By Sylvester's law of inertia, the number of positive entries is an invariant of the bilinear form, i . e . it does not depend on the choice of the diagonalizing basis, and is called the "'index "'of the Lie algebra \ mathfrak g.
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:: : : : : Also anything we build on earth is travelling the same speed as the earth ( law of inertia ) . the rockets will need to create momentum greater than the orbiting earth to knock it to a larger orbit.
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Sylvester understood the geometric meaning of positive-definite quadratic forms and proved his law of inertia in 1852, but it doesn't seem to have occurred to him ( or anyone else ) that there might be a more general kind of geometry corresponding to general quadratic forms.
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One possibility is to treat it as a modification to the classical law of inertia ( Newton's second law ), so that the force on an object is not proportional to the particle's acceleration "'a "'but rather to ? ( a / a 0 ) "'a " '.
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By the Galilean law of inertia as well as the principle Galilean invariance, also called Galilean relativity, for any object experiencing inertia, there is empirical justification of knowing only its being at " relative " rest or " relative " motion rest or motion with respect to another object.
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As an example, many students hold the aristotelian misconception that a net force is required to keep a body moving; instead, motion is modelled in modern physics with the Newton's First Law of inertia, stating that a body will keep its state of rest or movement unless a net force acts on the body.
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