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7.7: Generalized Energy and the Hamiltonian Function

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    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 \(\mathbf{p}\).

    This page titled 7.7: Generalized Energy and the Hamiltonian Function 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; a detailed edit history is available upon request.