By S. G. Rajeev

This path should be quite often approximately platforms that can not be solved during this approach in order that approximation equipment are important.

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**Example text**

Dt ∂q These are Hamilton’s equations. • Example Let L = 12 mq˙2 − V (q) , the Lagrangian of a partcle of mass m in a potential V (x) . The variable p is just the momentum p = mq˙ and H the energy: p2 + V (q). 2m Whenever the Lagrangian does not depend explicitly on time, the hamiltonian is the energy. The set of initial conditions is a manifold of dimension 2n , called the phase space. Through each point in this space is a unique curve which is the solution of the equations of motion. p2 • ExampleThe system with hamiltonian H = 2m + 12 kx2 is the simple harmonic oscillator.

03852. PHY411 S. G. Rajeev 37 The frequencies are given by ω2 = 1 1 1 ± [1 − 27ν(1 − ν)] 2 . 11 The libration periods as multiples of the period T are 1 √ 1 T1,2 = T √ [1 ± {1 − 27ν(1 − ν)}]− 2 . 90 years. 6 days. 13 Lagrange thought that these special solutions were artificial and that they would never be realized in nature. But we now know that there are asteroids ( Trojan asteroids) that form an equilateral triangle with Sun and Jupiter. 14 Lagrange discovered something even more astonishing: the equilateral triangle is an exact solution for the full three body problem, not assuming one of the bodies to be infinitesimally small.

Rajeev 37 The frequencies are given by ω2 = 1 1 1 ± [1 − 27ν(1 − ν)] 2 . 11 The libration periods as multiples of the period T are 1 √ 1 T1,2 = T √ [1 ± {1 − 27ν(1 − ν)}]− 2 . 90 years. 6 days. 13 Lagrange thought that these special solutions were artificial and that they would never be realized in nature. But we now know that there are asteroids ( Trojan asteroids) that form an equilateral triangle with Sun and Jupiter. 14 Lagrange discovered something even more astonishing: the equilateral triangle is an exact solution for the full three body problem, not assuming one of the bodies to be infinitesimally small.

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

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