Introduction to Lagrangian Mechanics

Introduction

The Euler-Lagrange Equation was developed by the mathematicians Leonhard Euler and Joseph-Louis Lagrange in the 1750s. This equation is a consequence of finding the stationary point of a functional $$S(q_j(x),q_j’(x),x)$$ and it is a differential equation which can be solved for the dependent variable $$q_j(x)$$. (For example, one stationary point of the quantity $$S(q_j(x),q_j’(x),x)$$ occurs when this quantity is minimized and another when this quantity is maximized.) It is practically useful because if, for example, the functional we are minimizing is a quantity known as the action, then it will allow us to derive the equations of motion of any system and does not depend on one’s choice of coordinates (and therefore reference frame). These coordinates may be chosen to be Cartesian coordinates but they may also be chosen to be polar or spherical coordinates, or moving coordinates, inertial or non-inertial coordinates. The Euler-Lagrange Equation, in this special case, may be expressed as a generalization of Newton’s second law which enables one to calculate the motion of any system using generalized coordinates (we’ll define this rigorously later on, but for now you can just think of them as, roughly, just many different kinds of possible coordinate systems) which do not necessarily have to be chosen to be Cartesian coordinates as I previously mentioned. For this reason, they are practically useful in an enormous range of problems as you’ll hopefully get a sense of as we solve problems. There are many problems in which the use of polar or spherical coordinates (or some other coordinate system) would be very convenient for describing the motion of an object. But to determine the motion of an object using Newton’s second law involves drawing free-body diagrams. Representing free-body diagrams using non-Cartesian coordinates would be very cumbersome and complicated. The Euler-Lagrange Equation (a generalization of Newton’s second law to generalized coordinates) allows one to calculate the motion of any system without using free-body diagrams or vector quantities and thereby circumvents this inconvenience. We have discussed the special case in which the function being minimized is the action; but more generally speaking, there are many other quantities—such as time, length, etc.—which can also be expressed as functionals and minimized. So more generally speaking, the solution $$q_j(x)$$ of this differential equation (the Euler-Lagrange equation) can be many different possible thing and not just motion—this is something to remember for later on when we solve problems.

To expand a little on the discussion above, the minimization of the quantity called the action is referred to as Hamilton’s Principle (also known as the so-called principle of least action). The action is defined as

$$A\equiv\int_{t_1}^{t_2}L(q_j, q_j’, t)dt\tag{1}$$

where a quantity called the Lagrangian is defined as

$$L(q_j(t), q_j’(t), t)\equiv{KE-PE(\vec{r})}.\tag{2}$$

We’ll see later that Hamilton’s principle can be proven correct by first “guessing it” ignorant of whether or not it actually is correct, and then discovering that the Euler-Lagrange equation actually simplifies to Newton’s second law: hence, when you solve the differential equation, you will indeed get the correct motion $$q_j(t)$$. For the moment, without concentrating on how these things were derived, let’s ponder the significance of Hamilton’s principle predicting the Euler-Lagrange equation (I said this in a more general way in the beginning of this writing) and Newton’s second law (as I’ll prove later on). In practice, Newton’s second law can be used to derive the laws of conservation of momentum and energy which have an enormous range of applications. But Newton’s second law predicts much more than that and it is one of the great generalizations which can, in principle, be used to accurately predict all physical phenomena in the classical universe. In a nutshell: Hamilton’s principle leads to Newton’s second law and, therefore, must also describe all physical phenomena in the classical universe. But it actually does more than that since we are using generalized coordinates—I’ll leave it there though for now.

In order for the action to be minimized, the Lagrangian has to be minimized. I told you (and will convince you later as we go through the proofs) that when the action is minimized, this essentially reproduces Newton’s laws; there’s really no escaping this fact that we must minimize the action because, if we didn’t, then we would produce laws of motion which are empirically incorrect. Thus, the Lagrangian or the combination $$KE-PE$$ must be minimized to produce laws of motion which agree with Newton’s laws and thus many other laws and predictions. Let’s interpret this peculiar fact that this combination must be minimized. It means that if a particle is moving from its initial position $$q_j(t_1)$$ to its final position $$q_j(t_2)$$, it’ll have to do so in a way in which it finds the right amount of kinetic and potential energy to minimize the Lagrangian. You could imagine one path where the particle shoots up very slowly to minimize $$KE$$ and to try to minimize the Lagrangian; but it does so too slowly to get a high amount of $$PE$$ and thus doesn’t minimize the Lagrangian that much. In another scenario, it might try to shoot up very fast to get a high amount of $$PE$$ to try to minimize the Lagrangian, but in doing so it acquired too much $$KE$$ and didn’t minimize the Lagrangian. The reason why, if I throw a rock, it’ll trace out a parabola is because that is the path which balanced the $$KE$$ and $$PE$$ in such a way that minimizes the Lagrangian and the action. It is very interesting how by just arguing with ourselves that this number $$L=KE-PE$$ has to minimized, we actually discover that this balancing act produces all the correct laws of motion. We made this discovery by using an argument which only involved the quantities $$KE$$ and $$PE$$—no vectors or free-body diagrams.