Symplectic Projector Research Articles

Local middle dimensional symplectic non-squeezing in the analytic setting

We prove the following middle-dimensional non-squeezing result for analytic symplectic embeddings of domains in $\mathbb{R}^{2n}$. Let $\varphi: D \hookrightarrow \mathbb{R}^{2n}$ be an analytic symplectic embedding of a domain $D \subset \mathbb{R}^{2n}$ and $P$ be a symplectic projector onto a linear $2k$-dimensional symplectic subspace $V\subset \mathbb{R}^{2n}$. Then there exists a positive function $r_0:D\rightarrow (0,+ \infty)$, bounded away from $0$ on compact subsets $K \subset D$, such that the inequality $Vol_{2k}(P\varphi (B_r(x)),\omega ^k _{0|V})\geq \pi^{k} r^{2k}$ holds for every $x \in D$ and for every $r < r_0(x)$. This claim will be deduced from an analytic middle-dimensional non-squeezing result (stated by considering paths of symplectic embeddings) whose proof will be carried on by taking advantage of a work by \'{A}lvarez Paiva and Balacheff.

Symplectic projector in constrained systems

In this work we develop a geometric formalism for constrained Hamiltonian systems. Using a symplectic projector, we show that the variables compatible with the constraints are written as functions of the arbitrary variables. Their Poisson bracket is just the Dirac bracket of the unconstrained variables. This approach yields in a natural way all properties associated to a generalized symplectic structure with a well-defined effective Hamiltonian.