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Chapter 6 The Lagrangian Method Copyright 2007 by David Morin, morin@physics.harvard.edu (draft version) In this chapter, we’re going to learn about a whole new way of looking at things. Consider the system of a mass on the end of a spring. We can analyze this, of course, by using F = ma to write down m¨ x = −kx. The solutions to this equation are sinusoidal functions, as we well know. We can, however, figure things out by using another method which doesn’t explicitly use F = ma. In many (in fact, probably most) physical situations, this new method is far superior to using F = ma. You will soon discover this for yourself when you tackle the problems and exercises for this chapter. We will present our new method by first stating its rules (without any justification) and showing that they somehow end up magically giving the correct answer. We will then give the method proper justification. 6.1 The EulerLagrange equations Here is the procedure. Consider the following seemingly silly combination of the kinetic and potential energies (T and V , respectively), L ≡ T − V. (6.1) This is called the Lagrangian. Yes, there is a minus sign in the definition (a plus sign would simply give the total energy). In the problem of a mass on the end of a spring, T = mx˙ 2 /2 and V = kx2 /2, so we have 1 1 L = mx˙ 2 − kx2 . (6.2) 2 2 Now write d ∂L ∂L = . (6.3) dt ∂ x˙ ∂x Don’t worry, we’ll show you in Section 6.2 where this comes from. This equation is called the EulerLagrange (EL) equation. For the problem at hand, we have ∂L/∂ x˙ = mx˙ and ∂L/∂x = −kx (see Appendix B for the definition of a partial derivative), so eq. (6.3) gives m¨ x = −kx, (6.4) which is exactly the result obtained by using F = ma. An equation such as eq. (6.4), which is derived from the EulerLagrange equation, is called an equation of motion.1 If the 1 The term “equation of motion” is a little ambiguous. It is understood to refer to the secondorder differential equation satisfied by x, and not the actual equation for x as a function of t, namely x(t) = A cos(ωt + φ) in this problem, which is obtained by integrating the equation of motion twice. VI1 VI2 CHAPTER 6. THE LAGRANGIAN METHOD problem involves more than one coordinate, as most problems do, we just have to apply eq. (6.3) to each coordinate. We will obtain as many equations as there are coordinates. Each equation may very well involve many of the coordinates (see the example below, where both equations involve both x and θ). A...
Chapter 6
The Lagrangian Method
Copyright 2007 by David Morin, morin@physics.harvard.edu (draft version)
In this chapter, we’re going to learn about a whole new way of looking at things. Consider
the system of a mass on the end of a spring. We can analyze this, of course, by using F = ma
to write down m¨x = −kx. The solutions to this equation are sinusoidal functions, as we well
know. We can, however, ﬁgure things out by using another method which doesn’t explicitly
use F = ma. In many (in fact, probably most) physical situations, this new method is far
superior to using F = ma. You will soon discover this for yourself when you tackle the
problems and exercises for this chapter. We will present our new method by ﬁrst stating its
rules (without any justiﬁcation) and showing that they somehow end up magically giving
the correct answer. We will then give the method proper justiﬁcation.
6.1 The EulerLagrange equations
Here is the procedure. Consider the following seemingly silly combination of the kinetic and
potential energies (T and V , respectively),
L ≡ T − V. (6.1)
This is called the Lagrangian. Yes, there is a minus sign in the deﬁnition (a plus sign would
simply give the total energy). In the problem of a mass on the end of a spring, T = m ˙x
2
/2
and V = kx
2
/2, so we have
L =
1
2
m ˙x
2
−
1
2
kx
2
. (6.2)
Now write
d
dt
∂L
∂ ˙x
=
∂L
∂x
. (6.3)
Don’t worry, we’ll show you in Section 6.2 where this comes from. This equation is called
the EulerLagrange (EL) equation. For the problem at hand, we have ∂L/∂ ˙x = m ˙x and
∂L/∂x = −kx (see Appendix B for the deﬁnition of a partial derivative), so eq. (6.3) gives
m¨x = −kx, (6.4)
which is exactly the result obtained by using F = ma. An equation such as eq. (6.4),
which is derived from the EulerLagrange equation, is called an equation of motion.
1
If the
1
The term “equation of motion” is a little ambiguous. It is understood to refer to the secondorder
diﬀerential equation satisﬁed by x, and not the actual equation for x as a function of t, namely x(t) =
A cos(ωt + φ) in this problem, which is obtained by integrating the equation of motion twice.
VI1
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