7.9: Generalized energy and total energy

The generalized kinetic energy, equation $(7.6.4)$, can be used to write the generalized Lagrangian as

$$L(\mathbf{q},\mathbf{ \dot{q}},t)=T_{2}(\mathbf{q},\mathbf{\dot{q}},t)+T_{1}(\mathbf{q},\mathbf{ \dot{q}},t)+T_{0}(\mathbf{q},t)-U(\mathbf{q},t)$$

If the potential energy $U$ does not depend explicitly on velocities $\dot{q }_{i}$ or time, then

$$\label{7.42} p_{j}=\frac{\partial L}{\partial \dot{q}_{j}}=\frac{\partial \left( T-U\right) }{\partial \dot{q}_{j}}=\frac{\partial T}{\partial \dot{q}_{j}}$$

Equation $\ref{7.42}$ can be used to write the Hamiltonian, equation $(7.7.6)$, as

$$H\left( \mathbf{q,p,}t\right) =\sum_{i}\left( \dot{q}_{j}\frac{\partial T_{2} }{\partial \dot{q}_{j}}\right) +\sum_{i}\left( \dot{q}_{j}\frac{\partial T_{1}}{\partial \dot{q}_{j}}\right) +\sum_{i}\left( \dot{q}_{j}\frac{ \partial T_{0}}{\partial \dot{q}_{j}}\right) -L(\mathbf{q},\mathbf{\dot{q}} ,t)$$

Using equations $(7.6.12)$, $(7.6.13)$, $(7.6.14)$ gives that the total generalized Hamiltonian $H\left( \mathbf{q,p,}t\right)$ equals

$$H\left( \mathbf{q,p,}t\right) =2T_{2}+T_{1}-(T_{2}+T_{1}+T_{0}-U)=T_{2}-T_{0}+U \label{7.44}$$

But the sum of the kinetic and potential energies equals the total energy. Thus Equation $\ref{7.44}$ can be rewritten in the form

$$H\left( \mathbf{q,p,}t\right) =(T+U)-(T_{1}+2T_{0})=E-(T_{1}+2T_{0})$$

Note that Jacobi’s generalized energy and the Hamiltonian do not equal the total energy $E$. However, in the special case where the transformation is scleronomic, then $T_{1}=T_{0}=0,$ and if the potential energy $U$ does not depend explicitly of $\dot{q}_{i}$, then the generalized energy (Hamiltonian) equals the total energy, that is, $H=E.$ Recognition of the relation between the Hamiltonian and the total energy facilitates determining the equations of motion.