1. Preview of Ch. 7 Hamilton’s Principle
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 = (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 Lagrangian L
L  T(xi) - U(xi) = L(xi,xi)
(1)  δ∫Ldt = (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.  The Lagrangian L = L(xi,xi) satisfies Euler’s equations!
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)] = (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!