Advanced classical mechanics: chaos by S. G. Rajeev

By S. G. Rajeev

This direction might be ordinarily approximately structures that can't be solved during this manner in order that approximation tools are beneficial.

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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. •An integrable system with n degrees of freedom will have n conserved quantities: the ‘action’ variables.

Then, x(V ) will be single valued function with range 0 < x(V ) < ∞ when V is in the range V0 < V < 0 . (Here, V0 = V (0) ). 0 T (E) = 2 V0 1 1 dx(V ) [E − V ]− 2 − E − 2 dV. dV ) which is a linear integral equation for dx(V . This can be solved using dV the theory of fractional integrals. Thus in this case the potential can be reconstructed from the scattering data. • Aside The fractional integral of order µ of a function f (x) is Rµ f (y) = 1 Γ(µ) y 0 f (x)(y − x)µ−1 dx. The path of integration in the complex x plane is the straight line connecting the origin to y .

In the case of the harmonic oscillator this frequency is even independent of energy or I ; but in general this is not true. Chapter 12 Hamilton-Jacobi Theory •Suppose we are presented a system with hamiltonian H(q, p) in some canonical co-ordinate system (q i , pi ) . If we can find a new canonical coordinate system (θi , Ii ) such that the hamiltonian depends only on the momentum variables Ii , then we have solved the equations of motion. For, in this system, the Hamiltons equations are dIi = 0, dt dθi = ω i (I) dt where the frequencies are ω i (I) = ∂H(I) .

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