Sketchy Notes on Lagrangian and Hamiltonian Mechanics Robert Jones Generalized Coordinates Suppose we have some physical system, like a free particle, a pendulum suspended from another pendulum, or a field that can take on a different value at every point in space. Doing physics means working out how the configuration of the system evolves in time: for a particle, we are interested in how its position changes in time; for our double pendulum, we are interested in how each of the two angles of the pendula change over time; and for a field, we are interested in how its field values at each point change in time. In our examples, the position of the free particle, the angles of the pendula, and the values of the fields provide all the information we need to know the configuration of the system. Note that knowing the configuration of the system at t = 0 is not enough to be able to predict its time evolution. Suppose we have a free particle and wish to know its motion. We cannot work this out if we only know where it is at t = 0, for we also need to know its initial velocity. The set of all configurations of a system is known as the configuration space, the dimension of the configuration space constitutes the number of degrees of freedom of the system, and coordinates on the configuration space are called generalized coordinates. As an example, consider a pendulum that is free to swing back and forth in a plane. While the bob is free to move in two dimensions of space, notice that since the length of the pendulum can t change, all of its possible configurations lie on a circle: a space of dimension. Thus the pendulum is said to have one degree of freedom. While a very natural choice of generalized coordinate for our system would be some sort of angle, like the angle the bob makes with the vertical, observe that we could also use a projective coordinate (as shown in the figure) or some other coordinate system on our configuration space. Each of these constitutes a choice of generalized coordinate. The key thing to realize is that: [dimension of configuration space] = [number of generalized coordinates] = [degrees of freedom]. If we are analyzing a system with Cartesian coordinates, we use F = ma. If we are analyzing a system in an angular coordinate system, we use τ = Iα. Lagrangian mechanics purpose is to fill in the analogous equations for an arbitrary choice of generalized coordinates. It is thus a very powerful mode of analysis for problems in classical and modern physics that resist nice familiar coordinate systems. Derivation of Lagrangian mechanics We begin with the idea of Fermat s principle, the idea that light moves in a way to minimize the time it takes to travel along its path. To formally state the principle, we let the path of the light particle be q(t) where q is a vector in configuration space (the set of all possible configurations, in this case positions, of the light particle) and we define the action of that path as: S light [q(t)] = dt. In order to calculate the path q(t) that is the minimum or maximum of the action, we must define a notion of taking the derivative of S with respect to q (so that we can eventually set it to zero). We recall that the basic idea of a derivative is to perturb the input of a function (let s say f) a small amount and to calculate the resulting perturbation in the output (f(x + dx) f(x) in our case). The derivative is then the ratio of that output perturbation to the input perturbation ( dx (f(x + dx) f(x)) in our case). However, the input
to S is itself a function, so when we perturb it we do so by adding an arbitrary function δq multiplied by a small real number ε that we will eventually shrink to zero: δs = lim (S[q(t) + εδq(t)] S[q(t)]). The value of δs will depend on our choice of perturbing function δq(t) in the same way that the derivative of a multivariate function depends on the direction you take the derivative. However in the case of a local max or min, the derivative in all directions (and analogously for all perturbing functions δq) will be zero. Physical matter clearly does not obey Fermat s principle, but many things (kinetic and potential energy chiefly among them) are variable for non-light objects that stay constant for light (remember that E = hf and f is the same no matter how the light is refracted, so the kinetic energy of light doesn t change under most circumstances). Therefore we suspect that the Fermat-like principle that matter obeys will be of the form S = L dt, where that L is a non-constant parameter we introduced that happens to be constant in the case of light. Our strategy will be to assume this and use it to derive an expression that we can compare with Newton s laws to get a physical interpretation of L. We assume L depends on generalized position, velocity, and time, and only work with one generalized coordinate for simplicity: S[q(t)] = L(q, q, t) dt. Next we take the variational derivative as we did before: δs = lim (S[q(t) + εδq(t)] S[q(t)]), ε = lim (L(q + εδq, q + εδ q, t) L(q, q, t)) dt. ε 0 In order to make this subtraction work, we ll series-expand L around (q, q, t). We anticipate that the ε and the lim ε 0 will get rid of the higher order terms in the series (so we won t bother even figuring out what they are). ( δs = lim L(q, q, t) + ) εδq + q q εδ q + O(ε2 ) L(q, q, t) dt, ( ) = lim δq + δ q + O(ε) dt, ε 0 q q ( ) = δq + q q δ q dt. This is pretty good, but because to compute δs we only need to specify a δq, we would like our answer to be only in terms of δq (no δ q s). Luckily this is easily done with an integration by parts: δs = δq dt + (δ q dt), q q [ ] t ( ) d = δq dt + q q δq But we choose our δq so that δq(t 0 ) = δq(t ) = 0 so that middle thing goes away and we are left with: ( δs = q d ) Note that this depends on our δq, which makes sense. If we picture some very high dimensional function space where each choice of q(t) is a point, then the δq(t) is the direction we want to take our derivative t 0 2
of S in, which uniquely determines our answer. However, if S[q(t)] is a local minimum, we expect that no matter what direction we take its derivative, we obtain zero: ( 0 = q d ) Here is where the fundamental lemma of the calculus of variations, which states that if x(t)y(t) dt = 0 for any choice of y(t), then x(t) = 0, is used to obtain: q = d. which is the famous Euler-Lagrange equation. To figure out what physical interpretation L should have, we compare it with Newton s Second Law ( U) x = d K dt ẋ. so we expect L = K U. Note that in our derivation we didn t rely on q having any physical meaning, so the Euler-Lagrange equation should hold for any generalized coordinate q. Note also that this derivation was done for a system of one degree of freedom, and that in a system with many degrees of freedom there would be one Euler-Lagrange equation for each generalized coordinate. Lastly, note that the d/dt introduced by the integration by parts is a total derivative. Example Next, we will work an example of the Lagrangian treatment of a system. A careful Newtonian analysis would also work for this system but it will illustrate the basic procedure. The system we are considering consists of a mass m 2 suspended by a spring of constant k 2 from a mass m, attached to a fixed point by a spring of constant k. We assume the system moves in one dimension and is influenced by gravity. Choose generalized coordinates. Note that the system has 2 degrees of freedom as we have two particles moving in one dimension each. One possible set of generalized coordinates would be the heights h and h 2 of each particle, with the advantage that it makes the kinetic energy terms non-complicated. Another might be the extensions x and x 2 of each spring which keeps potential terms simple. I choose x and x 2 for the rest of the analysis for teaching reasons even though h and h 2 are probably better. Calculate kinetic and potential energies. We begin with kinetic energy. The height of particle is h = x, and the height of particle 2 is h 2 = x x 2. Thus our velocities are ḣ = ẋ and ḣ2 = ẋ ẋ 2. We then write K = 2 m ḣ2 + 2 m 2ḣ2 2, = 2 m ẋ 2 + 2 m 2 (ẋ2 + 2ẋ ẋ 2 + ẋ 2 2), = 2 (m + m 2 )ẋ 2 + m 2 ẋ ẋ 2 + 2 m 2ẋ 2 2. In general for well behaved classical problems K will probably be quadratic in your generalized coordinates. However for generalized coordinates that are not simply the physical positions of the particles, I would recommend writing out the physical positions in terms of the generalized coordinates and differentiating (as we have done here) to make sure your K is calculated correctly. Potential energy is given as: U = (m + m 2 )gx m 2 gx 2 + 2 k x 2 + 2 k 2x 2 2. 3
Write the Lagrangian. Straightforwardly, L = 2 (m + m 2 )ẋ 2 + m 2 ẋ ẋ 2 + 2 m 2ẋ 2 2 + (m + m 2 )gx + m 2 gx 2 2 k x 2 2 k 2x 2 2. Write the Euler-Lagrange equations. Our equation in x is: and our equation in x 2 is: (m + m 2 )g k x = (m + m 2 )ẍ + m 2 ẍ 2, m 2 g k 2 x 2 = m 2 (ẍ + ẍ 2 ). To get the actual motion of the system we would solve these differential equations but this is the end of the Lagrangian method and the beginning of the analysis of the equations of motion phase of analyzing the system. Hamiltonian Mechanics Notice that the equations of motion we obtained were second order in x, which will be true in general of Euler-Lagrange equations. In many ways (computer implementation being one) it is advantageous to have a system of first order equations rather than a second order equation, and the reformulation of Lagrangian mechanics into first order equations is called Hamiltonian mechanics. To do so, we first introduce a quantity called the conjugate momentum to q, defined by: Then we have the first-order time-evolution equation Observe that if p = q. q = ṗ. q i = 0, then d dt p i = 0 so p i is conserved. In general, if the Lagrangian does not depend on a given q i, then it is called cyclic and its conjugate momentum p i is conserved. This is a special case of Noether s theorem. But we need an equation for q in order to have a complete system of time-evolution equations. To derive this, consider the total differential of the Lagrangian: dl = q dq + d q + q t dt, = dq + pd q + q t dt, = dq + d(p q) qdp + q t dt, d(p q L) = dq + qdp q t. 4
Here, we define a new quantity called the Hamiltonian H = p q L, whose physical interpretation we will arrive at later. For now, taking the total differential of the Hamiltonian, we have: dh = H q H H dq + dp + p t, which we can compare with the above expresion to obtain the first-order Hamiltonian equations of motion: H q = ṗ, H p = q. The specific derivation of the Hamiltonian formalism is not as important conceptually as the derivation of the Lagrangian formalism, and the important results are the definition of the Hamiltonian as H = i p i q i L and the above Hamilton s equations. Mathematically, the Hamiltonian is the Legendre transformation of the Lagrangian. For completeness (this isn t important to the physics), I state that the Lagrangian, being a function of q and q, is defined on the tangent bundle of the configuration space, whereas from its definition the set of all (q, p) pairs, where the Hamiltonian is defined, constitutes the cotangent bundle of the configuration space. The cotangent bundle of the configuration space is called the phase space. Notice that given a point in phase space, the future evolution of the system from that point on is completely determined by Hamilton s equations (since they are first-order). In fact, we can use Hamilton s equations to put a vector field on the phase space, called the Hamiltonian flow, such that the integral curves are solutions to Hamilton s equations. Notice that this vector field is divergence-free by commutativity of partial derivatives; this result is known as Liouville s theorem. An equivalent formulation of it is that given an initial chunk of volume in phase space, the volume of that chunk will not change as its points are time-evolved according to the Hamiltonian flow. 5