Hamiltonian form of Lagrange's equation

(Definition)

Modern conventions

We use generalized coordinates q_i, generalized velocities q_i, and for a natural mechanical system the Lagrangian

L(q,q˙,t) = T (q, ˙q,t) − V (q,t).

The canonical momenta are

∂L pi = ∂-˙q. i

When the Legendre transform is regular, the hamiltonian is

∑ n H (q,p,t) = pi ˙qi − L, i=1

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

If generalized nonconservative forces are present, they are denoted by Q_i^nc. Then Hamilton’s equations take the form

∂H ∂H nc ˙qi = ---, ˙pi = − ----+ Q i . ∂pi ∂qi

In the conservative case Q_i^nc = 0.

Relation with Lagrange’s equations

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

-d ∂L-− ∂L- = Qnc . dt ∂ ˙qi ∂qi i

Because p_i = ∂L∕∂q_i, these become

p˙− ∂L- = Qnc . (1) i ∂qi i

Now differentiate the Hamiltonian,

dH = ∑ ˙q dp − ∑ ∂L-dq − ∂L-dt. i i ∂qi i ∂t i i

Therefore

∂H--= q˙, ∂H-- = − ∂L-. (2) ∂pi i ∂qi ∂qi

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

˙q = ∂H-, ˙p = − ∂H--+ Qnc . (3) i ∂pi i ∂qi i

Autonomous natural systems

If the coordinate transformation is time-independent and

L = T(q,q˙) − V (q),

then the Hamiltonian coincides with the total energy,

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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