(1)

or integrating from t to t′,

Σ (δp′r·Δq′r − Δq′r·δq′r) = Σ (δpr·Δqr − Δpr·δqr).

(2)

If for shortness we write

(r, s) = ∂²S,   (r, s′) = ∂²S,
∂qr∂qs ∂qr∂q′s

(3)

we have

∂pr = −Σs (r, s) δqs − Σs (r, s′) δq′s

(4)