Convection Heat Transfer by Vedat S. Arpaci

By Vedat S. Arpaci

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Q˙ f are replaced by p1 , . . , p f , we do not refer to R f as a Routhian. Instead, it is the negative of the Hamiltonian H. 15. Derive the following set of equations of motion from the Routhian Rn : ∂ Rn ∂ Rn = p˙i , = −q˙i , i = 1, . . 179) ∂ qi ∂ pi and d dt ∂ Rn ∂ q˙i − ∂ Rn =0, ∂ qi i = n + 1, . . , f . 180) /// Equations of motion derived from a Routhian find practical applications in an analysis of the stability of a steady motion. 11 Poisson Bracket Quantities such as the energy and the total linear momentum of a system are called dynamical variables.

F . 154) Step 2: Solve these f equations for q˙1 , . . , q˙ f to express them as functions of q1 , . . , q f and p1 , . . , p f : q˙i = q˙i (q1 , . . , q f , p1 , . . , p f ) , i = 1, . . , f . 156) i=1 as a function of q1 , . . , q f and p1 , . . , p f . This makes H the negative of the Legendre transform of L. The function H(q1 , . . , q f , p1 , . . 157) is called the Hamiltonian of the system. 106), we see that Hamiltonian is the energy (function) of the system expressed as a function of q1 , .

Q f , q˙1 , . . 151) hold for any t A and t B . It follows that ∂L ≡0. 150) then indicates that E is a constant of motion. Energy is a constant of motion because “now” is as good as any other instant of time. One might argue that the outcome of the two experiments should be identical even if there is a time-dependent external field. But, this is so only if timing of the two experiments is properly chosen to make sure that they are in sync with the timedependent external field in an identical manner.

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