is moved to a new time-dependent position x(t), q:(t) = x(t) + r;(t) a) Show that the expressions for the kinetic energy are related by mi M T = x + M x · Q + , r 2 Here, M = E,m; and Q = M'E m;q; are the total mass and the center of
is moved to a new time-dependent position x(t), q:(t) = x(t) + r;(t) a) Show that the expressions for the kinetic energy are related by mi M T = x + M x · Q + , r 2 Here, M = E,m; and Q = M'E m;q; are the total mass and the center of
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Parts a to d
Transcribed Image Text:We consider a coordinate transformation where the origin of the coordinate systems
is moved to a new time-dependent position x(t),
q:(t) = x(t) + r;(t)
a) Show that the expressions for the kinetic energy are related by
M
x² + M x · Q + £ r
T =
2
Here, M = E;m; and Q = M¯' E, m;q; are the total mass and the center of
mass, respectively.
b) Show that the expressions for the total energy for motion in an external field
are related
E =T – M g · Q +E,(la. – qjl) = T – M g· Q + ® (Ir. – r;) – M g · x
i<j
i<j
c) Show that the angular momentum transforms as follows
L =Em; q; × ġ; = M x × Q + M (x + Q) × x + m; x; x *;
d) Show that conservation laws are mapped to conservation laws iff we consider a
Galilei transformation, i.e., a transformation where x = const.
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