7.7: Generalized Energy and the Hamiltonian Function

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 $18$.

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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 $\mathbf{p}$.