Calculus and linear algebra by Wilfred Kaplan, Donald J. Lewis

By Wilfred Kaplan, Donald J. Lewis

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5), the non-linearity is due to the presence of a square root. 49) 1+ε 1+ .

From the Euler–Lagrange equations, we may derive the Hamiltonian description by introducing the generalized momenta: ∂L · pi q, q = · ∂q , i = 1, n, pT = [p1 , · · · , pn ] . 24) The Hamiltonian is expressed as follows: n · · pi q 1 −L q, q, t . 25) i=1 The total derivative leads to: n dH = i ∂H ∂H dqi + dpi ∂qi ∂pi + ∂H dt. 25), we derive: · · q dpi + pi d q i − dH = i ∂L ∂L · dqi − · d q i ∂qi ∂ qi − ∂L dt. 24), yields5: ∂H ∂L =− , ∂t ∂t and  · ∂H   qi = , i = 1, n, ∂pi · ∂H   pi = − , i = 1, n.

Chemical reactors are used to manufacture a wide variety of materials. Consider a stirred tank reactor where neutralization takes place. In this chemical reaction, H + and OH − combine to form H2 O (water) molecules, and the remaining components lead to a salt. 14. The two input streams are sodium hydroxide (N aOH) and hydrochloric acid (HCl). The concentrations of N aOH and HCl in these inputs are C1 and C2 , respectively. The volume of the reactor is assumed to be constant and equal to V . 11.

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