On the mechanics with noncontinuous Hamiltonians. (English) Zbl 0606.58021

The authors attempt a formulation of mechanics with nondifferentiable or noncontinuous Hamiltonians. For this purpose they introduce the notion of an overflow, which is more general than the notion of a flow and more adequate to describe a dynamics of mechanical systems. For example, they consider the following Hamiltonians, given by (1) \({\mathbb{R}}^ 2\ni (x,p)\to H(x,p)=p^ 2/2m+V(x)\in {\mathbb{R}}^ 1\), where for all x: \[ {\mathbb{R}}^ 1\ni x\to V(x)= \begin{cases} 0 &\text{ for \(x\in {\mathbb{R}}^ 1\{0\}\)} \\ \infty &\text{ for \(x=0,\)} \end{cases} \] or \[ {\mathbb{R}}^ 1\ni x\to V(x)= \begin{cases} -\alpha x &\text{ for \(x\in (-\infty,0), \alpha >0\)} \\ \beta x &\text{ for \(x\in [0,\infty)\), \(\beta >0\);} \end{cases} \] (2) \({\mathbb{R}}^ 2\ni (x,p)\to H(x,p)=p^ 2/2m+V(x)\), where \[ {\mathbb{R}}^ 1\ni x\to V(x)= \begin{cases} 0 &\text{ for \(x\in (-\infty,0)\)} \\ E &\text{ for \(x\in [0,\infty)\)} \end{cases} \] or \[ V(x)= \begin{cases} 0 &\text{ for \(x\in {\mathbb{R}}^ 1-\{0\}\)} \\ E &\text{ for \(x=0,\)} \end{cases} \] and interesting results are obtained.


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H05 Hamilton’s equations