By S. G. Rajeev

This direction could be more often than not approximately platforms that can't be solved during this manner in order that approximation equipment are precious.

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**Sample text**

There are two ways this can happen: the asteroid can form an equilateral triangle with the Sun and Jupiter on either side of the line joining them. These are the Lagrange points L4 and L5 . These are actually maxima of the potential. In spite of this fact, they correspond to stable equilibrium points because of the effect of the velocity dependent forces. 10 Let us find the frequencies for small oscillations around these points. t. x, y give ∂ 2 V := KΩ2 = − √ ± 3 4 3 (1 − 2ν) 3 4 √ ± 3 4 3 (1 − 2ν) 9 4 Ω2 , B = 2Ω 0 −1 .

Since ∇2 V = 0 in a vacuum, at least one of the eigenvalues of ∂ 2 V will be negative. But a magnetic field can stabilize system. For example, if the magnetic field is along the third axis, 0 b 0 1 0 0 2 2 B = −b 0 0 , B = −b 0 1 0 . 0 0 0 0 0 0 PHY411 S. G. Rajeev 33 An electrostatic potential which has a minimum along the direction of the magnetic field can now provide a stable equilibrium point for the charged particles, provided that the magnitude of the negative eigenvalues are not too large.

Here, extrx denotes the extremum in the variable x . There is a unique extremum for the quantity in the brackets because f (x) is invertible everywhere. The extremum is the solution of the non-linear equations: f (x) = p which has a solution ( the inverse function theorem of calculus) in a domain where f (x) is non-zero. •The Legendre transform is itself a convex function. The transform is its own inverse: ˆ fˆ(x) = f (x). •A useful example is a quadratic function, 1 f (x) = xT Ax 2 for some positive matrix x .