PX285 - D3 - the conservation of energy

H = T + V

H = E

if energy is conserved, $\frac{d H}{d t} = 0$ , ie: $H$ is a constant of motion:

\int_{t_{0}}^{t_{1}} \frac{d H}{d t} d t = H (t_{1}) - H (t_{0}) = 0

constant energy is also defined as 'an integral of the motion'
this only contains the first-order time derivatives of the coordinates, whereas the euler-lagrange equation is second-order
a none on notation:
- $H$ is used, in place of $E$ , because it is reserved for the expression of energy in terms of coordinates
- $E$ stands for the constant value it keeps