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2.3.4 Two special cases

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

 (2.23)

Note that we are being somewhat informal in denoting the third argument of by , and also in omitting the -arguments. We now discuss two special cases in which (2.24) can be reduced to a differential equation that is of first order, and therefore more tractable.

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

 (2.24)

for various values of . We already encountered such a situation in Example 2.2, where we actually had . Due to the presence of the parameter , we expect that the family of solutions of (2.25) is rich enough to contain one (and only one) extremal that passes through two given points.

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.

Next: 2.3.5 Variable-endpoint problems Up: 2.3 First-order necessary conditions Previous: 2.3.3 Technical remarks   Contents   Index
Daniel 2010-12-20