V show that Y is Lagrangian; that is, show that Y = Y^\omega where Y^\omega is the symplectic complement. The Attempt at a Solution This is driving me crazy since I...
  24. E

    Symplectic integrators of the pendulum equation?

    In particular, a symplectic integrator to solve: \ddot{\theta} + \dfrac{g}{l} \sin(\theta) = 0 I'm currently using velocity verlet - by realizing that \ddot{\theta} = -\nabla (-cos(\theta)) = A(\theta(t)) ie. letting x = theta v = dtheta/dt a = d^2 theta /dt^2 is it safe to apply verlet...
  25. MathematicalPhysicist

    Proving the Symplectic Positive Definite Matrix Theorem

    I need any help with the next question I posted in math.stackexchange, thanks. http://math.stackexchange.com/questions/119105/symplectic-positive-definite-matrix
  26. W

    Invertibility of Symplectic Matrices

    Hi, All: The Wikipedia page on symplectic matrices: http://en.wikipedia.org/wiki/Symplectic_vector_space , claims that symplectic matrices are invertible , i.e., skew-symmetric nxn- matrix with entries w(b_i,b_j) , satisfying the properties: i)w(b_i,b_i)=0...
  27. N

    Why symplectic symmetry has N(N+1)/2 generators?

    Please teach me this: Why Sp(N) symmetry has N(N+1)/2 generators?(QFT of Peskin and Schroeder). Thank you very much for your kind helping.
  28. B

    Symplectic but Not Complex Manifolds.

    Hi, All: AFAIK, every complex manifold can be given a symplectic structure, by using w:=dz/\dz^ , where dz^ is the conjugate of dz, i.e., this form is closed, and symplectic. Still, I think the opposite is not true, i.e., not every symplectic manifold can be given a complex...
  29. B

    Quotients of Mapping Class Group Iso. to Symplectic Group.

    Hi, Again: I am a bit confused about this result: Mg/Mg^(2) ~ Sp(2g,Z) (group iso.) Where: i) Mg is the mapping class group of the genus-g surface, i.e., the collection of diffeomorphisms: f:Sg-->Sg , up to isotopy. ii)Mg^(2) is the subgroup of Mg of maps that...
  30. B

    Symplectic Basis on Sg and Non-Trivial Curves

    Hi, All: Let Sg be the genus-g orientable surface (connected sum of g tori), and consider a symplectic basis B= {x1,y1,x2,y2,..,x2g,y2g} for H_1(Sg,Z), i.e., a basis such that I(xi,yj)=1 if i=j, and 0 otherwise, where I( , ) is the algebraic intersection of (xi,yj), e.g...
  31. Steven Wang

    Symplectic Geometry: Learn, Understand & Expert Articles/Books

    I am interesting in symplectic geometry now. But I have only little knowledge about it. Can someone show me some materials or courses to learn or understand this subject. I want to know the classic articles and books about symplectic geometry and who are the experts in this field. Thank you .
  32. M

    Contraction of the canonical symplectic form by vertical vectors

    The canonical symplectic form on T^*M is the exterior derivative of the tautological 1-form: \omega=d\alpha where \alpha_p(X):=p(d\pi(X)) is the tautological 1-form. Let Y \in T_pT^*M a vertical vector, that is d\pi(Y)=0. It's trivial to prove using canonical coordinates that for...
  33. B

    Relationship Between Symplectic Group and Orthogonal Group

    Hi, All: Given a simplectic vector space (V,w), i.e., V is an n-dim. Vector Space ( n finite) and w is a symplectic form, i.e., a bilinear, antisymmetric totally isotropic and non-degenerate form, the simplectic groupSp(2n) of V is the (sub)group of GL(V) that...
  34. B

    Complex Hilbert Space as a Symplectic Space?

    Hi All: in the page: http://mathworld.wolfram.com/SymplecticForm.html, Complex Hilbert space, with "the inner-product" I<x,y> , where <.,.> is the inner-product Does this refer to taking the imaginary part of the standard inner-product ? If so, is I<x,y> symplectic in...
  35. B

    Surjection Between Mapping Class Grp. and Symplectic Matrices

    Hi, Everyone: I am reading a paper that refers to a "natural surjection" between M<sub>g</sub> and the group of symplectic 2gx2g-matrices. All I know is this map is related to some action of M<sub>g</sub> on H<sub>1</sub>(S<sub>g</sub>,Z). I think this action is...
  36. M

    Mass in Symplectic techniques in physics by Guillemin and Sternberg

    Mass in "Symplectic techniques in physics" by Guillemin and Sternberg Hi, have somebody read this book? Guillemin and Sternberg introduces the notion of mass on page 435. But I don't understand here everything. For example, I dont't know what is \sigma (it is introduced but isn't used) and...
  37. C

    How do symplectic manifolds describe kinematics/dynamics ?

    I understand that symplectic manifolds are phase spaces in classical mechanics, I just don't understand why we would use them. I understand both the mathematics and the physics here, it is the connection between these areas that is cloudy... What on Earth does the symplectic form have to do...
  38. R

    Is the One-Form d \theta Well-Defined at the North and South Pole on a 2-Sphere?

    Hi, The 2-sphere is given as example of symplectic manifolds, with a symplectic form \Omega = \sin{\varphi} d \varphi \wedge d \theta. Here the parametrization is given by (x,y,z) = (\cos{\theta}\sin{\varphi}, \sin{\theta}\sin{\varphi}, \cos{\varphi}) with \varphi \in [0,\pi],\ \theta \in...
  39. D

    Applications of Symplectic Geometry

    I am a mathematics graduate student who is doing research in symplectic geometry (specifically symplectic toric orbifolds, symplectic reduction, Hamiltonian actions of tori in the symplectic category, etc). I often have tried to convince others of the importance of symplectic geometry, so I...
  40. S

    Symplectic Integrators for N-Body Simulations

    Hi all, I'm currently working on some n-body simulations and I'm trying to implement higher order integrators. Currently, I have a Leapfrog and a Hermite integrator (From the student series on The Art of Computational Science). The results I get from Leapfrog are excellent, with respect to...
  41. N

    Easy-to-Follow Proofs for Symplectic Manifolds: A Comprehensive Resource

    What's a really good resources with numerous easy-to-follow proofs to theorems on symplectic manifolds? Arnold is too difficult.
  42. N

    Symplectic mechanics - elementary particles

    In Souriau's book of symplectic mechanics he describes an elementary dynamical system on which the Poincare group is dynamic and acts transitively. He then describes a massive particle with spin where the spin is some positive number. When we consider this particle in the presence of an external...
  43. M

    Symplectic geometry. What's this?

    My training is in mathematics. But during my free time I also try to understand fundamental physics. Recently I came across a material which said that the geometry of classical mechanics is symplectic. I'm not sure of the meaning. It was relating to the Hamiltonian which I'm also not...
  44. J

    How can I solve these Symplectic Notation problems?

    Homework Statement My problem is: ``For all eigenvalues \omega_j being distinct show that the normalization of the eigenvectors can be chosen in such a way that M has the properties of the Jacobian matrix.'' Another problem is to show that after this canonical transformation the new...
  45. J

    How Can Symplectic Eigenvector Normalization Influence Hamiltonian Forms?

    The Hamiltonian, H=\frac{1}{2}\vec{\varsigma}K\vec{\varsigma} is given. With K being a 2n \times 2n matrix with the entries: \[ \left( \begin{array}{cc} 0 & \tau \\ \vartheta & 0\end{array} \right)\] and \vec{\varsigma} being a 2n-dimensional vector with entries...
  46. P

    Symplectic Geometry in Physics: String Theory & Beyond

    How crucial is symplectic geometry to fundamental physics? Any examples? I know it is related to string theory.
  47. S

    Why are generalized momenta cotangent vectors in symplectic manifolds?

    I've been reading about the abstract formulation of dynamics in terms of symplectic manifolds, and it's amazing how naturally everything falls out of it. But one thing I can't see is why the generalized momenta should be cotangent vectors. I can see why generalized velocities are tangent...
  48. C

    Symplectic Integrator Research: Solver & Reference Suggestions

    This summer I am doing a physics research project involving some computer modeling. One of the main aspects of the simulation that I am doing is using a numerical ODE solver. I originally wrote my own solver using the 4th-order Runge-Kutta method with a variable step size just to get a feel...
  49. J

    Symplectic structure vs. metric structure

    Symplectic structure vs. metric structure A question about the relationship between the phase space of the Hamiltonian formulation of classical mechanics and of the Lagrangian formulation; that is, between the cotangent bundle of configuration space, T*Q, which has a natural symplectic...
  50. N

    Symplectic integrator/hamiltonian

    I'm trying to get a numerical solution for a hamiltonian mechanics problem. According to wikipedia, there's a method of solving the resulting differential equations called a symplectic integrator that's designed specifically for such problems, but my computational physics textbook doesn't...
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