Two-body and central forcemotion (Solution of the ditferential equation)Using a straightforward vectorapproachd'r2+rxr=0(1)rxdt?drh=rxDefiningangular momentum perunit mass asdtd?rddrdrdr-and=0(rx+Xdt2dtdtdtdtLeads to the conclusion that dh/ dt =0Thus, angular momentum is conserved and three integrals ofmotion areh=constantand this plane must be inertially fixed
Two-body and central force motion (Solution of the differential equation) Using a straightforward vector approach. 2 2 3 0 d dt r + = r r r r Defining angular momentum per unit mass as d dt = r h r and 2 2 ( ) 0 d d d d d dt dt dt dt dt = + = r r r r r r Leads to the conclusion that d dt h/ 0 = h = constant r Thus, angular momentum is conserved and three integrals of motion are . and this plane must be inertially fixed (1)
Two-bodyand central forcemotion (Solution of theditferential equation)Cross equation (1) withhd'rdrxh=-rxh=-rx(rxdt?rdtApplying the standard identity for triple vector products and noting thatdrdr-dt-dtd'r(2)deLeadstoxh=μdt?dtSinceh is constant,equation(2.)maybe integrated directlydxh=兰(r+re)(3)dte is a constant of integration and is called the eccentricity vector.Theorientation of einthis planeistaken as areference direction
Two-body and central force motion (Solution of the differential equation) Cross equation (1) with h 2 2 3 3 ( ) d d dt r r dt = − = − r r h r h r r Applying the standard identity for triple vector products and noting that d dr r dt dt = r r Leads to 2 2 ( ) d d dt dt r = r r h (2) Since h is constant, equation (2.) may be integrated directly ( ) d r dt r = + r h r e e is a constant of integration and is called the eccentricity vector. The orientation of e in this plane is taken as a reference direction (3)