In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex Hamiltonian systems with arbitrarily many degrees of freedom: $$ H(x,y)=h(y)+eps P(x,y), qquad xinmathbb{T}^n, yinmathbb{R}^n,quad ngeq 3. $$ Under typica
l perturbation $eps P$, the system admits connecting orbit that passes through any finitely many prescribed small balls in the same energy level $H^{-1}(E)$ provided $E>min h$.
In this paper, we show that there is a Cantor set of initial conditions in the planar four-body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlev{e} conjecture for the four-body case,
and thus settles the last open case of the conjecture.
In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in cite{P} in a sharp way.
As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*T^2$. This solves a problem of Arnold in cite{A}.
In this paper, we study a model of simplified four-body problem called planar two-center-two-body problem. In the plane, we have two fixed centers $Q_1=(-chi,0)$, $Q_2=(0,0)$ of masses 1, and two moving bodies $Q_3$ and $Q_4$ of masses $mull 1$. They
interact via Newtonian potential. $Q_3$ is captured by $Q_2$, and $Q_4$ travels back and forth between two centers. Based on a model of Gerver, we prove that there is a Cantor set of initial conditions which lead to solutions of the Hamiltonian system whose velocities are accelerated to infinity within finite time avoiding all early collisions. We consider this model as a simplified model for the planar four-body problem case of the Painlev{e} conjecture.
