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Physics LibreTexts

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(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=(TU)˙qj=T˙qj

Equation ??? can be used to write the Hamiltonian, equation (7.7.6), as

H(q,p,t)=i(˙qjT2˙qj)+i(˙qjT1˙qj)+i(˙qjT0˙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+T0U)=T2T0+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.


This page titled 7.9: Generalized energy and total energy is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

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