Euler–Lagrange equations

The Euler-Lagrange equations are an important part of Lagrangian mechanics, which is the cornerstone of classical mechanics. Understanding these equations helps solve problems involving the motion of particles and systems. This technique can reduce the complexity of solving mechanical problems with constraints.

Introduction to Lagrangian mechanics

Lagrangian mechanics is a reformulation of classical mechanics introduced by Joseph-Louis Lagrange in 1788. It introduces new ways of looking at dynamics by using scalar fields to describe systems relative to their configuration. Instead of using forces in Newtonian mechanics, Lagrangian mechanics uses the concept of energy to find the paths of particles.

The fundamental concept in Lagrangian mechanics is the "Lagrangian", a function represented by L(q, dot{q}, t), where:

• q - generalized coordinates, which represent the configuration of the system.
• dot{q} - generalized velocities, which are the time derivatives of the generalized coordinates.
• t – time.

Usually the Lagrangian is expressed as:

L = T - V

where T is the kinetic energy and V is the potential energy of the system. The advantage of using Lagrangian mechanics is that it makes it easier to deal with complex constraints and is highly useful in fields ranging from quantum mechanics to general relativity.

Euler–Lagrange equations

The Euler–Lagrange equation is used to find the path of the system that makes the action stationary (often a minimum), which is defined as the integral of the Lagrangian over time. Mathematically, the action S is given as:

S = int_{t_1}^{t_2} L(q, dot{q}, t) , dt

The Euler–Lagrange equation can be obtained by ensuring that the action S is an extremum. The condition giving the extremum is:

frac{d}{dt} left(frac{partial L}{partial dot{q}}right) - frac{partial L}{partial q} = 0

This differential equation must be satisfied for each coordinate q of the system. It indicates how the generalized coordinates must evolve over time in order to follow the path of a stationary action.

Derivation of the Euler–Lagrange equation

To obtain this integral condition, consider a path set with a slight variation. Let the actual path of the system be q(t), and consider a close path q(t) + epsilon eta(t) where epsilon is a small parameter and eta(t) is a function that vanishes at the boundaries of t_1 and t_2. The differential action is:

S'(epsilon) = int_{t_1}^{t_2} L(q + epsilon eta, dot{q} + epsilon dot{eta}, t) , dt

Expanding this to first order in epsilon gives:

S'(epsilon) = S(0) + epsilon int_{t_1}^{t_2} left(frac{partial L}{partial q} eta + frac{partial L}{partial dot{q}} dot{eta} right) dt + O(epsilon^2)

For the action to be at an extremum, its first derivative with respect to epsilon must be zero at epsilon = 0. Thus:

0 = left[ int_{t_1}^{t_2} left(frac{partial L}{partial q} eta + frac{partial L}{partial dot{q}} dot{eta} right) dt right] Bigg|_{epsilon=0}

Integrating the second term by parts and applying marginal conditions yields the Euler–Lagrange equations shown directly above.

Example

1. Free particle motion

Consider a free particle moving with constant kinetic energy in one dimension. The Lagrangian is:

L = frac{1}{2}mdot{x}^2

Where m is the mass and dot{x} is the velocity. Applying the Euler-Lagrange equation:

frac{d}{dt} left(m dot{x}right) = 0

This simplifies the equation of motion to:

m ddot{x} = 0

which tells us that the particle moves at a constant velocity, as expected.

2. Simple harmonic oscillator

The Lagrangian for a simple harmonic oscillator with mass m and spring constant k is:

L = frac{1}{2}mdot{x}^2 - frac{1}{2}kx^2

The Euler–Lagrange equation states that:

frac{d}{dt}(mdot{x}) + kx = 0

This yields the familiar differential equation of motion:

mddot{x} + kx = 0

which describes simple harmonic motion.

Visual representation

Consider a pendulum of length l and mass m in a uniform gravitational field. Its motion can be represented as:

    
        
            
                
                    lsin(theta)
                    -lcos(theta)
                
            
        
    

    

The Lagrangian is given by the difference of kinetic and potential energies expressed in terms of the generalized coordinate theta:

L = frac{1}{2}m(ldot{theta})^2 - mglcos(theta)

Applying the Euler-Lagrange equation, we obtain the equation of motion:

ml^2ddot{theta} + mglsin(theta) = 0

Normalized coordinates

The Euler-Lagrange equations use the concept of generalized coordinates. Unlike Cartesian coordinates, generalized coordinates can be angles, lengths, or any convenient measure that uniquely describes the configuration of a system. This is particularly useful for systems with constraints.

For a double pendulum, the two angles theta_1 and theta_2 are sufficient to completely describe the system. The challenge becomes clear when understanding coupled motions described through generalized coordinates.

Application

The application of the Euler–Lagrange equations is widespread in physics and engineering:

• Electromagnetic fields: In electromagnetism, the Lagrangian formalism deals elegantly with potential fields to derive Maxwell's equations.
• General relativity: Einstein's field equations were derived from the variational principle, using, among other techniques, the Euler–Lagrange method.
• Quantum mechanics: Path integrals in quantum mechanics serve as an extension of the action principle established in classical mechanics.
• Structural mechanics: In civil engineering, Lagrangian mechanics helps define the equations of motion for structures subjected to dynamic forces.

Conclusion

The Euler-Lagrange equations beautifully encapsulate the laws of motion for a wide range of physical systems within a simple but powerful framework. By focusing on energy rather than forces, the Lagrangian method not only simplifies complex problems but also provides unique insights into the underlying symmetries and dynamics of systems.

This knowledge continues to inform and advance not only physics but also fields as diverse as engineering, chemistry, and even economics, and demonstrates the far-reaching implications and utility of the Euler–Lagrange equations in classical mechanics.

Comments