Conservation properties

We first recall that energy-stepping [24] exactly conserves all the momentum maps and the symplectic structure of the original Lagrangian system because the stepwise approximation of the potential energy preserves all the symmetries of the system and the discrete trajectories are exact trajectories of a Lagrangian system. Likewise, as we show in this section, asynchronous energy-stepping exactly conserves all the momentum maps and the symplectic structure of the original local Lagrangian system.

In order to facilitate the analysis we recast Definition 4.2.1 into its sequential or synchronous form. Then, the potential energy is additively decomposed (4.5) and stepwise approximated (4.7) but a single sequence of times is employed (i.e., {𝑡_𝑘}=∪

𝑗{𝑡^(𝑗)_𝑘 }). Indeed, it bears emphasis that the asynchronicity of the method is an algorithmic consequence of its formulation, as opposed to the result of further approximations ora priori selection of different time steps.

Definition 4.3.1 Suppose (

𝑡_𝑘, 𝑞_𝑘,𝑞˙_𝑘⁺)

, the localization matrices 𝐿𝑜^(𝑗), and a piecewise-constant approximation of the potential energy𝑉_ℎ=∑

𝑗𝑉_ℎ^(𝑗) are given. Let𝑡_𝑘+1 and𝑞_𝑘+1 be the time and points of exit of the rectilinear trajectory 𝑞𝑘+ (𝑡−𝑡𝑘) ˙𝑞_𝑘⁺ from the set {𝑉^(𝑖)(𝐿𝑜^(𝑖)𝑞) = ℎ^(𝑖)ℤ}. Let Δ𝑉^(𝑖) be the energy jump at𝑞𝑘+1 in the direction of advance. The updated velocity is, then,

˙

𝑞_𝑘+1⁺ = ˙𝑞_𝑘⁺+𝜆𝑘+1𝑀⁻¹𝑛𝑘+1 (4.34)

where𝑛_𝑘+1= (𝐿𝑜^(𝑖))^𝑇𝑛^(𝑖)(𝐿𝑜^(𝑖)𝑞_𝑘+1) and

𝜆𝑘+1=

⎧















⎨















⎩

−2 𝑞˙⁺_𝑘 ⋅𝑛_𝑘+1 𝑛^𝑇_𝑘+1𝑀⁻¹𝑛_𝑘+1,

if (

˙

𝑞_𝑘⁺⋅𝑛𝑘+1)²

<2Δ𝑉^(𝑖)(

𝑛^𝑇_𝑘+1𝑀⁻¹𝑛𝑘+1) ,

−𝑞˙⁺_𝑘 ⋅𝑛𝑘+1+ sign(

Δ𝑉^(𝑖))√

(𝑞˙⁺_𝑘 ⋅𝑛𝑘+1

)2

−2Δ𝑉^(𝑖)(

𝑛^𝑇_𝑘+1𝑀⁻¹𝑛𝑘+1

) 𝑛^𝑇_𝑘+1𝑀⁻¹𝑛𝑘+1

,

otherwise,

(4.35)

It is now clear that we can take advantage of the symmetry considerations that have been discussed in Gonzalez et al. [24]. Since the piecewise constant approximation of the additively decomposed potential energy preserves all the symmetries of the local systems exactly, and the discrete trajectories are exact trajectories of a Lagrangian system, we have the following

Theorem 4.3.1 The asynchronous energy-stepping time-integration scheme is a symplectic-energy- momentum time-reversible integrator with automatic and asynchronous selection of the time step size in each subdomain. In particular, the scheme conserves exactly all the momentum maps of the original local Lagrangians. Moreover, if the process of localization preserves all the symmetries of the global system, asynchronous energy-stepping preserves all the momentum maps of the original global Lagrangian.

Symmetry or time-reversibility of asynchronous energy-stepping follows directly from the defini- tion of the scheme. The automatic and asynchronous time-step selection properties also follow by construction. In particular, in regions where the localized potential energy gradient∇𝑉^(𝑗) is steep,

the energy jumps are more closely spaced and the resulting time steps of the subdomain are small. By contrast, if the localized potential energy gradient is small, the resulting time steps of the subdomain are comparatively large. It bears emphasis that what is conserved along the trajectories of the ap- proximate Lagrangian𝐿ℎis theexact, time-continuous, momentum map of the original Lagrangian 𝐿. This is in contrast to discrete variational integrators, which conserve discrete forms of the mo-

mentum maps, instead of the exact, time-continuous, momentum maps of the original Lagrangian 𝐿. Thus, in particular: if𝑉 is invariant under translations then asynchronous energy-stepping con- serves the total linear momentum𝑝₁+⋅ ⋅ ⋅+𝑝_𝑁 of the system; and if𝑉 is invariant under rotations then asynchronous energy-stepping conserves the total angular momentum𝑞₁×𝑝₁+⋅ ⋅ ⋅+𝑞_𝑁×𝑝_𝑁 of the system. Throughout this chapter we consider configurations𝑞 ∈𝑄=𝐸(𝑛)^𝑁 and momenta 𝑝∈𝑇_𝑞^∗𝑄, where 𝐸(𝑛) is the Euclidean space of dimension 𝑛(i.e.,𝑑=𝑛𝑁).

4.3.1 Global and local conservation laws

In the context of continuum solid mechanics, the classical theorem of Noether states that each variational symmetry of a Lagrangian𝐿(𝜑,𝜑) leads to local and global conservation laws, where˙ 𝜑 is the deformation mapping [57]. If the system is free of external forces, theglobal momentum map 𝐽 defined by a symmetry is a constant of the motion, i. e., it remains constant along trajectories. If the conservation law is localized to the subdomain Ω^(𝑠), thelocal momentum map𝐽^(𝑠)defined by a symmetry is a constant of motion that additionally accounts for momentum fluxes across subdomain boundaries, i.e.,∂Ω^(𝑠)∖∂Ω.

Likewise, in the context of finite-dimensional Lagrangian systems 𝐿(𝑞,𝑞) (e. g., Lagrangian sys-˙ tems obtained from the finite-element discretization of the action of a continuum solid [51], or Lagrangian systems employed in molecular dynamics simulations), each symmetry of the system leads to local and global conservation laws in accordance with Noether’s theorem. We restrict atten- tion to those scenarios where the process of localization preserves all the symmetries of the global system, therefore asynchronous energy-stepping exactly conserves all global momentum maps of the system, as described above, and all local momentum maps as we examine next.

Let 𝐺be a Lie group with Lie algebra 𝔤=𝑇_𝑒𝐺. A left action of 𝐺on the local configuration space𝑄^(𝑠)is a mapping Φ^(𝑠):𝐺×𝑄^(𝑠)→𝑄^(𝑠), and the infinitesimal generator of Φ^(𝑠)corresponding to 𝜉∈𝔤is the vector field𝜉_𝑄(𝑠) ∈𝑇 𝑄^(𝑠). We say that the local Lagrangian𝐿^(𝑠)is invariant under the action Φ^(𝑠)if

𝐿^(𝑠)(Φ^(𝑠)(𝑔, 𝑞^(𝑠)), 𝑇Φ^(𝑠)(𝑔, 𝑞^(𝑠)) ˙𝑞^(𝑠)) =𝐿^(𝑠)(𝑞^(𝑠),𝑞˙^(𝑠)), ∀𝑔∈𝐺, (𝑞^(𝑠),𝑞˙^(𝑠))∈𝑇 𝑄^(𝑠) (4.36)

Then, the local momentum map 𝐽^(𝑠) : 𝑇 𝑄^(𝑠) → 𝔤^∗ defined by the action Φ^(𝑠), which expresses a symmetry of𝐿^(𝑠), follows from the identity

⟨𝐽^(𝑠), 𝜉⟩

𝑇 0

=⟨∂_𝑞˙(𝑠)𝐿^(𝑠), 𝜉_𝑄(𝑠)⟩

𝑇 0

+∑

𝑗∕=𝑠

[

⟨∂_𝑞˙(𝑠)𝐿^(𝑗), 𝜉_𝑄(𝑠)⟩

𝑇 0

−

∫ 𝑇 0

⟨∂_𝑞(𝑠)𝐿^(𝑗), 𝜉_𝑄(𝑠)⟩𝑑𝑡 ]

, ∀𝜉∈𝔤 (4.37) Naturally, the coupling terms in𝐿^(𝑗), i.e., terms which involve configurations that belong to Im(𝐿𝑜^{(𝑗)∩(𝑠)}𝑞^(𝑠)), result in momentum fluxes across subdomain boundaries. Classical examples include:

i) Local conservation of linear momentum. In this case, 𝑄^(𝑠) = 𝐸(𝑛)^𝑁𝑠, 𝐺 = 𝐸(𝑛) and Φ^(𝑠)(𝑢, 𝑞^(𝑠)) ={𝑞^(𝑠)₁ +𝑢, . . . , 𝑞^(𝑠)_𝑁

𝑠+𝑢}represents a rigid translation of the system by𝑢∈𝐸(𝑛).

The corresponding momentum map is the total linear momentum of the local system plus a linear momentum flux,

𝐽^(𝑠)

𝑇 0 =

𝑁_𝑠

∑

𝑎=1

𝑝^(𝑠)_𝑎

𝑇

0 +∑

𝑗∕=𝑠

∑

𝑎∈𝜔^(𝑗,𝑠)

[ 𝑝^(𝑗)_𝑎

𝑇 0 +

∫ 𝑇 0

∂_𝑞(𝑗) 𝑎 𝑉^(𝑗)𝑑𝑡

]

where 𝜔^(𝑗,𝑠)={𝑎:𝑞𝑎^(𝑗)∈Im(𝐿𝑜^{(𝑗)∩(𝑠)}𝑞^(𝑠))}.

ii) Local conservation of angular momentum. In this case, 𝑄^(𝑠) = 𝐸(𝑛)^𝑁𝑠, 𝐺 = 𝑆𝑂(𝑛) and Φ^(𝑠)(𝑅, 𝑞^(𝑠)) ={𝑅𝑞₁^(𝑠), . . . , 𝑅𝑞^(𝑠)_𝑁

𝑠}represents a rigid rotation of the system by𝑅∈𝑆𝑂(𝑛). The corresponding momentum map is the total angular momentum of the local system plus an

angular momentum flux,

𝐽^(𝑠)

𝑇 0

=

𝑁_𝑠

∑

𝑎=1

𝑞^(𝑠)_𝑎 ×𝑝^(𝑠)_𝑎

𝑇 0

+∑

𝑗∕=𝑠

∑

𝑎∈𝜔^(𝑗,𝑠)

[

𝑞^(𝑗)_𝑎 ×𝑝^(𝑗)_𝑎

𝑇 0

+

∫ 𝑇 0

𝑞^(𝑗)_𝑎 ×∂_𝑞(𝑗) 𝑎 𝑉^(𝑗)𝑑𝑡

]

iii) Local conservation of energy. In this case, a space-time configuration manifold is considered, i.e., ℚ^(𝑠) =ℝ×𝑄^(𝑠), 𝐺=ℝ and Φ^(𝑠)(𝑢,(𝑡^(𝑠), 𝑞^(𝑠))) = (𝑡^(𝑠)+𝑢, 𝑞^(𝑠)) represents a time-shift by𝑢∈ℝ. The corresponding momentum map is the total energy of the local system plus an energy flux,

𝐽^(𝑠)

𝑇 0

=−𝐸^(𝑠)

𝑇 0

−∑

𝑗∕=𝑠

∑

𝑎∈𝜔^(𝑗,𝑠)

[1

2𝑝^(𝑗)_𝑎 ⋅𝑞˙_𝑎^(𝑗)

𝑇

0

+

∫ 𝑇 0

∂_𝑞(𝑗)

𝑎 𝑉^(𝑗)⋅𝑞˙_𝑎^(𝑗)𝑑𝑡 ]

A particularly appealing property of the additive decomposition of the Lagrangian (4.3) assumed in this work and the piecewise approximation of the localized potential energies (4.7) is that they preserve all the symmetries of the original system exactly. To verify this, we simply observe that 𝑉_ℎ^(𝑗)has all the symmetries of𝑉^(𝑗)—which itself has all the symmetries of𝑉 by assumption—, that is

𝑉_ℎ^(𝑗)∘Φ^(𝑗)_𝑔 = (ℎ⌊ℎ⁻¹𝑉^(𝑗)⌋)∘Φ^(𝑗)_𝑔 =ℎ⌊ℎ⁻¹𝑉^(𝑗)∘Φ^(𝑗)_𝑔 ⌋=ℎ⌊ℎ⁻¹𝑉^(𝑗)⌋=𝑉_ℎ^(𝑗), ∀𝑔∈𝐺 (4.38)

where𝐺is a symmetry group of𝑉^(𝑗)and Φ^(𝑗)is an action that leaves𝑉^(𝑗)invariant, i.e.,𝑉^(𝑗)∘Φ^(𝑗)= 𝑉^(𝑗). Then, it follows from the regularization procedure of Theorem 2.2.1 (Theorem 2.1 in [24]), and a standard approximation argument, that the local momentum map 𝐽^(𝑠) is constant along asynchronous energy-stepping trajectories given by Definition 4.2.1.