We consider a natural Lagrangian system and show that from a point \$q_0\$ in n-space, where the potential energy \$V\$ has a (weak) maximum, one can go near the boundary of any compact ball where \$V(q)\le V(q_0)\$ with (arbitrarily small) nonvanishing initial speeds. The result holds true for sets which are \$C^2\$-diffeomorphic to a compact ball. This property is found as a simple consequence of the Hopf-Rinow theorem and of a theorem of Gordon. As a corollary we deduce a well known local result, namely a `converse' of the Lagrange-Dirichlet theorem, thus obtained via geometric arguments.

### How far can one move from a potential peak with small initial speed?

#### Abstract

We consider a natural Lagrangian system and show that from a point \$q_0\$ in n-space, where the potential energy \$V\$ has a (weak) maximum, one can go near the boundary of any compact ball where \$V(q)\le V(q_0)\$ with (arbitrarily small) nonvanishing initial speeds. The result holds true for sets which are \$C^2\$-diffeomorphic to a compact ball. This property is found as a simple consequence of the Hopf-Rinow theorem and of a theorem of Gordon. As a corollary we deduce a well known local result, namely a `converse' of the Lagrange-Dirichlet theorem, thus obtained via geometric arguments.
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Natural Lagrangian systems, Hopf-Rinow theorem, instability of the equilibrium
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11562/393336`
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