Classical Mechanics by Matthew J. Benacquista & Joseph D. Romano

Author:Matthew J. Benacquista & Joseph D. Romano
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

The Lagrangian for the system is given as usual by , where

(7.60)

and

(7.61)

As the Lagrangian is independent of and , the Euler-Lagrange equations for these coordinates imply

(7.62)

These equations can be inverted to yield

(7.63)

(7.64)

The above equations can be integrated to yield and once we know .

Exercise 7.9

Show that T has the above form in (7.60) using (6.​71) for the components of with respect to the prinicipal axes.

Rather than write down the Euler-Lagrange equation for , which will be a 2nd-order ordinary differential equation, we can obtain a 1st-order equation by using the fact that is conserved (since the Lagrangian does not depend explicitly on t). After some straightforward algebra, we find

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