As usual, the dual of E is denote by E ∗ ∗ . The fiber of π π over x ∈ ∈ M is denoted E x , and the restriction of L to E x is denoted by L | E x : E x → → R . The Legendre transformation of L is the smooth morphism F L : E → → E ∗ ∗ defined by F L ( v ) = d ( L | E x ) v ∈ ∈ E x ∗ ∗ , where x = π π ( v ) . Here we use the fact that since E x is a vector space, T v ( E x ) can be identified with E x . In other words, F L ( v ) ∈ ∈ E x ∗ ∗ is the covector that sends w ∈ ∈ E x to the directional derivative d d t | t = 0 L ( v + t w ) ∈ ∈ R .