7.9: Generalized energy and total energy
( \newcommand{\kernel}{\mathrm{null}\,}\)
The generalized kinetic energy, equation (7.6.4), can be used to write the generalized Lagrangian as
L(q,˙q,t)=T2(q,˙q,t)+T1(q,˙q,t)+T0(q,t)−U(q,t)
If the potential energy U does not depend explicitly on velocities ˙qi or time, then
pj=∂L∂˙qj=∂(T−U)∂˙qj=∂T∂˙qj
Equation ??? can be used to write the Hamiltonian, equation (7.7.6), as
H(q,p,t)=∑i(˙qj∂T2∂˙qj)+∑i(˙qj∂T1∂˙qj)+∑i(˙qj∂T0∂˙qj)−L(q,˙q,t)
Using equations (7.6.12), (7.6.13), (7.6.14) gives that the total generalized Hamiltonian H(q,p,t) equals
H(q,p,t)=2T2+T1−(T2+T1+T0−U)=T2−T0+U
But the sum of the kinetic and potential energies equals the total energy. Thus Equation ??? can be rewritten in the form
H(q,p,t)=(T+U)−(T1+2T0)=E−(T1+2T0)
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 T1=T0=0, and if the potential energy U does not depend explicitly of ˙qi, 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.