Derivation of the Hamilton principle for the discontinuous energy stepping

2.2 Energy-stepping integrators

2.2.2 Derivation of the Hamilton principle for the discontinuous energy stepping

The energy stepping potential (2.4) is not continuous and the corresponding action functional (2.6) is not Gateaux-differentiable on the space of curves in phase space. In particular, the first variation of 𝐼ℎ is not well-defined on this space and so the notion of critical points does not have a well- defined meaning in the classical sense. However, the calculations in the previous subsection can be given a rigorous interpretation by recourse to a smooth approximation procedure. In the spirit of identifying solutions to non-smooth differential equations by suitable selection principles, we will show that the energy-stepping dynamics for fixed ℎ is given by those solutions to (2.8) and (2.9) that are limiting trajectories of a nearby smooth dynamical system. It should be carefully noted that there are solutions to these equations other than those described in Section 2.2.1, as exemplified by a particle bouncing at an energy barrier that can be overcome energetically.

This approximation result has three important consequences:

(i) The notion of a critical point of𝐼ℎis given a rigorous definition.

(ii) The physical solutions to (2.8) and (2.9) are selected.

(iii) The existence of a nearby smooth system will prove extremely useful when investigating ana- lytical aspects of the energy-stepping trajectories such as conservation properties (cf. Section 2.3).

Here we call a solution𝑞to the equations (2.8) and (2.9)physicalif and only if𝑞behaves as described in Paragraph 2.2.1.1, 2.2.1.2 or 2.2.1.3. For later use we note that this is the case if and only if 𝑞 passes an energy jump surface precisely if this is possible without violating energy conservation. As a consequence, an energy-stepping trajectory is uniquely given by𝑞(0) and ˙𝑞(0).

For fixed ℎ >0 consider the smooth approximation𝜒_𝜀to the stepping function𝑡7→ℎ⌊ℎ⁻¹𝑡⌋as given by the convolution

𝜒(𝑡) =

∫

ℝ

𝜀⁻¹𝜂(𝜀⁻¹(𝑡−𝑠))ℎ⌊ℎ⁻¹𝑠⌋𝑑𝑠, 𝜀≪ℎ (2.22)

where𝜂 denotes a standard mollifier, i.e., 𝜂∈𝐶_𝑐^∞(−1,1),𝜂≥0 and∫1

−1𝜂= 1. Now define

𝐿_ℎ,𝜀(𝑞,𝑞) =˙ 1

2𝑞˙^𝑇𝑀𝑞˙−𝑉_ℎ,𝜀(𝑞) for 𝑉_ℎ,𝜀(𝑞) :=𝜒_𝜀(𝑉(𝑞)) (2.23)

Since 𝐿_ℎ,𝜀 is a smooth function, there is a well-defined classical dynamics corresponding to this Lagrangian. We will prove that, as𝜀→0, the critical points of the action functional

𝐼ℎ,𝜀(𝑞) =

∫ 𝑡₂ 𝑡0

𝐿ℎ,𝜀(𝑞,𝑞)˙ 𝑑𝑡 (2.24)

converge to the trajectories identified in Section 2.2.

More precisely, we have the following

Theorem 2.2.1 Suppose𝑞 is an energy-stepping trajectory satisfying ∇𝑉(𝑞) ∕= 0and (𝑛⋅𝑞˙⁻)² ∈/ {0,2ℎ𝑛^𝑇𝑀⁻¹𝑛}for𝑛=𝑛(𝑞) = _∣∇𝑉^{∇𝑉(𝑞)}_(𝑞)∣on every energy jump surface. Let𝑞𝜀be the smooth trajectory corresponding to 𝐿ℎ,𝜀 with the same initial conditions (𝑞0,𝑞˙0). Then 𝑞𝜀 converges to 𝑞 strongly in 𝑊^1,𝑝(0, 𝑇)for each𝑇 >0 and all𝑝 <∞and weak^∗ in𝑊^1,∞(0, 𝑇). Moreover, 𝑞˙𝜀→𝑞˙uniformly on [0, 𝑇]∖𝑆𝑟(𝑞)for every 𝑟 >0, where𝑆𝑟(𝑞)is the𝑟-neighborhood of the jump set of 𝑞.˙

It is worth noting that a proof for this result can be given that does not make use of the explicit formulae for𝑞obtained Section 2.2. Instead, we will see that limiting trajectories satisfy the discrete variational principle leading to (2.8) and (2.9) and thus give an a priori justification of the procedure employed in Section 2.2 to compute𝑞.

Proof. First note that as𝜀→0,𝜒_𝜀(𝑉(𝑞))→ℎ⌊ℎ⁻¹𝑉(𝑞)⌋=𝑉_ℎ(𝑞). In particular, 𝑞_𝜀(𝑡) is linear in time except in a small neighborhood of the energy jump surfaces. It suffices to consider the case when 𝑞 meets a jump surface Γ precisely once, at time 𝑡1 ∈ (𝑡0, 𝑡2), for example, and so we may

assume that𝑞is a physical solution to (2.8) and (2.9).

We begin by showing that 𝑞_𝜀 →𝑞˜as 𝜀 →0 strongly in 𝑊^1,𝑝(𝑡₀, 𝑡₂) for all𝑝 <∞ (and weak^∗ in 𝑊^1,∞(𝑡0, 𝑡2)), where ˜𝑞 is linear on (𝑡0, 𝑡1) and (𝑡1, 𝑡2). By Γ𝜀 we denote the neighborhood of Γ defined by

Γ𝜀={𝑥∈ℝ^𝑁 :𝑉2−𝜀≤𝑉(𝑥)≤𝑉2+𝜀} (2.25)

Note that as∇𝑉(𝑞(𝑡1))∕= 0 by assumption, in the vicinity of𝑞(𝑡1) Γ𝜀lies in an𝑂(𝜀)-neighborhood of Γ. 𝑞_𝜀being a critical point of𝐼_ℎ,𝜀 satisfies the Euler-Lagrange equation

¨

𝑞_𝜀=−𝑀⁻¹∇𝑉_ℎ,𝜀 (2.26)

Choose𝜏1,𝜀, 𝜏2,𝜀 such that𝑞𝜀enters Γ𝜀at time 𝑡1−𝜏1,𝜀 and leaves it at time𝑡1+𝜏2,𝜀. Writing

∇𝑉ℎ,𝜀(𝑥) =𝛼(𝑥)𝑛+𝛽(𝑥)𝑃 𝑥 (2.27)

for𝑛=𝑛(𝑞(𝑡₁)),𝑃 the projection onto the plane perpendicular to 𝑛and𝛼, 𝛽∈ℝ, we obtain that

∣𝛽(𝑥)∣ is bounded on 𝑟𝜀-neighborhoods of 𝑞(𝑡₁) independently of 𝜀 for each 𝑟 ∈ ℝ, whereas 𝛼(𝑥) scales with𝜀⁻¹. Splitting the trajectories accordingly as

𝑞𝜀(𝑡) =𝑞_𝜀^∥(𝑡) +𝑞^⊥_𝜀(𝑡) with 𝑞^∥_𝜀∈𝑞(𝑡1) +ℝ𝑀⁻¹𝑛, 𝑞^⊥_𝜀 ∈𝑀⁻¹𝑃ℝ^𝑑 (2.28)

we obtain that

¨

𝑞_𝜀^∥(𝑡) =−𝑀⁻¹𝛼(𝑞𝜀(𝑡))𝑛, 𝑞¨_𝜀^⊥(𝑡) =−𝑀⁻¹𝛽(𝑞𝜀(𝑡))𝑃 𝑞𝜀(𝑡) (2.29)

It is not hard to see that 𝜏_2,𝜀+𝜏_1,𝜀 =𝑂(𝜀) and so ∣¨𝑞^∥𝜀(𝑡) +𝑀⁻¹𝛼(𝑞𝜀^∥(𝑡))𝑛∣ ≤ 𝐶, ∣¨𝑞_𝜀^⊥(𝑡)∣ ≤ 𝐶, on (𝑡₁−𝜏_1,𝜀, 𝑡₁+𝜏_2,𝜀) for some suitable constant𝐶 >0. This proves that in fact ˙𝑞_𝜀^⊥(𝑡₁−𝜏_2,𝜀)−𝑞˙^⊥_𝜀(𝑡₁− 𝜏1,𝜀)→0 as𝜀→0 and, by energy conservation,

˙

𝑞𝜀(𝑡1−𝜏2,𝜀)−𝑞˙𝜀(𝑡1−𝜏1,𝜀)→𝑞˙⁺−𝑞˙⁻ (2.30)

where ˙𝑞⁺ is given by (2.15) resp. (2.18) resp. (2.21). From here it is now straightforward to obtain that𝑞_𝜀converges to ˜𝑞, where ˜𝑞is linear on (𝑡₀, 𝑡₁) and (𝑡₁, 𝑡₂) and satisfies the jump condition (2.15) resp. (2.18) resp. (2.21) at𝑡1.

Since we do not wish to use the fact that 𝑞 is explicitly given by these equations in our proof, we proceed as follows. As 𝑞𝜀 is a critical point of𝐼ℎ,𝜀, for all𝜑 ∈𝐶_𝑐^∞((𝑡0, 𝑡2),ℝ^𝑑×ℝ^𝑑), the first variation

𝛿𝐼_ℎ,𝜀(𝑞, 𝜑) =

∫ 𝑡₂ 𝑡₀

˙

𝑞^𝑇𝑀𝜑˙− ∇𝑉ℎ,𝜀(𝑞)𝜑 𝑑𝑡 (2.31)

vanishes. A standard approximation argument show that this remains true for all𝜑∈𝑊₀^1,∞((𝑡₀, 𝑡₂),ℝ^𝑑× ℝ^𝑑). Now suppose that 𝑞^′ is another piecewise linear trajectory with ˜𝑞(𝑡₀) = 𝑞^′(𝑡₀), ˜𝑞(𝑡₂) =𝑞^′(𝑡₂), which is nearby ˜𝑞and meets the same energy jump surface Γ at time𝑡^′₁. Choose𝜏_1,𝜀^′ , 𝜏_2,𝜀^′ , such that 𝑞^′ enters, resp. leaves, Γ𝜀 at time𝑡^′₁−𝜏_1,𝜀^′ resp.𝑡^′₁+𝜏_2,𝜀^′

Now construct the following particular approximation 𝑞_𝜀^′ to𝑞^′:

∙ 𝑞_𝜀^′(𝑡) is linear on (𝑡0, 𝑡^′₁−𝜏1,𝜀) and on (𝑡^′₁+𝜏2,𝜀, 𝑡2),

∙ 𝑞_𝜀^′(𝑡^′₁−𝜏_1,𝜀) =𝑞^′(𝑡^′₁−𝜏_1,𝜀^′ ) and𝑞_𝜀^′(𝑡^′₁+𝜏_2,𝜀)−𝑞^′(𝑡^′₁+𝜏_2,𝜀^′ )→0 as𝜀→0 and

∙ 𝑉_ℎ,𝜀(𝑞_𝜀^′(𝑡^′₁+𝑡)) =𝑉_ℎ,𝜀(𝑞_𝜀(𝑡₁+𝑡)) for 𝑡∈[−𝜏_1,𝜀, 𝜏_2,𝜀].

It is not hard to see that, as𝜀→0,𝑞_𝜀^′ →𝑞^′strongly in𝑊^1,𝑝for all𝑝 <∞(and weak^∗in𝑊^1,∞), and in particular𝐼ℎ,𝜀(𝑞𝜀)→𝐼ℎ(˜𝑞),𝐼ℎ,𝜀(𝑞_𝜀^′)→𝐼ℎ(𝑞^′). In fact, due to the careful definition of𝑞_𝜀^′, we also obtain control over the difference

𝐼ℎ,𝜀(𝑞𝜀)−𝐼ℎ,𝜀(𝑞_𝜀^′)→𝐼ℎ(˜𝑞)−𝐼ℎ(𝑞^′) (2.32)

To see this, it suffices to note that

∫ 𝑡^′₁+𝜏2,𝜀

𝑡^′₁−𝜏_1,𝜀

𝑉(𝑞^′_𝜀(𝑡))𝑑𝑡−

∫ 𝑡1+𝜏2,𝜀

𝑡₁−𝜏_1,𝜀

𝑉(𝑞_𝜀(𝑡))𝑑𝑡= 0.In fact, since𝑞_𝜀is a critical point of𝐼ℎ,𝜀, we have

∣𝐼ℎ,𝜀(𝑞𝜀)−𝐼ℎ,𝜀(𝑞_𝜀^′)∣=𝑜(∥𝑞𝜀−𝑞^′_𝜀∥_𝑊1,2) (2.33)

with a term 𝑜(⋅) independent of𝜀. (Consider a path [0,1]∋𝑠7→𝑞_𝜀(⋅, 𝑠) in the space of curves such that𝑞_𝜀(⋅,0) =𝑞_𝜀,𝑞_𝜀(⋅,1) =𝑞_𝜀^′ and𝑉_ℎ,𝜀(𝑞_𝜀(𝑡₁(𝑠) +⋅, 𝑠)) =𝑉_ℎ,𝜀(𝑞_𝜀(𝑡₁+⋅)) on [−𝜏_1,𝜀, 𝜏_2,𝜀], where𝑡₁(𝑠) interpolates between𝑡1and𝑡^′₁, i.e., 𝑡1(0) =𝑡1 and𝑡1(1) =𝑡^′₁.)

As a consequence we obtain

∣𝐼ℎ(˜𝑞)−𝐼ℎ(𝑞^′)∣=𝑜(∥𝑞˜−𝑞^′∥𝑊^1,2) (2.34)

and thus that ˜𝑞 is a critical point of the action functional 𝐼_ℎ on piecewise linear trajectories as identified in Section 2.2. Now since𝑞(0) = ˜𝑞(0) and ˙𝑞(0) = ˙˜𝑞(0), we obtain ˜𝑞=𝑞. Hence,𝑞 is the limit, as𝜀→0 of the smooth trajectories𝑞𝜀corresponding to the smooth Lagrangian𝐿ℎ,𝜀.