Geometrical Themes Inspired by the N-body Problem by Luis Hernández-Lamoneda Haydeé Herrera & Rafael Herrera

Author:Luis Hernández-Lamoneda, Haydeé Herrera & Rafael Herrera
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(7)

the sum being over all distinct pairs a, b. As is well known, the total energy

is conserved, where “conserved” means constant along solutions. (Another synonym for “conserved quantity” is “constant of the motion”.) We use

(8)

to measure the size of our configurations. Lagrange proved that

provided we are in center of mass coordinates: Σm a q a  = 0. Then

while

Exercise 1

Use the metric reformulation of Newton’s equations (3), the fact that U is homogeneous of degree − 1 and Euler’s identity for homogeneous functions to derive the “virial identity”, also known as the Lagrange-Jacobi identity: d 2(r 2)/dt 2 = 4H + 2U. Also show that 2H + U = H + K.

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