Read e-book online A Course in Mathematical Physics, Vol. 1: Classical PDF

By Walter E Thirring

ISBN-10: 0387814965

ISBN-13: 9780387814964

Mathematical Physics, Nat. Sciences, Physics, arithmetic

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Extra info for A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems

Example text

Stationarity of the action means that the action does not change to first order if a small variation of a path is made keeping the endpoints fixed. In order to see that the true classical path of the system is a stationary point of A, we need to consider a path Q(t) between points 1 and 2 and a second path, Q(t) + δQ(t), between points 1 and 2 that is only slightly different from Q(t). If a path Q(t) renders A[Q] stationary, then to first order in δQ(t), the variation δA of the action must vanish. This can be shown by first noting that the path, Q(t) satisfies the initial and final conditions: Q(t1 ) = Q1 , ˙ 1 ) = Q˙ 1 , Q(t Q(t2 ) = Q2 , ˙ 2 ) = Q˙ 2 .

5) becomes 1 L = μ˙r2 − V (r). 6) 2 In a two-dimensional space, the relative coordinate r is the two-component vector r = (x, y). However, if the distance between the two atoms is fixed at a value d, then there is a constraint in the form of x2 + y 2 = d2 . Rather than treating the constraint via a Lagrange multiplier, we could transform to polar coordinates according to x = d cos θ y = d sin θ. 9) where the notation, V (r) = V (x, y) = V (d cos θ, d sin θ) ≡ V˜ (θ), indicates that the potential varies only according to the variation in θ.

5) where M = 2m and μ = m/2 are the total and reduced masses, respectively. Note that in these coordinates, the center-of-mass has an equation of motion of the form ¨ = 0, which is the equation of motion of a free particle. As we have already seen, MR ˙ is a constant. According to the principle this means that the center-of-mass velocity R of Galilean relativity, the physics of the system must be the same in a fixed coordinate frame as in a coordinate frame moving with constant velocity. Thus, we may transform to a coordinate system that moves with the molecule.

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A Course in Mathematical Physics, Vol. 1: Classical Dynamical Systems by Walter E Thirring

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