The Lagrangian Formulation of Mechanics

Hamilton’s principe

The action is defined by

and the Euler-Lagrange equation of motion is

Conservation of energy

For Lagrangian, we have

$$\frac{\mathrm{d}L}{\mathrm{d}t}=\sum_{i}\left[\frac{\partial L}{\partial q_{i}}\dot{q}{i}+\frac{\partial L}{\partial\dot{q}{i}}\ddot{q}_{i}\right],$$

and then

Thus

remains constant during the motion, and which is called the energy of the system.

The generalised momenta, , are defined by

The Hamiltonian of a system is defined by

In terms of and , the energy equation for a closed system becomes

Continuous systems

Consider a flexible string, we have

Remember the extension of a string is , and the potential energy is , hence

,

where

is called Lagrangian density.

The corresponding action is

where , .

Also, we have

yield

insert above Lagrangian density equation, yield

which is the familiar wave equation for small amplitude waves on a string.

We can define the momentum density

and the Hamiltonian density

Note that the Lagrangian density does not depend explicitly on , it follows that

Lorentz covariant field theory

We have

$$ S=\int \mathcal L\mathrm{d}x\mathrm{d}y\mathrm{d}z\mathrm{d}t=\int \mathcal L\mathrm{d}x^0\mathrm{d}x^1\mathrm{d}x^2\mathrm{d}x^3.$$

Note the ‘volume element’ is a Lorentz invariant.

Consider a Lorentz invariant Lagrangian density of the form

where is a scalar field. Finally, we find the field equation

the Klein-Gordon equation

We have already know the Lorentz invariant Lagrangian density is

where is a real scalar field, and it is a particular case of above equation. The above field equation becomes

or

where , is called d’Alembert operation. The equation is known as the Klein-Gordon equation.

The general solution of above equation is a superposition of such plane waves:

The energy-momentum tensor

Consider a infinitesimal space-time displacement

where does not depend on . And the corresponding change in is

Vary the Lagrangian density

yield

We have also

So, since the are arbitrary, it follows on comparing these expressions for that

or

where

is called the energy-momentum tensor. And the component

corresponds to the Hamiltonian density defined in equation , and is interpreted as the energy density of the field. Then, the component of the above equation may be written

integrating over all space and using the divergence theorem yields

This equation expresses the overall conservation of energy. And we can see that the energy is conserved when the field is vanished at large distances.

Similarly, the overall total momentum of the field given by

As with the energy, the total momentum of the field is conserved if the field vanishes at large distances.

According to the Klein-Gordon Lagrangian density,

and the energy density

Expressing in terms of the field amplitudes and , and integrating over all space, give the total field energy

Similarly, the total momentum of the field can be shown to be

Complex scalar fields

The real Lagrangian density of complex scalar fields is

$$\mathcal{L}=\partial_\mu\Phi^\partial^\mu\Phi-m^2\Phi^\Phi.$$

Note that the Lagrangian density is the sum of contributions from the scalar fields and :

$\mathcal{L}=\partial_\mu\Phi^\partial^\mu\Phi-m^2\Phi^\Phi$

Then the general solution is

where and are now independent complex numbers. The field energy is

$$H=\sum_\mathrm{k}\left(a_\mathrm{k}^a_\mathrm{k}+b_\mathrm{k}^b_\mathrm{k}\right)\omega_\mathrm{k}.$$

We shall see that we can interpret this expression as being made up of the distinct contributions of positively and negatively charged fields. (The and mesons are composite particles whose overall motion is described by complex scalar fields).