# Classical mechanics and Geommetry

1. Aug 7, 2007

### Klaus_Hoffmann

Given the Hamiltonian of a system $$\mathcal H$$ , could we obtain the curves solution to Hamilton equations X(t) Y(t) Z(t) as the Geodesic of a certain surface with Christoffle symbols $$\Gamma ^{i} _{jk}$$ i mean the curve X(t) satisfies the equation:

$$\nabla _{x(t)} X(t)=0$$ (covariant derivative vanishes)

Also given the 1-form $$\theta =p^{i}dq^{i}-Hdt$$ and some elements of Diff. Geommetry how could solve our physical system ?? or reduce the solutions for x,y,z to 'Quadratures' ?? thanks.