What Is Lagrange Equation

This section describes the Lagrange Equation states that the time derivative of the partial derivative Lagrangian against velocity equals to the partial derivative Lagrangian against position. A proof is provided to show that the Lagrange Equation is equivalent to the Hamilton Principle.

What Is Lagrange Equation? Lagrange Equation states that the time derivative of the partial derivative Lagrangian L against velocity x' equals to the partial derivative Lagrangian L against position x, if x is a true position function of a system.

d(∂L/∂x')/dt = ∂L/∂x                             (G.5)
  # L = L(x,x',t) is the Lagrangian function
  # t is time
  # x is the position of the object
  # x' is the velocity of the object

Lagrange Equation is actually equivalent to the Hamilton's Principle. Here is a simple proof.

According the Hamilton's Principle, if x(t) is a true position function of the system, it must be a stationary point on the Action S[x(t)].

S[x(t)] =  L(x,x',t)dt                     (G.2)
  # Between t1 and t2

Now if we introduce a small variation of x(t) as xs(t) with an error function e(t) that vanishes at t1 and t2. A small parameter s is used to control the scale of the variation.

xs(t) = x(t) + s*e(t)                       (G.6)
  # s is small parameter
  # e(t) is any function with e(t1) = e(t2) = 0

or:
  xs = x + s*e
    # (t) omitted

Its velocity function will be xs'(t):

xs'(t) = x'(t) + s*e'(t)                    (G.7)

or:
  xs' = x' + s*e'
    # (t) omitted

This will result a new Action value Ss:

Ss = S[xs(t)]

or:
  Ss =  L(xs,xs',t)dt                       (G.8)

or:
  Ss =  L(x+s*e,x'+s*e',t)dt                (G.9)
    # G.6 and G.7 applied

Now we can expand the Lagrangian function with its first order partial derivatives, because the parameter s is very small.

L(x+s*e,x'+s*e',t) = L(x,x',t) + s*e*∂L/∂x + s*e'*∂L/∂x'      (G.10)

Apply G.10 to G.9. We have:

Ss = ∫ (L(x,x',t) + s*e*∂L/∂x + s*e'*∂L/∂x')dt

or:
  Ss = ∫ L(x,x',t)dt + ∫ (s*e*∂L/∂x + s*e'*∂L/∂x')dt          (G.11)

According the Hamilton's Principle, x(t) is a stationary point on the Action S[x(t)]. So the small variation x(t) + s*e(t) should result the same Action S value:

S[xs(t)] - S[x(t)] = 0
  # Since x(t) is a stationary point and s is small

or:
  Ss - S = 0

or:
  ∫ L(x,x',t)dt + ∫ (s*e*∂L/∂x + s*e'*∂L/∂x')dt - ∫ L(x,x',t)dt = 0
    # G.2 and G.11 applied

or:
  ∫ (s*e*∂L/∂x + s*e'*∂L/∂x')dt = 0                      (G.12)

Further reduction of G.12 requires the rule of integration by parts:

s*e*∂L/∂x'| = ∫ (s*e'*∂L/∂x')dt + ∫ (s*e*(∂L/∂x')')dt           (G.13)
  # Between t1 and t2

or:
  0 = ∫ (s*e'*∂L/∂x')dt + ∫ (s*e*(∂L/∂x')')dt
    # Since s*e*∂L/∂x'| = 0
    # Remember that e(t1) = e(t2) = 0

or:
  ∫ (s*e'*∂L/∂x')dt = - ∫ (s*e*(∂L/∂x')')dt                   (G.14)
  # Between t1 and t2

Applying G.14 to G.12, we have:

∫ (s*x*∂L/∂x)dt - ∫ (s*x*(∂L/∂x')')dt = 0
  # Between t1 and t2

∫ s*x*(∂L/∂x - (∂L/∂x')')dt = 0                              (G.15)
  # Between t1 and t2

Now here is a big jump. According to many text books, for equation G.15 to be true for any s, the following must be true.

∂L/∂x - (∂L/∂x')' = 0

or:
  ∂L/∂x - d(∂L/∂x')/dt = 0
    # Since ()' = d()/dt

or:
  d(∂L/∂x')/dt = ∂L/∂x                                    (G.16)

Bingo! Equation G.16 is exactly the Lagrange Equation G.5. So we have approved that Hamilton's Principle is equivalent to Lagrange Equation.

By the way, the Lagrange equation is also called the Euler-Lagrange equation.

Table of Contents

 About This Book

 Introduction of Space

 Introduction of Frame of Reference

 Introduction of Time

 Introduction of Speed

 Newton's Laws of Motion

 Introduction of Special Relativity

 Time Dilation in Special Relativity

 Length Contraction in Special Relativity

 The Relativity of Simultaneity

 Introduction of Spacetime

 Minkowski Spacetime and Diagrams

 Introduction of Hamiltonian

Introduction of Lagrangian

 What Is Lagrangian

 Action - Integral of Lagrangian

 Action - Functional of Position Function x(t)

 Hamilton's Principle - Stationary Action

What Is Lagrange Equation

 Other Proofs of the Lagrange Equation

 Lagrange Equation on Free Fall Motion

 Lagrange Equation on Simple Harmonic Motion

 Lagrangian in Cartesian Coordinates

 Lagrange Equations in Cartesian Coordinates

 Introduction of Generalized Coordinates

 Phase Space and Phase Portrait

 References

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