D.3 Lagrangian mechanics

This note gives the derivations for the addendum on the Lagrangian equations of motion.

D.3.1 Lagrangian equations of motion

To derive the nonrelativistic Lagrangian, consider the system to be
build up from elementary particles numbered by an index

Newton’s second law says that the motion of each individual
particle

where the derivative of the potential

Now consider an infinitesimal virtual displacement of the system from
its normal evolution in time. It produces an infinitesimal change in
position

In the equation of motion for the correct position

Multiply out and integrate the first term by parts:

The virtual displacements of interest here are only nonzero over a limited range of times, so the integration by parts did not produce any end point values.

Recognize the first two terms within the brackets as the virtual
change in the Lagrangian due to the virtual displacement at that time.
Note that this requires that the potential energy depends only on the
position coordinates and time, and not also on the time derivatives of
the position coordinates. You get

(D.3) |

In case that the additional forces

Unchanging action is an integral equation involving the Lagrangian.
To get ordinary differential equations, take the virtual change in
position to be that due to an infinitesimal change

The integrand in the final term is by definition the generalized force

Now suppose that there is any time at which the expression within the
square brackets is nonzero. Then a virtual change

D.3.2 Hamiltonian dynamics

To derive the Hamiltonian equations, consider the general differential
of the Hamiltonian function (regardless of any motion that may go on).
According to the given definition of the Hamiltonian function, and
using a total differential for

The sums within parentheses cancel each other because of the definition of the canonical momentum. The remaining differences are of the arguments of the Hamiltonian function, and so by the very definition of partial derivatives,

Now consider an actual motion. For an actual motion,

It is still to be shown that the Hamiltonian of a classical system is
the sum of kinetic and potential energy if the position of the system
does not depend explicitly on time. The Lagrangian can be written out
in terms of the system particles as

where the sum represents the kinetic energy. The Hamiltonian is defined as

and straight substitution shows the first term to be twice the kinetic energy.

D.3.3 Fields

As discussed in {A.1.5}, the Lagrangian for fields
takes the form

Here the spatial integration is over all space. The first term depends only on the discrete variables

where

where

The action is

where the time range from

Consider now first an infinitesimal deviation

After an integration by parts of the second and fourth terms that becomes, noting that the deviation must vanish at the initial and final times,

This can only be zero for whatever you take

Next consider an infinitesimal deviation

Now integrate the derivative terms by parts in the appropriate direction to get, noting that the deviation must vanish at the limits of integration,

Here

The canonical momenta are defined as

For Hamilton’s equations, assume at first that there are no
discrete variables. In that case, the Hamiltonian can be written in
terms of a Hamiltonian density

Take a differential of the Hamiltonian density

The first and third terms in the square brackets cancel because of the definition of the canonical momentum. Then according to calculus

The first of these expressions gives the time derivative of

If there are discrete variables, this no longer works. The full
Hamiltonian is then

To find Hamilton’s equations, the integrals in this Hamiltonian must be approximated. The region of integration is mentally chopped into little pieces of the same volume

Here

The differential of this approximate Hamiltonian is

The

The

For the field, consider an position

Of course, in real life you would not actually write out these limits.
Instead you simply differentiate the normal Hamiltonian

Then you think to yourself that you are not really evaluating this, but actually

where

even though the left hand side would mathematically be nonsense without discretization and division by