2

0. Some Aspects of Theoretical Mechanics

I

or in the local coordinates as

t i—• qi(t) with qi(t0) = g°, i = 1,..., n.

Here physical principles must be found which allow one to give the curve as a

solution to a differential equation. The starting point for this determination

is the classical mechanical principle of least action. For this it is assumed

that the system has a Lagrange function L of the form

L = L(q,q,t),

which is gotten as the difference of the kinetic and the potential energies,

L = -Eton

_

^pot

which is also written as

L = T - V.

The principle of least action now says that the change in the system

proceeds so that the curve 7 minimizes the path integral

ti

Ldt.

to

The variational calculus now says (see

COURANT-HILBERT

[CH], p. 170)

that for the minimum curve 7 — q(t) the system satisfies the Euler-Lagrange

equations

m d dL dL

dt dqi dqi

This can be seen as a system of ordinary differential equations in a

2n-dimensional space TQ with local coordinates

9i • • • 9n? Qi, • • • An

(which can be understood as the tangent bundle over the configuration space

Q (see Section A.3)). The desired curve 7 on Q is the projection of the

solution curve 7 of (1) onto TQ.

0.2. Hamilton's equations

Classical mechanics now takes the following formulation: for a given La-

grange function L the coordinates position and velocity, (g, g), are replaced

by the coordinates position and momentum (g, p) made possible by the

transformation

dL .

dqi