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Ch01 Newtonian mechanics

The total energy of the ball i e the sum of potential

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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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