bradley j. nartowt Tuesday, December 31, 2013, 19:47:16
PHYS 6246 – classical classical mechanics Dr. Whiting Whiting
jerky lagrangian mechanics: The term generalized mechanics has come to designate a variety of classical mechanics in which the Lagrangian contains time-derivatives of qi higher than the first. Problems for which x f ( x, x, x; t ) have
been referred to as “jerky” mechanics. Such equations of motion have interesting applications in chaos theory. By applying applying the methods methods of the calculus of variations: variations: show that that if there there is a Lagrangian Lagrangian of the form L L( qi , qi , qi ; t ) , and hamilton’s principle holds with the zero variation of ( qi , qi ) at the end-points, then the EL equations are,
d 2 L
d L L 0 dt 2 qi dt qi qi
i 1,2,..., n
[I.1]
Summation over i is implied. Minimize action of the functional L( qi , qi , qi ; t ) using the total differential,
0d
Notice that, that, by the product rule of
d L
dt qi
2
1
d dt
L( qi , qi , qi ; t ) dt
L L L d q d q d q dt 1 qi i qi i qi i dt 2
[I.2]
(to prepare for partial integration),
d L L d q dqi i d t q q i i
dqi
d L
dt qi
d L L d q dqi i d t q q i i
dqi
[I.3]
Use [I.3] to integrate the second and third terms of [I.2], yielding, 2
2 L L d L d L L 0 dqi dq dqi dqi d q i dqi dt qi qi 1 1 qi dt qi dt qi
The terms
2
1 vanish by hypothesis (zero variations at path-endpoints). Look at the term
L qi
[I.4]
dqi --we eventually want to
partial integrate integrate that, that, so write the product product rule again again as,
d d L
d 2 L d L d q d q i 2 i dqi dt dt qi d t d q d t d q i i Using [I.5] on the term ( dtd
L qi )dqi
[I.5]
in [I.4], we get,
2
2 2 L 2 L d L d L d 2 L d L d L 0 dqi 2 dqi 1 dqi 2 dqi dt 1 dqi dt [I.6] d t q q d t q d t d q q d t q d t d q i i i i i i 1 i
Since our hypothesis treated the function L L(q, q, q; t ) , the q, q, q are as completely-independent degrees of freedom of the functional L. This is the reasoning behind the fundamental fundamental lemma of variational calculus calculus (Goldstein [2.10]), which allows us to immediately immediately write,
L d L d 2 L 0 qi dt qi dt 2 dqi
[I.7]
Apply [I.1] to the following Lagrangian, and anticipate a familiar equation of motion.
L 12 mqq 12 kq 2
Computing the derivatives,
[I.8]
d 2 L
d2 1 1 2 mq 0 mq 2 dt q dt 2 2 That would be Hooke’s
law — clearly, q is oscillatory.
d L
0 dt q
L kq q
1 2
mq kq
[I.9]
