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Euler-Lagrange and Hamilton's Equations

Newton's second law of motion is  
 \begin{displaymath}
{\bf F} = \frac{d{\bf p}}{dt}\end{displaymath} (139)
or in component form (for each component Fi)  
 \begin{displaymath}
F_{i} = \frac{d{p_{i}}}{dt}\end{displaymath} (140)
where $p_{i} = m \dot{q}_{i}$ (with qi being the generalized position coordinate) so that $\frac{d{p_{i}}}{dt} = \dot{m} \dot{q}_{i} + m \ddot{q}_{i}$. If $\dot{m} = 0 $ then $F_{i} = m
\ddot{q}_{i} = m a_{i}$. For conservative forces ${\bf F} = - \bigtriangledown V $ where V is the scalar potential. Rewriting Newton's law we have  
 \begin{displaymath}
-\frac{dV}{d q_{i}} = \frac{d}{dt}(m \dot{q}_{i})\end{displaymath} (141)
Let us define the Lagrangian $L(q_{i} , \dot{q}_{i}) \equiv T - V$ where T is the kinetic energy. In freshman physics $T = T(\dot{q}_{i}) = \frac{1}{2} m \dot{q}_{i}^{2}$ and V=V(qi) such as the harmonic oscillator $V(q_{i}) = \frac{1}{2} k q_{i}^{2}$. That is in freshman physics T is a function only of velocity $\dot{q}_{i}$ and V is a function only of position qi. Thus $L(q_{i} , \dot{q}_{i}) = T(\dot{q}_{i}) - V(q_{i})$. It follows that $\frac{\partial L}{\partial q_{i}} =-\frac{dV}{d q_{i}} $ and $\frac{\partial L}{\partial \dot{q}_{i}}
=\frac{dT}{d \dot{q}_{i}} = m \dot{q}_{i} = p_{i}$. Thus Newton's law is

with the canonical momentum [1] defined as  
 \begin{displaymath}
p_{i} \equiv \frac{\partial L}{\partial \dot{q}_{i}}\end{displaymath} (142)
The second equation of (4.4) is known as the Euler-Lagrange equations of motion and serves as an alternative formulation of mechanics [1]. It is usually written  
 \begin{displaymath}
\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_{i}}) - \frac{\partial L}{\partial q_{i}} = 0\end{displaymath} (143)
or just  
 \begin{displaymath}
\dot{p}_{i} = \frac{\partial L}{\partial q_{i}}\end{displaymath} (144)
We have obtained the Euler-Lagrange equations using simple arguments. A more rigorous derivation is based on the calculus of variations [1] as discussed in Section 7.3.

We now introduce the Hamiltonian H defined as a function of p and q as  
 \begin{displaymath}
H( p_{i} , q_{i}) \equiv p_{i} \dot{q}_{i} - L( q_{i} , \dot{q}_{i} )\end{displaymath} (145)
For the simple case $T= \frac{1}{2} m \dot{q}_{i}^{2}$ and $V \neq V(\dot{q}_{i})$ we have $p_{i}\frac{\partial L}{\partial \dot{q}_{i}} = m \dot{q}_{i}$ so that $ T = \frac{p_{i}^{2}}{2m}$and $p_{i} \dot{q}_{i} = \frac{p_{i}^{2}}{m}$ so that $H( p_{i} , q_{i}) = \frac{p_{i}^{2}}{2m} + V(q_{i}) = T + V $ which is the total energy. Hamilton's equations of motion immediately follow from (4.8) as  
 \begin{displaymath}
\frac{\partial H}{\partial p_{i}} = \dot{q}_{i}\end{displaymath} (146)
because $L \neq L(p_{i}) $ and $\frac{\partial H}{\partial q_{i}} = - \frac{\partial L}{\partial q_{i}} $ so that from (4.4)  
 \begin{displaymath}
- \frac{\partial H}{\partial q_{i}} = \dot{p}_{i} .\end{displaymath} (147)


next up previous contents
Next: Classical Field Theory Up: ENERGY-MOMENTUM TENSOR Previous: ENERGY-MOMENTUM TENSOR
John Norbury
12/9/1997