The Lagrangian is defined as
where
and
is a function of
Hence
The first of Hamilton's equations givesso the bracketed term vanishes and leaves
The other of Hamilton's equations is
and we can use
to give
This is Lagrange's equation of motion. It is a second order differential equation when we use
instead of
to represent the generalised velocity:
Since the Lagrangian is a function of
we have by the chain rule,
so Lagrange's equation can also be written
(1)
which is clearly second order. This is equivalent to Hamilton's coupled first order equations. It is important that
The addition to
of a function depending only on the time does not affect the equation of motion which is instant from (1). neither does the addition of a total time derivative of a function
and
Differentiating with respect toand
commutes: 
obtaining
The equation of motion is then
Similarly if two Lagrangians differ by a function