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**Unformatted text preview: **PHYS4221 Quantum Mechanics I Fall 2010 Review of Analytic Classical Mechanics Lagrangian Formulation of Classical Mechanics : Given a particle whose kinetic energy T and potential energy V are known, the classical trajectory x cl ( t ) of the particle can be determined as follows: 1. Define a Lagrangian function L ( x, ˙ x ) = T- V . 1 2. For each path x ( t ) connecting the two space-time points ( x i ,t i ) and ( x f ,t f ), where t i ≤ t f , calculate the functional S [ x ( t )] defined by S [ x ( t )] = Z t f t i L ( x, ˙ x ) dt . (1) This functional is known as the action and depends upon the entire path x ( t ). 3. The classical path x cl ( t ) is the one on which S is a minimum . Along the x cl ( t ), the Euler-Lagrange equation is obeyed: 2 d dt ∂L ( x, ˙ x ) ∂ ˙ x ( t ) !- ∂L ( x, ˙ x ) ∂x ( t ) = 0 for t i ≤ t ≤ t f . (2) Since T = 1 2 m ˙ x 2 and V = V ( x ), we get ∂L ∂ ˙ x = ∂T ∂ ˙ x = m ˙ x (3) and ∂L ∂x =- ∂V ∂x . (4) Accordingly, the Euler-Lagrange equation becomes d dt ( m ˙ x ) =- ∂V ∂x (5) which is just Newton’s equation of motion. 1 The Lagrangian function will also have explicit t dependence if the particle is in an external time- dependent field. 2 Detailed derivations can be found in any standard textbook on Analytic Classical Mechanics . 1 For a system of N particles, the same procedure yields d dt ∂L ( { x i } , { ˙ x i } ) ∂ ˙ x i ( t ) !...

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