In Taylor, read sections 7.1 for today, and 7.2 to 7.3 for Monday.
From the calculus of variations, minimizing ∫ f(qi, dtqi, t) dt implies
for every qi. This is the generalized form of the Euler-Lagrange equations for an arbitrary number of variables.
Now we are ready to begin Lagrangian mechanics. Lagrangian mechanics can be derived in a number of ways. It can be shown to be equivalent to Newtonian mechanics, and it can be shown to be equivalent to Hamiltonian mechanics. Our interest is to learn what it is and how to use it, with only passing interest in how it can be derived and demonstrating its equivalence to alternative formulations of mechanics. (The first ten pages of Landau & Lifshitz, Mechanics contains an interesting discussion that justifies the form of the Lagrangian.) We'll begin with a definition of the "action", and show how it is used to determine the dynamics of a particle. This is the basis of Lagrangian mechanics. We'll show show its equivalence to Newtonian mechanics for some simple examples, and begin applying it to problems.
The Principle of Least Action, also known as Hamilton's principle, states that a system will follow a path such that the time
integral of the difference between the kinetic and potential energy is an extremum.
The difference between the kinetic and potential energy is called the Lagrangian L = T - U.
Since the kinetic energy is a function of the velocities, and the (conservative) potential energy is a function of the coordinates, the Lagrangian is a function of the coordinates, velocities, and possibly time, L = L(qi, dtqi, t).
The use of the symbols qi for coordinates and dtqi for velocities is a generalization that will be elaborated on later.
For now, take these to be the Cartesian coordinates of a particle.
The integral of the Lagrangian over the path is called the
hence the name Principle of Least Action. This is a variational problem of the sort examined in chapter 6, and results in a set of differential equations, the Euler-Langrange equations, but in the context of Lagrangian mechanics they are known as Lagrange's equations:
Let's apply Lagrange's equations to a simple problem.
One end of a spring is fixed to a fixed, vertical wall while the other end is attached to a mass that is free to slide horizontally. Use Lagrange's equations to solve for the motion of the system.
The kinetic energy is T = ½ m (dtx)² and the potential energy is ½ k x², so L = T - U = ½ m (dtx)² - ½ k x². Applying Lagrange's equations yields the equation of motion
This is identical to the equation of motion obtained using Newton's Laws, m dt²x = -kx, and has the well known solution x = A cos(ω t + δ), where ω = √(k/m).
The question now is why is L = T - U, kinetic minus potential energy expressed in coordinates relative to an inertial frame the correct form? I won't derive that result, but let's motivate it from some simple arguments.
First, notice that the variation of S is unchanged by adding a total time derivative of a function of position and time to the Lagrangian:
Upon integration, the time derivative gives the difference of the function at the endpoints of the path:
Since this derivative term changes the action by a constant, it doesn't change where the action is stationary (minimum, maximum or inflection point). This result will be used in the argument below.
We know that in an inertial frame, a free particle will move with constant velocity, both direction and magnitude. And the uniformity of space means there is no preferred direction of the velocity, so the Lagrangian can depend on the magnitude, or square of the velocity, L = L(v²). Now suppose that we observe the particle from another inertial frame moving with a small relative velocity to the first, δu, so that the particle is observed to move with constant velocity v' = v + δu according to the principle of Galilean relativity. This second frame is still an inertial frame, so the Lagrangian must have the same form in this frame as the first, that is L' = L(v'²) = L(v² + 2v ⋅ δu + δu²). Expand this expression to first order in δu:
This change of frames shouldn't change the path that minimizes the action. Therefore, L(v'²) must differ from L(v²) by, at most, a total time derivative of a function of the coordinates and time, that is, the remaining term must be a total time derivative of some function of the position and time. Such a time derivative is, at most, linear in the velocity [df(q,t)/dt = ∂f/∂t + ∂f/∂q (dq/dt)]. Therefore, since (∂L/∂v²) 2v &sdot δu contains one term of velocity explicitly, we can infer that ∂L/∂v² is independent of the velocity. Since L is a function of v², it must contain at most a term proportional to v² which we can write as ½mv².
By straightforward extension, we conclude that for n free and non-interacting particles, the lagrangian is L = ∑ ½mava². This is, of course, the kinetic energy of the particles, which we denote by T.
If the particles are interacting, then the Lagrangian is modified by a function U that can depend on the coordinates and time, but not the velocities
Of course, this function is the potential energy of the system. To see this, put the Lagrangian into Lagrange's equations:
and likewise for y and z components of each particle, and take the appropriate derivatives to find the equations of motion:
and likewise for y and z. This is Newton's second law, where -∂U/∂xa is the x component of the force on particle a, Fxa.
This simple argument justifies the form of the Lagrangian, L = T - U, at least for cartesian coordinates. The full derivation is more complex, because the Lagrangian can be expressed in a wide variety of coordinates. Notice also that this example requires conservative forces, that is, forces derivable from a potential energy. As you may guess from the form of the Lagrangian, that is generally true. The Lagrangian formalism can be adapted to include non-conservative forces, but if these are important, perhaps Newtonian mechanics is a better approach to solve the problem.
Write down the Lagrangian for a particle moving in two dimensions under the influence of a central force, in polar coordinates. Find the equations of motion and relate them to known results.
In polar coordinates, the position of the particle is r = rrhat, and its velocity is rdot = rdotrhat + rφdotφhat. The Lagrangian is
This Lagrangian has two independent coordinates, r and φ, so there will be two equations.
Lagrange's equation for the r coordinate is
or
We can rewrite this, calling -∂U/∂r = Fr to get
where the last step relies on noting that the radial component of the acceleration is just the term in parentheses.
Lagrange's equation for the φ coordinate is
or
The left side of this equation is the torque, Γ, in two dimensions (Fφ = -(1/r)∂U/∂φ, so Γ = rFφ = -∂U/∂φ). The right side of the equation is the time derivative of the angular momentum, L. Therefore, this is equivalent to the familiar expression Γ = dL/dt.
The derivatives of the Lagrangian are known as a generalized force (∂L/∂qi = Fi) and a generalized momentum (∂L/∂qidot = pi). When thought of this way, Lagrange's equations can be expressed in the form
an expression that looks deceivingly like Newton's second law. But notice that the generalized momentum and force can be completely different from a traditional linear momentum and force in Newtonian mechanics. They are just the appropriate derivatives of the Lagrangian. For example the Lagrange equation in φ yielded a torque for the generalized force, and an angular momentum for the generalized momentum.
These ideas are easily extended to a system with many paricles. If there are N particles moving freely in 3-dimensions then there are 3N coordinates, qi, and 3N Lagrange equations.