LEAST ACTION PRINCIPLE. We continue with the notation from Pollard, section 2.1. Newton s equations for N-body motion are found in Pollard section 2.1, eq. 1.1 there. In section 2.2 they are succinctly reformulated as: m k r k = U (2.2) where U = U(r 1,..., r n ) is a function of all the positions (see eq. (2.1) p. 60) where U = GΣ m im j r ij r ij = r i r j is the distance between mass i and mass j, and where the sum defining U is over all distinct unordered pairs ij = ji, i j, i, j = 1,..., n of masses. See the first paragraph of 2.2 for the notation - it is the gradient with respect to the vector r k. The constant G in front of the sum defining U is called the gravitational constant. The gravitational potential, usually denoted V is the negative of U. Note: U 0. To bring this into the form needed for calculus of variations set and (0.1) (0.2) (0.3) (0.4) T = 1 2 Σm k v k 2 = kinetic energy L = L(r 1,..., r n, v 1,..., v n ) = Lagrangian = kinetic energy potential energy = T + U. A computation. The equations (2.2) are equivalent to the system: d L = L (3) dt v k supplemented with the understanding that the velocities are the time derivatives of the positions: ṙ k = v k. Definition 0.1. Let x(t) = (r 1 (t),...,..., r n (t)) be a curve for all n-bodies: that is a specification of how each body moves over time, with the time varying over an interval [a, b]. Then the number expression A(x( )) = b is called the action of the path x(t). a L(r 1 (t),..., r n (t), ṙ 1 (t),..., ṙ n (t))dt 1
2 LEAST ACTION PRINCIPLE. Meaning of action. Let x(t) = (r 1 (t),..., r n (t)), a t b be a curve of possible motions for the n bodies. We do not assume this curve is a solution. Then the curve defines values for the vectors r k (t) and ṙ k (t) so that the integrand of the action is a function of t times dt. We integrate it from a to b to get a real number, the action. Thus the action is a function whose domain is the space of all paths in R 3 ) n and whose range is the real numbers (plus the value if the integral is infinite). Principle of Least Action. The principle of least action asserts that if a possible path x(t) minimizes the action then it solves the Euler-Lagrange equations, and hence Newton s equations. Like any principle in mathematics or physics it is not true in general, but rather will only be true under further hypothesis, or assumptions in place. Notation. In symbols the principle asserts that da(x( )) = 0 implies that x(t) satisfies (2.1). Here da is the total differential of the action, or: its derivative with respect to (all) paths. In physics they write δa(x(t)) δx(t) in place of da(x( ). Here is perhaps the simplest true version of the principle. Theorem 0.2 (Principle of least action for fixed endpoints). Suppose that the curve x(t) = (r 1 (t),..., r n (t)) minimizes the action among all competing curves w(t) which satisfy w(a) = x(a), w(b) = x(b) and which are absolutely continuous [See your real analysis class for a definition]. Suppose in addition that x is continuously differentiable and has no collisions in the interval [a, b]. (A collision in the interval is a time t and a pair of distict indices i, j such that r i (t) = r j (t).) Then the curve x(t) satisfies Newton s equations of motion for the n-bodies. History. The principle goes back to Maupertuis. Euler and Lagrange used it, hence the phrase Euler-Lagrange equations. Its formulation is often attributed to Hamilton. Probably Feynman s main contributions is realizing that quantum mechanics can be formulated in terms of integrating exp(iaction (path) h) over all paths connecting two points. Notation. Formalism. Note we are writing simply x for a point in the (large!) space R 3n. So: the single point x encodes the location of all n masses in space. The curve x(t) is a map [a, b] R 3n. Write Ω for the space of all absolutely continuous paths in R 3n, defined on the interval [a, b] and taking a to x(a), b to x(b). Then the action is a function Ω with values in the extended reals [0, ]. (There are differentiable paths with infinite action.) The action is always nonnegative for the n-body problem because both T and U are non-negative. Then, in this notation, the assertion of the theorem is that if x(t) Ω is collisionfree, and continuously differentiable, and if A(x( )) A(w( )) for all w(t) Ω then x(t) is a solution to Newton s equations. The idea behind the proof is precisely the same idea that you learned in first quarter calculus: to minimize a function, you set its derivative equal to zero. So, we have to figure out how to differentiate functions A of paths. The theorem is proved by showing that the vanishing of this derivative for all admissible variations w of x is the Euler-Lagrange equation. Variations. Write x ɛ (t) = x(t) + ɛ(δx(t)) (V AR) where ɛ is a small number, allowed to vary, and δx(t) : [a, b] Q is another path.
LEAST ACTION PRINCIPLE. 3 Definition 0.3. An expression such as (VAR) is called a variation of the path x(t). Admissible variations. In order that x ɛ ( ) lies in Ω, i.e. satisfies x ɛ (a) = x(a), x(b) = x ɛ (b) we must have that δx(a) = δx(b) = 0, the 0 vector in Q = R 3n. Hence we call such variations δx admissible variations. We will also write δr k for the vector components of δx. We are now in a position to differentiate the action: Proposition 0.4. d dɛ ɛ=0a(x ɛ ) = b a Σ( d dt ( L v k (x(t), ẋ(t)) + L (x(t), ẋ(t)) δr k (t)dt provided both x and δx are continuously differentiable and satisfy the boundary conditions above. One proves the proposition by a computation involving differentiating under the integral sign and integration by parts. We will do this computation in class. The proof of the theorem now follows immediately from the previous propositon and the Lemma 0.5 (Fundamental Lemma of the Calculus of Variations). Suppose that e : [a, b] R D is a continuous function such that for all continuous differentiable functions η : [a, b] R D which vanish at the endpoints a, b we have that b a Σ αe α (t) η α (t)dt = 0 where the e α, η α denote the components of the e(t), η(t). Then e(t) = 0. Note that in applying the lemma to get the theorem we use the case of dimension D = 3n. Extremals = Solutions. An extremal of a function is a point in its domain where the (total) derivative is zero. The points in the domain for the action function are curves so that an action extremal is is a curve along which the derivative of the action (with boundary conditions are imposed before taking the derivative) is zero. The lemma and proposition taken together imply that soutions to the Euler- Lagrange equations (so, to Newton s equations for our Lagrangian) are precisely the extremals of the action. That is to say, solutions need not correspond to actionminimizers. Any extremals of the action yield solutions. Other boundary conditions. The fixed endpoint boundary condition of the theorem can be replaced by other endpoint conditions. If we are want periodic orbits of period P we impose the conditions x(t + P ) = x(t). We can design solutions with various appealing shapes by imposing symmetries. The figure eight orbit can be achieved by looking for solutions satisfying the following two symmetries: (r 1 (t), r 2 (t), r 3 (t)) = (R(r 3 (t P/6)), R(r 1 (t P/6)), R(r 2 (t P/6)) (r 1 (t), r 2 (t), r 3 (t)) = ( r 1 ( t), r 3 ( t), r 2 ( t)) Here the r i are vectors in the xy-plane and in the first condition R(x, y) = (x, y) is reflection about the y-axis. Designer orbits obtained by imposing such symmetry constraints and then using the direct method of the calculus of variations became a minor industry in mathematics after my paper with Chenciner in 2000. Direct Method.
4 LEAST ACTION PRINCIPLE. A minimizing sequence for A is an infinite sequence x n, n = 1, 2, 3,... of paths satisfying the imposed boundary conditions and such that lim n A(x n ()) is trying to go to the minimum of A. Recall: the definition of infimum of a set of real numbers. Thus a minimizing sequence is a sequence of curves such that lim n A(x n ( )) = inf x Ω plus boundary conditions A(x( )) The example of a function having a nice infimum but which is not realized. The main technical problem: showing that your sequence of curves converges to a nice curve. The direct method in the calculus of variations consists of three steps. Step 1. Choose a minimizing sequence. Step 2. Show this sequence converges in some sense to SOME path x. Show that the limit path x is sufficiently regular that the analogue of the theorem holds: it is continuous, continuously differentiable and satisfies the equations of motion. The method is also called Hilbert s direct method in honor of the man who pioneered its use as a rigorous method of existence proof. Fourier series. A practical way to implement the direct method numerically, when one is interested in periodic paths is by way of Fourier series. They are most simply explained in the plane. A vector in the plane is a complex number. Write z = x + iy and identify the number z with the vector (x, y). A periodic curve in the plane with period 2π is a map R C satsifying z(t + 2π) = z(t). The imaginary exponentials e ikt, k an integer are periodic curves. Fourier series asserts that any continous curve can be represented as an infinite Fourier sum: z(t) = Σ + a k e ikt (F 1) The complex numbers a k can be computed by the formula a k = 1 2π 2π 0 z(t)e ikt dt (F 2). The Fourier coefficients a k should be thought of as alternative coordinates on the space of all periodic curves. The Fourier inversion formulas (F1), (F2) describes the change of coordinates from the usual representation z(t) [a vector in C indexed by the continuous set of t s] to the Fourier representation as an infinite list a k of complex numbers [a vector in C indexed by the integers]. So, we can view the action as a function A = A(..., a 1, a 0, a 1,..., a n,..., ) of the Fourier coefficients of a periodic curve. We can then apply various minimization schemes to vary the a k so as to tend towards the infimum of the action. Here is a very rough sketch behind some numerical algorithms such as Nauenberg s for finding designer periodic solutions. Step 1. Impose some symmetry such as that of the eight and in this way impose constraints on the Fourier coefficients. Step 2. Draw or guess, SOME curve satisfying 1 and that symmetry. compute its Fourier coefficients up to some finite order N Step 3. Compute the action to the order N as a function of the Fourier coefficients. Apply a minimization algorithm so as to yield a new curve of lesser action. Repeat until the action is barely changing.
LEAST ACTION PRINCIPLE. 5 Step 4. Call it a victory. (Test accuracy of solution by increasing N or by ODE solvers if desired.)