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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

Source:
The Lazy Universe
Author(s):

Jennifer Coopersmith

Publisher:
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:

(A6.5.1)
i=1NFiδri=j=1Mi=1NFiriqjδqj=j=1MQjδqj

Therefore the generalized forces, Qj, are given by:

(A6.5.2)
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:

(A6.5.3)
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

(A6.5.4)
Qj=Vqj+ddtVq˙jj=1 to M

and the Lagrange Equations become

(A6.5.5)
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.