Derive lagrange's equation from hamilton's principle for holonomic system
Answers
We now use the fact that we can “cancel the dots” and ∂x˙
A/∂q˙a = ∂xA/∂qa which
we can prove by substituting the expression for ˙x
A into the LHS. Taking the time
derivative of (6.48) gives us
d
dt
∂L
∂q˙a
=
d
dt
∂L
∂x˙
A
∂xA
∂qa
+
∂L
∂x˙
A
∂
2x
A
∂qa∂qb
q˙b +
∂
2x
A
∂qa∂t
(6.49)
So combining (6.47) with (6.49) we find
q˙a −
∂L
∂qa
=
d
dt
∂L
∂x˙
A
−
∂L
∂xA
∂xA
∂qa
(6.50)
Equation (6.50) is our final result. We see that if Lagrange’s equation is solved in
the x
A coordinate system (so that [. . .] on the RHS vanishes) then it is also solved in
the qa coordinate system. (Conversely, if it is satisfied in the qa coordinate system,
so the LHS vanishes, then it is also satisfied in the x
A coordinate system as long as
our choice of coordinates is invertible: i.e det(∂xA/∂qa) 6= 0)