We wish to minimize subject to a given power(constant in time) 1 mc 1 Fc nd te write(1)as which eliminates thrust Next, treat (3)as a dynamic constraint, and append it to the cost through a time dependent Lagrange multiplier i(t) Define the hamiltonian H=2nP m dt or, using(2) H 2nP( 2nP, dm (6) and minimize(unconstrained )the integral J Hdt To do this, perturb about the optimum solution 8+2a+2m+nmj=0 dt d Integrate last term by parts: 16.522, Space Propulsion Lecture 2 Prof. manuel martinez-Sanchez Page 10 of 1916.522, Space Propulsion Lecture 2 Prof. Manuel Martinez-Sanchez Page 10 of 19 We wish to minimize tf 0 F V = dt m ⎛ ⎞ ∆ ∫ ⎜ ⎟ ⎝ ⎠ (1) subject to a given power (constant in time) 2 1 m c 1 Fc p= = 2 2 η η i (2) and to dm m=- dt i (3) write (1) as tf 0 2 P V = dt mc η ∆ ∫ (4) which eliminates thrust. Next, treat (3) as a dynamic constraint, and append it to the cost through a time￾dependent Lagrange multiplier λ (t). Define the Hamiltonian 2 P dm H = - m+ mc dt η ⎛ ⎞ λ ⎜ ⎟ ⎝ ⎠ i (5) or, using (2), 2 2 P 2 P dm H= - + mc dt c η η ⎛ ⎞ λ ⎜ ⎟ ⎝ ⎠ (6) and minimize (unconstrained) the integral tf ∫0 Hdt .To do this, perturb about the optimum solution: t t f f 0 0 H H H dm Hdt = c + m + dt = 0 c m dt dm dt ⎡ ⎛⎞ ∂∂ ∂ ⎤ δ δ δ δ ∫ ∫ ⎢ ⎜ ⎟⎥ ⎣ ⎝⎠ ∂ ∂ ⎛ ⎞ ⎦ ∂ ⎜ ⎟ ⎝ ⎠ ( ) d m dt δ Integrate last term by parts: