9.1 Euler-Lagrange Equations and Functionals

Calculus of variations is a powerful tool in physics, finding the optimal path a system takes between two points. It introduces the Euler-Lagrange equation, which helps derive equations of motion for various physical systems.

The Euler-Lagrange equation is key to solving variational problems. It's used in classical mechanics, field theory, and beyond, providing a way to find extremal functions that minimize or maximize functionals in physics.

Calculus of Variations and the Euler-Lagrange Equation

Derivation of Euler-Lagrange equation

Top images from around the web for Derivation of Euler-Lagrange equation

• 

  EulerEquations | Wolfram Function Repository View original

  Is this image relevant?

• 

  On the Principle of Least Action View original

  Is this image relevant?

• 

  Euler-Lagrange equation View original

  Is this image relevant?

• 

  EulerEquations | Wolfram Function Repository View original

  Is this image relevant?

• 

  On the Principle of Least Action View original

  Is this image relevant?

1 of 3

Top images from around the web for Derivation of Euler-Lagrange equation

• 

  EulerEquations | Wolfram Function Repository View original

  Is this image relevant?

• 

  On the Principle of Least Action View original

  Is this image relevant?

• 

  Euler-Lagrange equation View original

  Is this image relevant?

• 

  EulerEquations | Wolfram Function Repository View original

  Is this image relevant?

• 

  On the Principle of Least Action View original

  Is this image relevant?

1 of 3

• Principle of least action states the path taken by a system between two points minimizes the action functional $S[q]$
  • Action functional defined as integral of Lagrangian $L(q, \dot{q}, t)$ over time: $S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt$
• To find path minimizing action, consider small variations of path $q(t) \to q(t) + \delta q(t)$
  • Variation of action given by $\delta S = \int_{t_1}^{t_2} \left(\frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q}\right) dt$
  • Integration by parts on second term yields $\int_{t_1}^{t_2} \frac{\partial L}{\partial \dot{q}} \delta \dot{q} dt = \left.\frac{\partial L}{\partial \dot{q}} \delta q\right|_{t_1}^{t_2} - \int_{t_1}^{t_2} \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) \delta q dt$
• For action to be stationary, $\delta S = 0$ for any variation $\delta q(t)$ vanishing at endpoints
  • Leads to Euler-Lagrange equation: $\frac{\partial L}{\partial q} - \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) = 0$
  • Necessary condition for path to extremize action functional

Applications in variational problems

• Euler-Lagrange equation derives equations of motion for various physical systems
• Particle moving in potential $V(q)$ with Lagrangian $L = \frac{1}{2}m\dot{q}^2 - V(q)$
  • Applying Euler-Lagrange equation yields Newton's second law: $-\frac{\partial V}{\partial q} - m\ddot{q} = 0$
• Stretched string with fixed endpoints and Lagrangian $L = \frac{1}{2}\int_0^L \left(\mu \dot{y}^2 - T \left(\frac{\partial y}{\partial x}\right)^2\right) dx$
  • Euler-Lagrange equation leads to wave equation: $\mu \frac{\partial^2 y}{\partial t^2} - T \frac{\partial^2 y}{\partial x^2} = 0$
• Provides powerful tool for deriving governing equations in classical mechanics, field theory, and beyond

Concept of functionals

• Functional maps functions to real numbers, unlike functions mapping numbers to numbers
  • Action functional $S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt$ maps paths $q(t)$ to real numbers
• Calculus of variations seeks function extremizing (minimizing or maximizing) given functional
• Euler-Lagrange equation provides necessary condition for function to extremize functional
  • Derived by setting variation of functional equal to zero: $\delta S[q] = 0$
• Functionals play central role in formulating variational principles in physics (principle of least action)

Extremal functions for functionals

• Extremal functions solve Euler-Lagrange equation for given functional
• Steps to find extremal functions:
  1. Write functional in form $J[y] = \int_a^b F(x, y, y') dx$
  2. Apply Euler-Lagrange equation: $\frac{\partial F}{\partial y} - \frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) = 0$
  3. Solve resulting differential equation for $y(x)$, subject to boundary conditions
• Example: Extremal function for $J[y] = \int_0^1 (y'^2 + y^2) dx$ with $y(0) = 0$ and $y(1) = 1$
  • Euler-Lagrange equation yields $y'' + y = 0$
  • Solution satisfying boundary conditions is $y(x) = \frac{\sin(\pi x)}{\sin(\pi)}$, minimizing the functional
• Extremal functions correspond to stationary points (minima, maxima, or saddle points) of functionals