By S. G. Rajeev

This path could be usually approximately platforms that can't be solved during this manner in order that approximation tools are priceless.

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N=0 where |A| = supu ||Au|| is the norm of the matrix. ) •There is a similar convergent power series expansion for the solution of the time dependent linear system. This is in fact a way to prove the existence of solutions of such a syste,. It also proves the real analyticity of the solution as a function of time, provided A(t) itself is real analytic. •The trick is to write the ODE as a system of integral equations: t z(t) = z(0) + A(t1 )z(t1 )dt1 0 and iterate it to get an infinite series: t t z(t) = z(0) + 0 A(t1 )z(0)dt1 + t 0 tn−1 dtn−1 0 t + 0 tn dtn 0 dtn−2 · · · dtn−1 · · · 0 t2 0 0 t2 t2 dt2 0 dt1 A(t2 )A(t1 )z(0) + · · · dt1 A(tn−1 ) · · · A(t2 )A(t1 )z(0) dt1 A(tn ) · · · A(t2 )A(t1 )z(0) + · · · .

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 .

This is called the ‘Jacobi integral’ in classical literature. 6 The hamiltonian is of the form H = T + V where T is the kinetic energy and V is an effective potential energy: V (r, χ) = −G[M + m] r2 1−ν ν + + 3 2R r1 r2 It consists of the gravitational potential energy plus a term due to the centrifugal barrier, since we are in a rotating co-ordinate system. 36 PHY411 S. G. 7 The effective potential V (r, χ) is conveniently expressed in terms of the distances to the massive bodies, V (r1 , r2 ) = −G M r12 1 + 2R3 r1 +m r22 1 + 2R3 r2 using the identity 1 2 1 2 1 r1 + r2 = r 2 + R2 .

### Advanced classical mechanics: chaos by S. G. Rajeev

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