Abstract
After the publication of the first edition of Mécanique analytique, the most important step forward in the transformation of the differential equation of motion was made by Poisson in a paper which deals with the method of variation of constants and which appears in Volume 15 of the Polytechnique Journal. Here Poisson introduces the quantity \(p = \frac{{\partial T}} {{\partial q'}}\), in place of the quantity q′; now since, as already remarked, T is a homogeneous function of the second degree in the quantities q′ whose coefficients depend on q, p is a linear function of the quantities q′; for the definition of p one has the k equations of the form \({p_i} = {\tilde \omega _i}\), where \({\tilde \omega _i}\) is linear with respect to q′1, …, q’ k . If one solves these linear equations for the quantities q′, one then obtains equations of the form q′ i = K i where the K i ’s are linear expressions in p whose coefficients depend on the q. We shall insert these expressions for q′ i in the equation (8.10) of Lecture 8, i.e., in the equation
, where \(\frac{{\partial U}} {{\partial {q_i}}}\) contains only q, while \(\frac{{\partial T}} {{\partial {q_i}}}\) is, besides, a function of the quantities q′, indeed, a homogeneous function of the second degree of these quantities. If we set q′1 = K i , then \(\frac{{\partial T}} {{\partial {q_i}}}\) is a homogeneous function of second degree in the quantities p i . Hence the above equations will be of the form
, where P i is an expression in p and q and in fact of the second degree with respect to p.
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Clebsch, A. (2009). Hamilton’s Form of the Equations of Motion. In: Clebsch, A. (eds) Jacobi’s Lectures on Dynamics. Texts and Readings in Mathematics. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-62-0_9
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DOI: https://doi.org/10.1007/978-93-86279-62-0_9
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-91-3
Online ISBN: 978-93-86279-62-0
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