NOTES ON HAMILTONIAN SYSTEMS JONATHAN LUK These notes discuss first integrals and Hamiltonian systems. It is some variation of §10, 12 of the textbook, with a small amount of related material that is not in the book. This is a preliminary version: If you have any comments or corrections, even very minor ones, please send them to me. In these notes, we will be studying autonomous ODEs, i.e., u0 (t) = F (u(t)) n 1 (0.1) n where F : U → R is a C function, U ⊂ R an open set. By the existence and uniqueness theorem (Exercise: Why does it apply?), given any initial datum, there always exists a unique solution local solution to (0.1) achieving the given initial condition. (In fact the solution is unique as long as it exists.) 1. First integrals 1 Definition 1.1. A C function f : V → R for some open set V ⊂ U is a first integral of motion for the ODE u0 (t) = F (u(t)) if n X ∂f hF, ∇f i = F i i = 0. ∂x i=1 First integrals of motion are important because of the following Lemma 1.2. Suppose u : I → U is a solution to (0.1). Then f (u(t)) = f (u(s)) for every t, s ∈ I. Proof. We compute d f (u(t)) = h(∇f )(u(t)), u0 (t)i = h(∇f )(u(t)), F (u(t))i = 0. dt For instance, any constant function is a first integral of motion. Less trivially, we have the following Proposition 1.3. Let F be as in (0.1) and suppose n ≥ 2 and F (u0 ) 6= 0. Then there exists an open set V ⊂ U containing u0 and a non-constant function f : V → R which is a first integral of motion. Proof. We will omit the proof. If you are interested, consult the textbook. 2. Hamiltonian systems A particularly interesting class of ODEs which arises in physics is the class of Hamiltonian systems. They also provide examples of ODEs with non-trivial first integrals. Definition 2.1. Let H : R2n → R be a C 2 function of 2n variables (p1 , p2 , . . . , pn , q1 , . . . , qn ). The system of Hamilton canonical equations with Hamiltonian H is given by ( p0i (t) = − ∂H ∂qi (p(t), q(t)) (2.1) ∂H 0 qi (t) = ∂pi (p(t), q(t)). It is helpful to have something more concrete to think about. A prototypical example is the Newton’s equation y 00 (t) = −(∇V )(y(t)), (2.2) n 2 where V : R → R is a C function and the unknown is y : I → Rn for some interval I. Writing n = N d, this is the equation which determines the motion of N particles in d dimensions. In this (1) (1) (1) (2) (2) (N ) (N ) (k) case, say, y = (y1 , y2 , . . . , yd , y1 , . . . , yd , . . . , y1 , . . . , yd ), y` gives the `-th coordinate of the k-th 1 2 JONATHAN LUK particle. The particles move according to Newton’s second law, where the acceleration is proportional to the force. In this case, the force is given by −∇V , and it depends on the positions of the particles. One easily checks that (2.2) can be reformulated as a Hamiltonian system: Proposition 2.2. Suppose y : I → Rn satisfies (2.2). Then p, q : I → Rn defined by q(t) = y(t) and p(t) = y 0 (t) satisfies (2.1) with H(p, q) = 12 kpk2 + V (q). Proof. This is a computation. Theorem 2.3. Let (p, q) = (p1 , p2 , . . . , pn , q1 , . . . , qn ) be a solution to (2.1) on I. Then H(p(t), q(t)) = H(p(s), q(s)) for every t, s ∈ I. Proof. By Lemma 1.2, it suffices to show that H is a first integral of motion in the sense of Definition 1.1. Let u = (p, q) = (p1 , p2 , . . . , pn , q1 , . . . , qn ). Then (2.1) can be written in the form of (0.1) with ( i = 1, . . . , n − ∂H i ∂qi , F (u) = ∂H i = n + 1, . . . , 2n ∂pi−n , Hence, hF, ∇Hi = − n n X ∂H ∂H X ∂H ∂H + = 0. ∂qi ∂pi ∂pi ∂qi i=1 i=1 Remark 2.4. In the case of (2.2), the Hamiltonian, which is independent of t, is often called the energy. Moreover, 21 kpk2 is called the kinetic energy and V (q) is called the potential energy. Theorem 2.5. Suppose V ≥ 0. Then any solution y(t) to (2.2) (a priori defined on some open interval) can be defined for all time. Proof. We rewrite (2.2) into (2.1) using Proposition 2.2, i.e., we let p = y 0 and q = y. Let J be the maximal interval of existence. Without loss of generality, assume 0 ∈ J. Suppose J = (T− , T+ ) 6= R. If T+ is finite, then by the extension theorem, lim inf (kp(t)k + kq(t)k) = +∞. t→T+ (2.3) On the other hand, by Proposition 2.2 and Theorem 2.3, we know that 21 kp(t)k2 + V (q(t)) = 12 kp(0)k2 + V (q(0)) =: E is finite. √ Since V ≥ 0, this√implies kp(t)k ≤ 2E for all t ∈ J. Now since q 0 (t) = p(t), we also know that kq(t)k ≤ kq(0)k + T+ 2E for all t ∈ [0, T+ ). These obviously contradict (2.3). This shows that T+ = +∞. The fact that T− = −∞ can be shown in a similar manner. This implies that J = R. Remark 2.6. The condition that V ≥ 0 is necessary. Exercise: Find a counter-example with V < 0. 3. Stability of equilibrium points It turns out that for some equilibrium points of Hamiltonian systems, it is easy to check their stability. For this let us recall Definition 3.1. A solution u : [0, ∞) → Rn to (0.1) with initial data u0 is (Lyapunov) stable if for every > 0, there is a δ > 0 such that for any ũ0 ∈ B(u0 , δ), the solution ũ to ( ũ0 (t) = F (ũ(t)), (3.1) ũ(0) = ũ0 , exists for all t ≥ 0 and ũ(t) ∈ B(u(t), ) for all t ≥ 0. The main theorem is as follows: ∂H Theorem 3.2. Suppose H has a strict local minimum at (p̄, q̄) and (− ∂H ∂qi , ∂pi )(p̄, q̄) = 0. Then (p(t), q(t)) = (p̄, q̄) is a stable equilibrium solution to (2.1). NOTES ON HAMILTONIAN SYSTEMS 3 We prove this as a special case of a more general theorem: Theorem 3.3 (Lyapunov). Let ū be an equilibrium solution to (0.1). Suppose L is a C 1 function on B(ū, r) for some r > 0 such that • L has a strict minimum at ū, • h∇L, F i ≤ 0 on B(ū, r). Then ū is (Lyapunov) stable. Proof. Let > 0 such that < r. Since L has a strict minimum at ū, we have L(ū) < inf L. ∂B(ū,) By continuity, there is a δ ∈ (0, ) such that sup L < B(ū,δ) inf L. ∂B(ū,) We claim that for every u0 ∈ B(ū, δ), the unique solution u(t) to (0.1) exists for all t ≥ 0 and such that u ∈ B(ū, ). Suppose not. Then there exists u0 ∈ B(ū, δ) such that we can define a finite τ satisfying τ = inf{t ∈ [0, ∞) : u(t) exists and u(t) ∈ / B(ū, )}. This implies u(τ ) ∈ ∂B(ū, ) and u(t) ∈ B(ū, ) for t ∈ (0, τ ). On the other hand, since B(ū, ) ⊂ B(ū, r), for every t ∈ (0, τ ), we have d L(u(t)) = h∇L, u0 (t)i = h∇L, F (u(t))i ≤ 0, dt i.e., L(u(t)) is decreasing for t ∈ (0, τ ). Therefore, inf ∂B(ū,) L ≤ L(u(τ )) ≤ L(u0 ) ≤ sup L, which is a contradiction. B(ū,δ) We can specialize to prove Theorem 3.2 as follows: Proof of Theorem 3.2. Let L = H. We check that (with the same computation as in the proof of Theorem 2.3) h∇L, F i = h∇H, F i = 0. Hence, we can apply Theorem 3.3.