# 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.