We know from the discussion given towards the end of Section 2.3.1--see the formula (2.19) in particular--that the Euler-Lagrange equation (2.18) is a second-order differential equation for ; indeed, it can be written in more detail as
SPECIAL CASE 1 (``no "). This refers to the situation where the Lagrangian does not depend on , i.e., . The Euler-Lagrange equation (2.18) becomes which means that must stay constant. In other words, extremals are solutions of the first-order differential equation
The quantity , evaluated along a given curve, is called the momentum. This terminology will be justified in Section 2.4.
SPECIAL CASE 2 (``no "). Suppose now that the Lagrangian does not depend on , i.e., . In this case the partial derivative vanishes from (2.24), and the Euler-Lagrange equation becomes
Multiplying both sides by , we have
where the last equality is easily verified (the terms cancel out). This means that must remain constant. Thus, similarly to Case 1, extremals are described by the family of first-order differential equations
parameterized by .
The quantity is called the Hamiltonian. Although its significance is not yet clear at this point, it will play a crucial role throughout the rest of the book.