The Lagrangian - A Qualitative Analysis

The Lagrangian is defined as,

L = T - V

where T is the kinetic energy and V is the potential energy of a system.

The Lagrangian is useful because you can find the equations of motion for a system using the relation:

where the notation x_i is for generalized coordinates. In this notation, instead of coordinates of x, y, z, you would have coordinates of x₁, x₂, x₃, just different names for the same measured quantity. Since I'm only going to talk about the one dimensional case, lets just pretend the x_i in the equation above is x. Also notice the
in the equation above. It's just another way of writing dx/dt which is velocity. For our purposes, lets just call it v. So the equation above simplified a bit becomes:

[^d/_dt(^dL/_dv)] - (^dL/_dx) = 0

Example:

Consider an object at height x above the Earth initially at rest. The Potential Energy (V) of that object would be V=mgx. The Kinetic Energy (T) of a mass once it starts to fall will be ¹/₂mv². So,

L = T - V = ¹/₂mv² - mgx

Using the relation;

[^d/_dt(^dL/_dv)] - (^dL/_dx) = 0

we get the equation of motion;

m(^dv/_{dt^{) +}} mg = 0
(^dv/_{dt^{) =}} -g
∫ dv = ∫ -g dt
v = -gt

The velocity increases linearly over time.

The solution to the differential equation involving the Lagrangian is the equation of motion for the system. Why? Lets take another look at the Lagrangian.

The Lagrangian

The Lagrangian L is the difference between the Kinetic and Potential energy. The relation:

[^d/_dt(^dL/_dv)] = (^dL/_dx)

Notice one side of the equation above is the Lagrangian with respect to position (dL/dx) and on the other side of the equation is the Lagrangian with respect to velocity (dL/dv).

Lets take a step back to Physics 101. Physicists generally divide our energy up into two categories, first Kinetic Energy which is energy of motion, and second Potential Energy which is energy of position. The Lagrangian is the difference between these two types of energy, a way of expressing the two types of energy in one term. The equation:

[^d/_dt(^dL/_dv)] = (^dL/_dx)

is basically saying, you've got potential and kinetic energy. Either potential energy is going to increase and kinetic energy is going to decrease or potential energy is going to decrease and the kinetic energy is going to increase. Either way (or both in an oscillating system) this transfer of energy is will occur in a very particular way (only one way). Why only one way? Because of the principle of least action.

The principle of least action is an old idea. The principle basically says that nature is efficient. If the potential energy is going to be turned into kinetic energy, its going to occur in the most efficient way possible which means there is only one path an object can take through space to get this done (classical physics). That path represents the most efficient conversion of T to V or V to T.

Example of the Principle of Least Action

Lets look at that stationary object position x above the ground from the earlier example above. The Potential Energy is mgx and the Kinetic Energy (when it starts to move) is ¹/₂mv². The principle of least action says that the object will fall in a path like this:

not like this:

Why not this second path? The starting and end points are the same, so the same amount of potential energy has been turned into kinetic energy for each path. Energy is conserved, so whats the problem? The problem is that the curvy path is inefficient. The principle of least action says that only the straight path is possible. Its a good thing too, because you would never know what to expect when you dropped something if there wasn't a principle of least action. An object could take a minute, a year, or forever to fall, depending on how "curvy" its path was to the ground.

So Lagrangian Mechanics exploit this principle of least action. It takes account of all the potential energy and kinetic energy in a given system, and through calculus of variations finds the motion that produces the most efficient transfer of kinetic energy to potential energy or vice versa.

A Closer Look

So lets look at the equation again:

[^d/_dt(^dL/_dv)] = (^dL/_dx)

The right side is the derivative of the Lagrangian with respect to position (Potential Energy Part). The left side is the derivative of the Lagrangian with respect to velocity (Kinetic Energy Part). Essentially we are measuring the change in potential energy and also measuring the change in kinetic energy, but we are not measuring those changes in the same way (one with a change in velocity, the other with a change in position). To set the rates equal, you have to covert one of the rates (the kinetic energy rate of change) by multiplying it with a d/dt. This puts the rates on equal footing in the equation and allows them to be related.

If you look at the units (SI of course), you get:

[^d/_dt(^dL/_dv)] = (^dL/_dx)
1/s (J/m/s) = J/m
J/m = J/m

Remember, L = T-V, so L has units of Joules.

Conclusion

So an equation of motion can be thought of as the most efficient path by which potential energy can be converted to kinetic energy or vice versa for a given system.

Thus by using the Lagrangian, an expression of the Kinetic and Potential energy of a system, we can obtain the equations of motion through the relation:

[^d/_dt(^dL/_dv)] = (^dL/_dx)


Next time I will discuss the Hamiltonian, which is:

H = T + V

first as its used in classical physics and then its application to quantum mechanics.

Special thanks to the following websites:

http://en.wikipedia.org/wiki/Lagrangian
http://scienceworld.wolfram.com/physics/Lagrangian.html