156 CHAPTER 6. HAMILTON’S EQUATIONS later subdivide these into coordinates and velocities. We will take the space in which x takes values to be some general n-dimensional space we call , M which might be ordinary Euclidean space but might be something else, like the surface of a sphere1. Given a differentiable function f of n independent variables xi, the differential is Chapter 6 n ∂f df = dxi. (6.1) ∂xi Xi=1 Hamilton’s Equations What does that mean? As an approximate statement, this can be regarded as saying n ∂f df ∆f f(xi +∆xi) f(xi)= ∆xi + (∆xi∆xj), ≈ ≡ − ∂xi O We discussed the generalized momenta Xi=1 ∂L(q, q,˙ t) with some statement about the ∆xi being small, followed by the dropping of pi = , the “order (∆x)2” terms. Notice that df is a function not only of the point ∂q˙i x , but also of the small displacements ∆x . A very useful mathematical ∈M i and how the canonical variables q ,p describe phase space. One can use language emerges if we formalize the definition of df , extending its definition { i j} phase space rather than q , q˙ to describe the state of a system at any to arbitrary ∆x , even when the ∆x are not small. Of course, for large { i j} i i moment. In this chapter we will explore the tools which stem from this ∆xi they can no longer be thought of as the difference of two positions in phase space approach to dynamics. and df no longer has the meaning of the difference of values of f at M two different points. Our formal df is now defined as a linear function of these ∆xi variables, which we therefore consider to be a vector ~v lying in 6.1 Legendre transforms an n-dimensional vector space Rn.Thusdf : Rn R is a real-valued M× → function with two arguments, one in and one in a vector space. The dx M i The important object for determining the motion of a system using the La- which appear in (6.1) can be thought of as operators acting on this vector grangian approach is not the Lagrangian itself but its variation, under arbi- space argument to extract the i0th component, and the action of df on the trary changes in the variables q andq ˙, treated as independent variables. It argument (x,~v)isdf (x,~v)= i(∂f/∂xi)vi. is the vanishing of the variation of the action under such variations which This differential is a special case of a 1-form, as is each of the operators determines the dynamical equations. In the phase space approach, we want P dxi. All n of these dxi form a basis of 1-forms, which are more generally to change variablesq ˙ p, where the p are components of the gradient of → i the Lagrangian with respect to the velocities. This is an example of a general ω = ωi(x)dxi, procedure called the Legendre transformation. We will discuss it in terms of Xi the mathematical concept of a differential form. where the ω (x) are n functions on the manifold . If there exists an ordi- Because it is the variation of L which is important, we need to focus i M our attention on the differential dL rather than on L itself. We first want nary function f(x) such that ω = df , then ω is said to be an exact 1-form. to give a formal definition of the differential, which we will do first for a 1Mathematically, is a manifold, but we will not carefully define that here. The M function f(x1, ..., xn)ofn variables, although for the Lagrangian we will precise definition is available in Ref. [16]. 155 6.1. LEGENDRE TRANSFORMS 157 158 CHAPTER 6. HAMILTON’S EQUATIONS Consider L(qi,vj,t), where vi =˙qi. At a given time we consider q and v where d¯Q is not an exact differential, and the heat Q is not a well defined as independant variables. The differential of L on the space of coordinates system variable. Though Q is not a well defined state function, the differential and velocities, at a fixed time, is d¯Q is a well defined 1-form on the manifold of possible states of the system. ∂L ∂L ∂L It is not, however, an exact 1-form, which is why Q is not a function on that dL = dqi + dvi = dqi + pidvi. manifold. We can express d¯Q by defining the entropy and temperature, in ∂qi ∂vi ∂qi Xi Xi Xi Xi terms of which d¯Q = TdS, and the entropy S and temperature T are well If we wish to describe physics in phase space (qi,pi), we are making a change defined state functions. Thus the state of the gas can be described by the of variables from vi to the gradient with respect to these variables, pi = two variables S and V , and changes involve an energy change ∂L/∂vi, where we focus now on the variables being transformed and ignore the fixed q variables. So dL = p dv , and the p are functions of the v dE = TdS pdV. i i i i i j − determined by the function L(vi). Is there a function g(pi) which reverses P We see that the temperature is T = ∂E/∂S . If we wish to find quantities the roles of v and p, for which dg = i vidpi? If we can invert the functions |V appropriate for describing the gas as a function of T rather than S, we define p(v), we can define g(pi)= i pivi(pj) L(vi(pj)), which has a differential P− the free energy F by F = TS E so dF = SdT pdV , and we treat dg = p dv + vPdp dL = p dv + v dp p dv − − − − i i i i − i i i i − i i F as a function F (T,V ). Alternatively, to use the pressure p rather than V , Xi Xi Xi Xi Xi we define the enthalpy X(p, S)=Vp+E, dX = Vdp+TdS. To make both = vidpi changes, and use (T,p) to describe the state of the gas, we use the Gibbs Xi free energy G(T,p)=X TS = E + Vp TS, dG = Vdp SdT. Each of − − − as requested, and which also determines the relationship between v and p, these involves a Legendre transformation starting with E(S, V ). ∂g Unlike Q, E is a well defined property of the gas when it is in a volume vi = = vi(pj), V if its entropy is S,soE = E(S, V ), and ∂pi giving the inverse relation to pk(v`). This particular form of changing vari- ∂E ∂E T = ,p= . ables is called a Legendre transformation. In the case of interest here, ∂S ∂V V S the function g is called H(qi,pj,t), the Hamiltonian, ∂2E ∂2E ∂T ∂p H(qi,pj,t)= pkq˙k(qi,pj,t) L(qi, q˙j(q`,pm,t),t). (6.2) As = we can conclude that = . We may also k − ∂S∂V ∂V ∂S ∂V ∂S X S V consider the state of the gas to be described by T and V ,so Then for fixed time, dH = (dp q˙ + p dq˙ ) dL = (˙q dp p dq ) , ∂E ∂E k k k k − k k − k k dE = dT + dV k k ∂T ∂V X X V T ∂H ∂H =˙qk, = pk. (6.3) 1 p 1 ∂E 1 ∂E ∂p ∂q − k q,t k p,t dS = dE + dV = dT + p + dV, T T T ∂T "T ∂V !# V T Other examples of Legendre transformations occur in thermodynamics. The energy change of a gas in a variable container with heat flow is sometimes from which we can conclude written ∂ 1 ∂E ∂ 1 ∂E = p + , dE =d¯Q pdV, ∂V T ∂T ! ∂T "T ∂V !# − V T T V 6.1. LEGENDRE TRANSFORMS 159 160 CHAPTER 6. HAMILTON’S EQUATIONS and therefore In Section 2.1 we discussed how the Lagrangian is unchanged by a change ∂p ∂E T p = . of generalized coordinates used to describe the physical situation. More pre- ∂T − ∂V V T cisely, the Lagrangian transforms as a scalar under such point transforma- tions, taking on the same value at the same physical point, described in the This is a useful relation in thermodynamics. Let us get back to mechanics. Most Lagrangians we encounter have the new coordinates. There is no unique set of generalized coordinates which describes the physics. But in transforming to the Hamiltonian language, decomposition L = L2+L1+L0 into terms quadratic, linear, and independent of velocities, as considered in 2.4.2. Then the momenta are linear in velocities, different generalized coordinates may give different momenta and different Hamiltonians. An nice example is given in Goldstein, a mass on a spring at- pi = j Mijq˙j + ai, or in matrix form p = M q˙ + a, which has the inverse 1 · 1 1 tached to a “fixed point” which is on a truck moving at uniform velocity v , relationq ˙ = M − (p a). As H = L L , H = (p a) M − (p a) L . T P · − 2 − 0 2 − · · − − 0 As a simple example, with a = 0 and a diagonal matrix M, consider spherical relative to the Earth. If we use the Earth coordinate x to describe the mass, coordinates, in which the kinetic energy is the equilibrium position of the spring is moving in time, xeq = vT t, ignoring a 1 2 1 2 negligible initial position.