7.9: Generalized energy and total energy
- Page ID
- 14077
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.