S = L(q(t),(t),t) dt
| (14) |

q_{}(t) = q(t) + q(t)
| (15) |

S = _{ = 0}L(q_{}(t),(t),t) dt.
| (16) |

The classical calculation yields

_{ = 0}L(q_{}(t),(t),t) dt
| = |
_{ = 0}L(q_{}(t),(t),t) dt
| |

= |
dt.
| (17) |

(t) = (q(t))
| (18) |

S
| = |
q(t) dt
| |

+ | (19) |

The above extreme property of the solutions of the Lagrange equations
5 shows the invariance of these equations under
coordinate changes: if we use a time-dependent substitution *q* = *F*(*Q*,*t*) where
*F* : `x` is a change of variables then the new
Lagrangian with respect to coordinates *Q* is

K(Q,,t) = L(F(Q),DF(Q,t) + (Q,t),t)
| (20) |