Symplectic Definition and 42 Threads

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...

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...

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...

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...

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)...

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...

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...

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...

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 :=...

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...

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...

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...

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...

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...

(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...

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...

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 ...

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...

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...

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...

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

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...

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.

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...

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 .

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...

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...

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...

Hi, Everyone: I am reading a paper that refers to a "natural surjection" between M_g and the group of symplectic 2gx2g-matrices. All I know is this map is related to some action of M_g on H₁(S_g,Z). I think this action is...

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...

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...

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

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...

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...

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

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...

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...

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...

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...

Hi, I have this hamiltonian K = \frac{1}{2}(P_1^2 + P_2^2) + P_1 Q_2 - P_2 Q_1 - (\frac{1-\mu}{R_1}+\frac{\mu}{R_2}) (see this link if there is any latex problem) with non separable variables. I am looking for a symplectic algorithm (runge kutta would be good) to solve the correspondant...

Hi ! I'm trying to solve the restricted problem of three bodies, where a negligeable mass particule is moving in the gravitationnal field of two heavy objects which are in circular orbit around their common center of mass. this is a plane problem... I describe the mouvment in the mobile...

What is "symplectic"? I was interested in learning the meaning of this word as I saw it used in Goldstein. Looking in the web it appears this word has to do with a "symplectic group". I don't know much about group theory, but I was wondering if someone could give an explanation "for the...