By J. Paull, D. Paull
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Extra info for Simple Mechanics [gradeschool level]
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Yij = 1/2 = pJ~ , (3IJ.. rjh (i,j,k,h = 1, 2, ... ,n), 48 the other invariants - for n > 2 - are expressed with their help. Therefore we have H =H (ai, ~ij' 'Yij' 'Yijh • 'Yik,jh ), where i,j,h,k == 1,2, ... n. 2nd, The expressions of the impulses Pj and of H must satisfy the law of motion, which require that the (invariant) geometrical derivation of the system (p 1 , p 2 , ... , Pn•H) should vanish. yd. The components of the geometrical derivative of the system (p 1 , p 2 , ... , Pn, H) are, by definition, the coefficients of the vartattons lix 1 , ...
5. : In accordance with principle 4, the expression of H must reduce under Newtonian conditions to the form HNewton = 1/2 m~ v~ + 1/2 m~ v~- momo f - 1- 2 r +c. (12) Since HNewton is defmite except the constant, and since whenwe pass to the invariant mechanics we fmd for any material particle instead of 1/2 m~v 2 + C the expression mc 2, we write (13) 52 where the brackets include besides the individual masses also the interaction masses required by the expressions (12 ). In this form H must satisfy the conditions required by the principle 2.
Motion of the Stable Particles in a Field 1. The Potential Form. n 6
. In accordance with the inertial form (21, §1) of n 6(i) the field is defined by two vector potentials A(A 1 , A2 , A 3 ) and B(B 1 , B2 , B3 ) relative to the position of P and to the body orientation respectively, and by a scalar potential C. Hence we have n,(P) 0 = ~ ~ A. -1 8x. - C8t . J ~ J (1) J A priori, we must consider A, B and C as functions of all the variables x 1 , x 2 , x 3 , a 1 , a 2 , a 3 , t. However, if we confine ourselves to the classical fields and adopt for A, B and C the calculations indicated by the nature of these fields - as we shall do in the next paragraph - then A, B and C must be considered as functions of x 1 , x 2 , x 3 and t only.