7.7: Generalized Energy and the Hamiltonian Function
- Page ID
- 14075
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consider the time derivative of the Lagrangian, plus the fact that time is the independent variable in the Lagrangian. Then the total time derivative is
\[\frac{dL}{dt}=\sum_{j}\frac{\partial L}{\partial q_{j}}\dot{q}_{j}+\sum_{j} \frac{\partial L}{\partial \dot{q}_{j}}\ddot{q}_{j}+\frac{\partial L}{ \partial t} \label{7.32}\]
The Lagrange equations for a conservative force are given by equation \((6.5.12)\) to be
\[\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_{j}}-\frac{\partial L}{ \partial q_{j}}=Q_{j}^{EXC}+\sum_{k=1}^{m}\lambda _{k}\frac{\partial g_{k}}{ \partial q_{j}}(\mathbf{q},t) \label{7.33}\]
The holonomic constraints can be accounted for using the Lagrange multiplier terms while the generalized force \(Q_{j}^{EXC}\) includes non-holonomic forces or other forces not included in the potential energy term of the Lagrangian, or holonomic forces not accounted for by the Lagrange multiplier terms.
Substituting Equation \ref{7.33} into Equation \ref{7.32} gives
\[\begin{align} \frac{dL}{dt} &=&\sum_{j}\dot{q}_{j}\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_{j}}-\sum_{j}\dot{q}_{j}\left[ Q_{j}^{EXC}+\sum_{k=1}^{m}\lambda _{k} \frac{\partial g_{k}}{\partial q_{j}}(\mathbf{q},t)\right] +\sum_{j}\frac{ \partial L}{\partial \dot{q}_{j}}\ddot{q}_{j}+\frac{\partial L}{\partial t} \notag \\ &=&\sum_{j}\frac{d}{dt}\left( \dot{q}_{j}\frac{\partial L}{\partial \dot{q} _{j}}\right) -\sum_{j}\dot{q}_{j}\left[ Q_{j}^{EXC}+\sum_{k=1}^{m}\lambda _{k}\frac{\partial g_{k}}{\partial q_{j}}(\mathbf{q},t)\right] +\frac{ \partial L}{\partial t}\end{align}\]
This can be written in the form \[\frac{d}{dt}\left[ \sum_{j}\left( \dot{q}_{j}\frac{\partial L}{\partial \dot{ q}_{j}}\right) -L\right] =\sum_{j}\dot{q}_{j}\left[ Q_{j}^{EXC}+ \sum_{k=1}^{m}\lambda _{k}\frac{\partial g_{k}}{\partial q_{j}}(\mathbf{q},t) \right] -\frac{\partial L}{\partial t}\]
Define Jacobi’s Generalized Energy1 \(h(\mathbf{q},\mathbf{ \dot{q}},t)\) by
\[h(\mathbf{q},\mathbf{ \dot{q}},t)\equiv \sum_{j}\left( \dot{q}_{j}\frac{\partial L}{\partial \dot{q }_{j}}\right) -L(\mathbf{q},\mathbf{\dot{q}},t)\]
Jacobi’s generalized momentum, equation \(7.2.3,\) can be used to express the generalized energy \(h(q,\dot{q},t)\) in terms of the canonical coordinates \( \dot{q}_{i}\) and \(p_{i}\), plus time \(t\). Define the Hamiltonian function to equal the generalized energy expressed in terms of the conjugate variables \((q_{j},p_{j})\), that is,
\[H\left( \mathbf{q,p,}t\right) \equiv h(\mathbf{q},\mathbf{\dot{q}},t)\equiv \sum_{j}\left( \dot{q}_{j}\frac{\partial L}{\partial \dot{q}_{j}}\right) -L( \mathbf{q},\mathbf{\dot{q}},t)=\sum_{j}\left( \dot{q}_{j}p_{j}\right) -L( \mathbf{q},\mathbf{\dot{q}},t)\]
This Hamiltonian \(H\left( \mathbf{q,p,}t\right)\) underlies Hamiltonian mechanics which plays a profoundly important role in most branches of physics as illustrated in chapters \(8,15\) and \(18\).
1Most textbooks call the function \(h(\mathbf{q},\mathbf{\dot{q}},t)\) Jacobi’s energy integral. This book adopts the more descriptive name Generalized energy in analogy with use of generalized coordinates \( \mathbf{q}\) and generalized momentum \(\mathbf{p}\).