8
votes
0answers
186 views

What's the physical intuition for symplectic structures?

I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector ...
6
votes
2answers
187 views

Are Poisson brackets of second-class constraints independent of the canonical coordinates?

Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\{\chi_N(q,p),\chi_M(q,p)\}_P$ . Is ...
6
votes
5answers
1k views

What does symplecticity imply?

Symplectic systems are a common object of studies in classical physics and nonlinearity sciences. At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context ...
7
votes
4answers
468 views

Hamiltonian and the space-time structure

I'm reading Arnold's "Mathematical Methods of Classical Mechanics" but I failed to find rigorous development for the allowed forms of Hamiltonian. Space-time structure dictates the form of ...
4
votes
3answers
230 views

What are some mechanics examples with a globally non-generic symplecic structure?

In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
2
votes
1answer
212 views

A question regarding particle trajectories in the symplectic manifold formalism

How to solve a free particle on a 2-sphere using symplectic manifold formalism of classical mechanics ? Is there a way to get coriolis effect directly, without going into Newton mechanics? And is ...
7
votes
2answers
541 views

Lorentz invariance of the 3 + 1 decomposition of spacetime

Why is allowed decompose the spacetime metric into a spatial part + temporal part like this for example $$ds^2 ~=~ (-N^2 + N_aN^a)dt^2 + 2N_adtdx^a + q_{ab}dx^adx^b$$ ($N$ is called lapse, $N_a$ is ...
4
votes
3answers
926 views

Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics?

Books such as Mathematical methods of classical mechanics describe an approach to classical (Newtonian/Galilean) mechanics where Hamiltonian mechanics turn into a theory of symplectic forms on ...