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The Lagrangian Equation of Motion

The Lagrangian is defined aswhere andis a function ofHence

The first of Hamilton's equations givesso the bracketed term vanishes and leaves

The other of Hamilton's equations isand we can useto give

This is Lagrange's equation of motion. It is a second order differential equation when we use instead ofto represent the generalised velocity:

Since the Lagrangian is a function ofwe 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 toof 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 toandcommutes:

obtaining

The equation of motion is then

Similarly if two Lagrangians differ by a function