The Hamilton-Jacobi Equation
This blog has posted more than a few times in the past about classical mechanics. Luckily, classical mechanics can be approached in several ways. This approach, which uses the Hamilton-Jacobi equation (HJE), is one of the most elegant and powerful methods.
Why is the HJE so powerful? Consider a dynamical system with a Hamiltonian H=H(q,p,t). Suppose we knew of a canonical transformation (CT) that generated a new Hamiltonian K=K(Q,P,t) which (for a local chart on phase space) vanishes identically. Then the canonical equations would give that the transformed coordinates (Q,P) are constant in this region. How easy it would be to solve a system where you know that most of the important quantities are constant!
The rub is in finding such a canonical transformation. Sometimes it can’t even be done analytically, but nevertheless this is the goal of the Hamilton-Jacobi method of solving mechanical systems. In the equation given above, S is the generating function of the CT. Coincidentally, it often comes out to just equal the classical action up to an additive constant! This is due to the connection between canonical transformations and mechanical gauge transformations; it turns out that the additive function used to define the latter is the generating function of the former. In general the HJE is a partial differential equation that might be solvable by additive separation of variables… but don’t get too hopeful! Oftentimes the value of the HJE comes not in finding the actual equations of motion but in revealing symmetry and conservation properties of the system.
Originally, in the Newtonian formulation of classical mechanics, equations of motion were determined by summing up vector forces (à la free body diagrams). Is there a different way to find the equations of motion?
In place of drawing a free body diagram, we can represent a system more rigorously by describing its configuration space. The configuration space (often denoted Q) of a system is a mathematical space (a differential manifold) where every point in the space represents a particular state or configuration of the system. A curve drawn through a configuration space, then, represents the evolution of a system through a sequence of configurations.
Consider a rod along which a pendulum can slide. We need two numbers to describe the state of this system: the angle of the swinging pendulum and the position of the pendulum’s base along the rod. These two numbers are generalized coordinates for our system. Just like a traditional, linear vector space has a coordinate basis (like x, y, and z), our configuration space can use our generalized coordinates as a basis; let’s choose to name the position on the rod x and the angle of the pendulum φ. Since x can take any real value and φ can take any value from 0 to 2π (or 0 to 360o, if you like), the x dimension can be represented by a line (R1) and the φ dimension by a circle (or a one-sphere, S1). When we combine these dimensions, our new space — the configuration space of this system — is shaped like an infinite cylinder, R1 x S1. Just imagine connecting a circle to every point on a line… or, conversely, a line to every point on a circle.
The general process of examining a system and the constraints on its movement is a standard first step for solving mechanics problems analytically. After accounting for the constraints on a system, the ways a system can vary are called the degrees of freedom. Their count is often represented by the variable s. Notice: s = dim(Q).
Now that we’ve represented the configuration of our system, we need to talk about the forces present. There are several different ways that we can set up scalar fields on our configuration manifold that represent quantities related to the energy of the system. The simplest to deal with is often the Lagrangian, L = T - V = (Total kinetic energy) - (Total potential energy). Some fancy mathematics (a.k.a. calculus of variations) shows that when we define the Lagrangian in terms of our coordinates and their time derivatives, we can easily derive the equations of motion using the Euler-Lagrange equation.
For more complicated systems, configurations spaces may look different. A double pendulum (a pendulum on a pendulum) would have the topology S1 x S1 = T2, the torus (as pictured). Many systems will have higher dimensions that prevent them from being easily visualized.
Exercise left to the reader: the Lagrangian explicitly takes the time derivatives of the coordinates as arguments; information about the velocities of the system is needed to derive the equations of motion. But this information isn’t included in Q, so Lagrangian dynamics actually happens on TQ, the tangent bundle to Q. This new manifold includes information about how the system changes from every given configuration; since it needs to include a velocity coordinate for each configuration coordinate, dim(TQ) = 2s. TQ is also called Γv, the velocity phase space. T*Q, the cotangent bundle to Q, is the dual of TQ, and is traditionally just called the phase space, Γ; this is where Hamiltonian mechanics takes place.
FBDs: An Intro to Newton’s Laws
To describe some truths of how and why objects move, Newton formulated three laws of motion. Practitioners of the science which resulted from these laws (sometimes called mechanics or kinematics) began and spread the use of free body diagrams (FBDs) to aid in developing a sense of how motion works.
The first law defines inertia, the property of any object with mass to retain a constant velocity unless prompted by an external force. Mass is actually the measure of an object’s inertia, which has an SI unit kilogram (kg). When we draw a free body diagram, we draw only the object (the body) and vectors to indicate forces. Thus the start of an FBD looks like
and, if no force vectors are added, means that the object is moving at a constant (perhaps zero) velocity. This might describe a meteorite in space far away from any planets or stars - since it doesn’t “feel” any forces, it will continue traveling at the same speed in the same direction.
Of course, the very existence of other objects with mass in the universe means at least that any object we choose to study will feel a gravitational attraction to the other objects, and often other forces as well - the normal force from the ground, drag from fluid viscosity, friction between surfaces, pushing from a hand, tension through a rope. The second law then provides for how our body of interest reacts to the application of one or more external forces. An FBD simplifies the bookkeeping of these forces; again, it isolates the body from its surroundings and visualizes only the forces as vectors acting on it:
Newton’s second law states that this object will experience a net acceleration (change in velocity) proportional to the sum of the forces,
F1 + F2 + ··· + Fn = ΣiFi = m anet
and one notices that the proportionality constant is the object’s mass, or inertia. A more massive body accelerates less when acted on by the same set of forces than a less massive body; in other words, it better resists a change to its initial velocity.
The third law, sometimes called the law of reciprocal forces, states that if one object exerts a force on another object, the second object will exert a force of the same magnitude and the opposite direction on the first.
This can be helpful in checking that your FBD has all right forces, and only the right forces. But exercise caution. The third law deals with two forces on two different objects, whereas an FBD is made to deal with all the forces on one object. For instance, it is impossible to see a third law force pair in a free body diagram of a single body.
The ultimate goal of mechanics, which got its start in Newton’s laws and free body diagrams, is to solve the equations of motion for a given system. Eventually the more advanced tools of Lagrangian and Hamiltonian mechanics would develop, to be used respectively to analyze classical and quantum mechanics.