7.9: Generalized energy and total energy

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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.