Action Principles and Euler-Lagrange Equations

In this section, we will review the Lagrangian formalism, which is the basis of classical field theory.

Table of Contents

• Table of Contents
• Action Principles
• Euler-Lagrange Equations
• Coordinate Systems
  • Canonical Momentum
• Example Problem: Simple Harmonic Oscillator
• Exercises
• Summary and Next Steps

Action Principles

In classical mechanics, the motion of a particle is determined by the principle of stationary action (also known as the principle of least action). Historically, this principle was first formulated by Pierre Louis Maupertuis, where the action is defined as the product of mass, velocity, and distance.

Modern physics uses a more general form of the principle of stationary action, known as the Hamilton's principle. William Rowan Hamilton wrote that the action can be expressed as the integral of a quantity called the Lagrangian over time:

where is the path taken by the particle, is the Lagrangian, and is the derivative of with respect to time. For a more detailed explanation of action principles, I like this Veritasium video.

The principle of stationary action states that the path taken by a particle between two points in space and time is such that the action is stationary. If is the actual path taken by the particle, then if we perturb the path by a small amount, the action should not change to first order in . This is analogous to how a function has a minimum at a stationary point - if you move a little bit away from the stationary point, the function should not change much. Mathematically, this is expressed as .

In simple cases, the Lagrangian is defined as the difference between the kinetic and potential energies of the system:

Lagrangian: The Lagrangian of a system is defined as the difference between the kinetic and potential energies:

Euler-Lagrange Equations

The principle of stationary action can be used to derive the Euler-Lagrange equations, which are the equations of motion in the Lagrangian formalism.

The following derivation tries not to use any calculus of variations to keep it simple.

Suppose is the path taken by the particle, and is a nearby path. We let be a function that vanishes at the endpoints of the path, i.e. , and is a small parameter that we will later set to zero. This effectively means that is a small perturbation of the path that does not chnage the endpoints. Call this new path .

According to the principle of stationary action, the action should be at a stationary value, hence the derivative of the action with respect to should vanish at :

Plugging the definition (Equation ) of the action into the above equation, we get:

is a function of , , and , so we can use the chain rule as follows:

(there is no explicit dependence of on ; changes only through and ). Since , we have and :

We can use the product rule to simplify the right-most term in the above equation. Doing so and factoring out , we get:

Plugging this back into the principle of stationary action, we get:

The first term in the above equation can be simplified using the fundamental theorem of calculus to get . But since , this term vanishes. Hence, the above equation simplifies to:

By the principle of stationary action, this should vanish for all , which implies that the integrand should vanish for all . Since is arbitrary, the integrand should vanish for all , which gives us the Euler-Lagrange equations:

Euler-Lagrange Equations: For a Lagrangian defined by , the Euler-Lagrange equations are given by:

info

The Euler-Lagrange equations are the equations of motion in the Lagrangian formalism. We specifically used a derivation that does not impose a specific form of the Lagrangian, so the above equation is quite general.

In principle, it would be easier to plug in the value of the Lagrangian and derive the Euler-Lagrange equations. The action becomes:

Then, if varies by to , the resulting change in the action with respect to is:

Since , then . For the term, .

Then, the change in the action is:

Using the product rule, :

Just like before, the first term, when integrated, gives , which vanishes since . Hence, the above equation simplifies to:

Since this has to be zero for all , the integrand should vanish for all . This gives the following:

Notice that and . Hence:

which is Newton's second law.

It is important to notice the parallel between the Euler-Lagrange equations and Newton's second law:

Euler-Lagrange	Newton's Second Law

In other words, Newton's second law is a special case of the Euler-Lagrange equations:

However, we did not assume that the Lagrangian was of the form in the derivation of the Euler-Lagrange equations, so the above equation is quite general. In fact, the Lagrangian can be any function of , , and , and applying Hamilton's principle will lead to the Euler-Lagrange equations.

Coordinate Systems

Note that the Euler-Lagrange equations are written in terms of the generalized coordinates . These are coordinates that describe the configuration of the system, and they are not necessarily the Cartesian coordinates. As long as a set of coordinates can uniquely describe the configuration of the system, they can be used as generalized coordinates.

Given a set of coordinates , the space that these coordinates span is called the configuration space. It contains all possible configurations of the system; each point in configuration space corresponds to a unique physical state.

Canonical Momentum

In Newtonian mechanics, the momentum of a particle is defined as . Of course, this relies on a few assumptions of the -coordinate. If it was another coordinate, such as for a pendulum, the momentum would be . We need to form a more general definition of momentum that works for any coordinate system.

The canonical momentum that is conjugate to a generalized coordinate is defined as:

Example Problem: Simple Harmonic Oscillator

Consider a spring-mass system with mass and spring constant . Using the Lagrangian formalism, derive the equations of motion for the system.

The Lagrangian for this system is defined as the difference between the kinetic and potential energies:

Next, we just need to find the two terms in the Euler-Lagrange equations and plug them in.

First, is:

Second, is:

Plugging these into the Euler-Lagrange equations, we get:

which is the equation of motion for a simple harmonic oscillator.

Exercises

1. Show that the Euler-Lagrange equations lead to Newton's second law for a simple one-dimensional system. (i.e. )

   Solution

   Let , where and . We just need to find and and plug them into the Euler-Lagrange equations.

   First, is:

   Second, is:

   Thus, the Euler-Lagrange equations become:

   Since is just the force acting on the particle, we get:

   which is Newton's second law.

2. Derive the equations of motion for a simple pendulum using the Lagrangian formalism.

   Solution

   Let's consider a simple pendulum with mass and length . The Lagrangian for this system is defined as the difference between the kinetic and potential energies:

   Next, we just need to find the two terms in the Euler-Lagrange equations and plug them in.

   First, is:

   Second, is:

   Plugging these into the Euler-Lagrange equations, we get:

Summary and Next Steps

In this section, we briefly reviewed the Lagrangian formalism, which is the basis of most classical field theories.

Here are the key points to remember:

• The principle of stationary action states that the path taken by a particle between two points in space and time is such that the action is stationary.

• The Lagrangian is defined as the difference between the kinetic and potential energies of the system.

• The Euler-Lagrange equations are the equations of motion in the Lagrangian formalism:

  The Euler-Lagrange equations apply to any system as long as Hamilton's principle applies.

• The Euler-Lagrange equations are written in terms of the generalized coordinates , which can be any set of coordinates that describe the configuration of the system.

• The canonical momentum is defined as:

• The configuration space is the space spanned by the generalized coordinates.

The next section is skippable - we apply the Lagrangian formalism to the double pendulum system to demonstrate its power.