Florida State University
Department of Physics
PHY 5246
Assignment # 4 (Due Friday, 23rd October, 2009)
(1) (10 points) If
L
is a Lagrangian for a system of
n
degrees of freedom satisfying Lagrange’s
equations, show by direct substitution that
L
′
=
L
+
dF
(
q
1
,...,q
n
,t
)
dt
also satisfies Lagrange’s equations, where
F
is any arbitrary, but differentiable, function of its
arguments.
(2) (10 points) Let
q
1
,...,q
n
be a set of independent generalized coordinates for a system
of
n
degrees of freedom, with a Lagrangian
L
(
q,
˙
q,t
). Suppose we transform to another set of
independent coordinates
s
1
,...,s
n
, by means of transformation equations
q
i
=
q
i
(
s
1
,...,s
n
,t
)
, i
= 1
,...,n.
(Such a transformation is called a
point transformation
.) Show that if the Lagrangian function
is expressed as a function of
s
j
,
˙
s
j
and
t
through the equations of transformation, then
L
satisfies Lagrange’s equations with respect to the
s
coordinates:
d
dt
parenleftBigg
∂L
∂
˙
s
j
parenrightBigg
−
∂L
∂s
j
= 0
.
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 Fall '09
 ROBERTS
 Physics, mechanics, Equations, Lagrangian mechanics, Euler–Lagrange equation

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