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Unformatted text preview: . The total energy of the ball, i. e. the sum of potential and kinetic energy © E. F. Schubert 2 remains constant during the oscillatory motion as long as no external forces act on the object. 1.3 Hamiltonian formulation of newtonian mechanics
Equations (1.1) and (1.2) are known as the newtonian formulation of classical mechanics. The
hamiltonian formulation of classical mechanics has the same physical content as the newtonian
formulation. However, the hamiltonian formulation focuses on energy. The hamiltonian
function H ( x , p) is defined as the total energy of a system
H ( x, p ) = p2
+ U ( x) .
2m (1.7) The partial derivatives of the hamiltonian function with respect to x and p are given by
∂
H ( x, p ) =
∂x d
U ( x)
dx (1.8) ∂
H ( x, p ) =
∂p p
.
m (1.9) Employing these partial derivatives and Eqs. (1.1) and (1.4), one obtains two equations, which
are known as the hamiltonian equations of motion:
∂
H ( x, p )
∂p dx
dt
dp
dt © E. F. Schubert = =− ∂
H ( x, p ) .
∂...
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This note was uploaded on 02/05/2013 for the course CHEM 131A taught by Professor Rentzepis during the Fall '12 term at UC Irvine.
 Fall '12
 Rentzepis

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