Question

1. Conservation of energy Consider a Lagrangian which depends on the generalized coordinates q_i(t) and their time derivatives dot{q}_i(t), but which does not depend explicitly on time. Use the Beltrami identity to show that the Hamiltonian H = (sum_{i} frac{partial mathcal{L}}{partial dot{q}_i} dot{q}_i) - mathcal{L}, is a conserved quantity. To understand the significance of this result, consider the case where the Lagrangian is that of a particle moving in 3 dimensions, so that the generalized coordinates may be taken to be x, y, and z. Show that in this case H = T + U.

          1. Conservation of energy
Consider a Lagrangian which depends on the generalized coordinates q_i(t) and their time derivatives dot{q}_i(t), but which does not depend explicitly on time. Use the Beltrami identity to show that the Hamiltonian
H = (sum_{i} frac{partial mathcal{L}}{partial dot{q}_i} dot{q}_i) - mathcal{L},
is a conserved quantity. To understand the significance of this result, consider the case where the Lagrangian is that of a particle moving in 3 dimensions, so that the generalized coordinates may be taken to be x, y, and z. Show that in this case
H = T + U.

$1. Conservation of energy Consider a Lagrangian which depends on the generalized coordinates qi(t) and their time derivatives dotqi(t), but which does not depend explicitly on time. Use the Beltrami identity to show that the Hamiltonian H = (sumi fracpartial mathcalLpartial dotqi dotqi) - mathcalL, is a conserved quantity. To understand the significance of this result, consider the case where the Lagrangian is that of a particle moving in 3 dimensions, so that the generalized coordinates may be taken to be x, y, and z. Show that in this case H = T + U.$

Added by Stanley W.

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