ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 3 (January 20, 2010)

Solution for assignment 11a:

For a bilinear kinetic Energy we find by differentiation and carrying the sum over i out

∑_{i} q̇_{i} \frac{∂T}{∂q̇_{i}} = ∑_{i} q̇_{i} \frac{∂}{∂q̇_{i}} ∑_{j,k} a_{jk} q̇_{j}q̇_{k} = ∑_{i} q̇_{i} ∑_{j,k} a_{jk} (δ_{ij} q̇_{k} + q̇_{j} δ_{ik})
= ∑_{j,k} a_{jk} (q̇_{j} q̇_{k} + q̇_{j} q̇_{k}) = 2 T .

Solution for assignment 11b: Motivation of generalized momentum definition.

\frac{∂}{∂ẋ_{i}} (\frac{1}{2} m v^{2} − V (x⃗)) = m ẋ_{i} = p_{i} .