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.
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...
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...
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...
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...
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...
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...
(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...
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...
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...
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...
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...
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...
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<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...
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...
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...
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...
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...
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...
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...
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...
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...