Preview of Ch. 7

1. Preview of Ch. 7 • The use of this abstract calculus of variations in mechanics! More details next time! Hamilton’s Principle • Of all of the possible paths of a mechanical system, the path actually followed is the one which minimizes the time integral of the difference in the kinetic & potential energies. That is, the actual path is the one which makes the variation of the following integral vanish: δ∫[T - U] dt = 0(limits t1 < t < t2)

2. δ∫[T - U] dt = 0 (limits t1 < t < t2)(1) Here: T = T(xi) & U = U(xi), • N particles in 3d, i = 1,2, ..(3N) • Define: The LagrangianL L  T(xi) - U(xi) = L(xi,xi) (1) δ∫Ldt = 0 (2) • This is identical to the abstract calculus of variations problem of Ch. 6 with the replacements: δJ  δ∫Ldt , x  t , yi(x)  xi(t) yi(x)  (dxi(t)/dt) = xi(t), f[yi(x),yi(x);x]  L(xi,xi;t) (1) The Lagrangian L satisfies Euler’s equations with these replacements!

3. δ∫Ldt = 0 (limits t1 < t < t2)(2)  The Lagrangian L = L(xi,xi) satisfies Euler’s equations! Or, with the changes noted: (L/xi) - (d/dt)[(L/xi)] = 0 (3) N particles in 3d: i = 1,2, ..(3N)  Lagrange Equations of Motion • (3) enables us to get equations of motion without (explicitly) using Newton’s 2nd Law!