Advanced classical mechanics: chaos by S. G. Rajeev

By S. G. Rajeev

This direction can be commonly approximately platforms that can't be solved during this manner in order that approximation equipment are priceless.

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This completes the proof. •The solution of the equation Dws = Rs is facilitated by diagonalizing D . Being anti-symmetric, it is diagonalized by complex linear combinations zj = qj + ipj , z¯j = qj − ipj : Dz A z¯B = i [ai − bi ]ωi z A z¯B . i Thus the pseudo-inverse of D may be defined to be 0 D−1 z A z¯B = i if − bi ]ωi = 0 otherwise i [ai 1 z A z¯B [a −b ]ω i i i i . Then we just have to express H s as a polynomial in z, z¯ and then set ws = −D−1 H s . •This will give us the hamiltonian as a function of (q, P ) .

One of the simplest mechanical systems is the simple harmonic oscillator with Lagrangian 1 1 L = mx˙ 2 − kx2 . 2 2 •The equations of motion m¨ x + kx = 0 have the well known solutions in terms of trigonometric functions: x(t) = a cos[ω0 (t − t0 )] √ where the angular frequency ω0 = (k/m) . The constants a and t0 are constants of integration. They have simple physical meanins: A is the maximum displacement and t0 is a time at which x(t) has this maximum value. The energy is conserved and has the value 12 ma2 .

Rajeev 57 As long as the potential is of the form V = a(r) + c(φ) b(θ) + r2 r2 sin2 θ a solution of the form W = R(r) + Θ(θ) + Φ(φ) exists. This is the method of separation of variables. The ordinary differential equations for the functions R, Θ, Φ can be solved in terms of elliptic functions. For details see Landau and Lifshitz. •If we can solve the H-J equations, and determine the normal co-ordinates, we can solve the equations of motion completely. Such systems are said to be integrable. It turns out that not all systems are integrable.

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