Let \(M\) be a smooth manifold and \(\pi:TM\to M\) its tangent bundle. The vertical subbundle \(V\subset T(TM)\) is \(\text{Ker} D\pi\) and a supplement of it is a horizontal bundle. A linear connection in \(V\) is good if it can be canonically prolonged to \(TM\). A Finsler function on \(TM\) provides a Riemannian metric for \(V\) and the Cartan connection appears as a good metrical connection with some torsions vanishing. The author provides similar characterisations for the Chern and the Berwald connections. He reobtains a result of the reviewer that the Chern connection coincides with the Rund connection [Contemp. Math. 196, 171-176 (1996; Zbl 0868.53050)]. A minimal compatibility condition between a vertical connection and a Finsler metric is described by using the symplectic structure canonically associated to a Finsler metric. Reviewer's remark: The name of Berwald is mistakenly printed Bernwald in the whole paper.