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The Lazy UniverseAn Introduction to the Principle of Least Action$

Jennifer Coopersmith

Print publication date: 2017

Print ISBN-13: 9780198743040

Published to Oxford Scholarship Online: June 2017

DOI: 10.1093/oso/9780198743040.001.0001

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(p.228) Appendix A6.5 Generalized Forces

(p.228) Appendix A6.5 Generalized Forces

The Lazy Universe

Jennifer Coopersmith

Oxford University Press

What is the connection between the potential energy function, V, the force, F, and the generalized force, Q? The virtual work, for N particles, may be transformed from rectangular to generalized coordinates as follows:


Therefore the generalized forces, Qj, are given by:

Qj=i=1NFiriqjj=1 to M

Note how the upper limits, N, or M, are not the same. In the especially simple case of a ‘central potential’, that is, V=V(r1,r2,rN), we have:

Fi=Vrii=1 to N,andQj=i=1NVririqj=Vqjj=1 to M

More generally we have V=V(r1,r2,rN;r˙1,r˙2,r˙N;t), and then

Qj=Vqj+ddtVq˙jj=1 to M

and the Lagrange Equations become

ddtLq˙jLqj=Qjj=1 to M

(p.229) where L includes only the conservative forces, and Qj comprises the non-conservative forces such as frictional forces. (Note that these non-conservative forces are still to be expressed in functional form, for example, Qj=F/q˙j where F is Rayleigh’s dissipation function.) In summary, the non-conservative forces introduce a ‘right-hand side’ to the Lagrange Equations.