Preliminary to this proof, it is important to address an ambiguity in the related mathematical notation. While a change of variables can be used to equate L ( p , q , t ) = L ( q , q ˙ ˙ , t ) , it is important to note that ∂ ∂ L ( q , q ˙ ˙ , t ) ∂ ∂ q ˙ ˙ i ≠ ≠ ∂ ∂ L ( p , q , t ) ∂ ∂ q ˙ ˙ i . In this case, the right hand side always evaluates to 0. To perform a change of variables inside of a partial derivative, the multivariable chain rule should be used. Hence, to avoid ambiguity, the function arguments of any term inside of a partial derivative should be stated.