Modern conventions

We use generalized coordinates $q_i$

, generalized velocities $\dot q_i$ , and for a natural mechanical system the Lagrangian

$\displaystyle L(q,\dot q,t)=T(q,\dot q,t)-V(q,t).$

The canonical momenta are

$\displaystyle p_i=\frac{\partial L}{\partial \dot q_i}.$

When the Legendre transform is regular, the Hamiltonian is

$\displaystyle H(q,p,t)=\sum_{i=1}^{n} p_i\dot q_i-L,$

with the velocities expressed in terms of $(q,p,t)$

If generalized nonconservative forces are present, they are denoted by $Q_i^{\mathrm{nc}}$ . Then Hamilton's equations take the form

$\displaystyle \dot q_i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q_i}+Q_i^{\mathrm{nc}}.$

In the conservative case $Q_i^{\mathrm{nc}}=0$ .

Relation with Lagrange's equations

Start from the Euler–Lagrange equations with nonconservative generalized forces,

$\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot q_i}-\frac{\partial L}{\partial q_i} =Q_i^{\mathrm{nc}}.$

Because $p_i=\partial L/\partial \dot q_i$ , these become

$\displaystyle \dot p_i-\frac{\partial L}{\partial q_i}=Q_i^{\mathrm{nc}}. \tag{1}$

Now differentiate the Hamiltonian,

$\displaystyle dH=\sum_i \dot q_i\,dp_i-\sum_i \frac{\partial L}{\partial q_i}\,dq_i- \frac{\partial L}{\partial t}\,dt.$

Therefore

$\displaystyle \frac{\partial H}{\partial p_i}=\dot q_i, \qquad \frac{\partial H}{\partial q_i}=-\frac{\partial L}{\partial q_i}. \tag{2}$

Combining (1) and (2) yields Hamilton's equations

$\displaystyle \dot q_i=\frac{\partial H}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H}{\partial q_i}+Q_i^{\mathrm{nc}}. \tag{3}$

Autonomous natural systems

If the coordinate transformation is time-independent and

$\displaystyle L=T(q,\dot q)-V(q),$

then the Hamiltonian coincides with the total energy,

$\displaystyle H=T+V.$

This is the standard case behind many of Byerly's examples, even though he originally formulated the discussion in terms of a momentum-version of the kinetic energy alone.

Interpretation

Hamilton's equations replace $n$

second-order Euler–Lagrange equations by $2n$

first-order equations in phase-space variables $(q_i,p_i)$

. They are not usually computationally shorter for elementary problems, but they are the natural entrance to canonical transformations, phase space, Poisson brackets, and modern analytical mechanics.

Modern notation references

The notation and terminology in this modernized transcription follow standard analytical mechanics conventions, especially:

H. Goldstein, C. Poole, and J. Safko, classical mechanics, 3rd ed., Addison–Wesley, 2002.
C. Lanczos, The Variational Principles of Mechanics, 4th ed., Dover, 1986.
J. R. Taylor, Classical Mechanics, University Science Books, 2005.

Source

This article is a modernized restatement of the corresponding portion of William Elwood Byerly, An Introduction to the Use of Generalized Coördinates in Mechanics and Physics, Ginn and Company, 1916, Chapter II. The 1916 source work is in the public domain in the United States.

"Hamiltonian form of Lagrange's equation" is owned by bloftin.
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