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Reactions annihilating the virtual displacements are transpose multiples, with a positive definite multiplier matrix

Statement

Let $G : O \to R^{k}$ be a $C^{2}$ regular holonomic constraint with $G_{q} (t, q)$ of rank $k$ on $O \subseteq R \times R^{n}$ (A regular holonomic constraint, its virtual displacements and the ideal-reaction law), and let $M \in M_{n} (R)$ be symmetric positive definite. Fix $(t, q) \in O$ and write $V = \ker G_{q} (t, q)$ for the virtual displacements. Then:

for every $R \in R^{n}$ , the condition $R \cdot δ = 0$ for all $δ \in V$ holds if and only if there is a unique $λ \in R^{k}$ with

$R = G_{q} (t, q)^{T} λ;$

the $k \times k$ matrix

$C (t, q) : = G_{q} (t, q) M^{- 1} G_{q} (t, q)^{T}$

is symmetric and positive definite, hence invertible; in particular for every $R$ orthogonal to $V$ the multiplier of clause 1 is $λ = (G_{q} M^{- 1} G_{q}^{T})^{- 1} G_{q} M^{- 1} R$ , and the multiplier is a $C^{1}$ function of $(t, q, R)$ .

Facts & Assumptions

Given: The regular constraint $G$ with $k \times n$ Jacobian $G_{q} (t, q)$ of rank $k$ at the fixed point, the symmetric positive definite matrix $M$ , and a vector $R \in R^{n}$ .

[F1]

The multiplier lemma (A linear functional annihilating the kernel of a surjection is a unique transpose multiple): for a surjective linear map $A : R^{n} \to R^{k}$ and a linear functional $ℓ : R^{n} \to R$ vanishing on $\ker A$ , there is a unique $λ \in R^{k}$ with $ℓ (v) = λ \cdot A v$ for all $v$ . Here $A = G_{q} (t, q)$ is surjective because its rank is $k$ ( $C^{k}$ Euclidean maps and diffeomorphisms, The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case, Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ ).

[F2]

The inverse of a symmetric positive definite matrix is symmetric and positive definite, and the matrix $M$ is invertible (Positive definite symmetric matrices have positive definite square roots, and the congruence M^{-1/2} K M^{-1/2} preserves symmetry and positivity clause 1); matrix transposition and multiplication satisfy $(A B)^{T} = B^{T} A^{T}$ and $(G_{q}^{T} λ) \cdot v = λ \cdot G_{q} v$ (Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes, The transpose $A^{T}$ of a matrix, The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , Linear map between vector spaces over the same field).

[F3]

A linear map of a finite-dimensional space with trivial kernel is injective (The kernel and image are linear subspaces, and a linear map is injective if and only if its kernel is trivial), and an injective linear endomorphism of $R^{k}$ is invertible: Rank-nullity: $\dim_{F} V = nullity T + rank T$ gives image dimension $k$ , hence surjectivity, and the inverse of a linear bijection is linear and two-sided (Invertible matrices and the general linear group ${GL}_{n} (F)$ ).

Proof

technique · direct; translate the orthogonality condition into the multiplier lemma and verify positive definiteness of the multiplier matrix by a quadratic-form computation

1.1F1F2algebra

Clause 1, forward: assume $R \cdot δ = 0$ for all $δ \in V = \ker G_{q} (t, q)$ . Apply [F1] to the surjective $A = G_{q} (t, q)$ and the linear functional $ℓ (v) : = R \cdot v$ , which vanishes on $\ker A = V$ by hypothesis: there is a unique $λ \in R^{k}$ with $R \cdot v = λ \cdot G_{q} (t, q) v = (G_{q} (t, q)^{T} λ) \cdot v$ for every $v \in R^{n}$ . Hence $R = G_{q} (t, q)^{T} λ$ .

1.2F1F2algebra

Clause 1, converse: if $R = G_{q}^{T} λ$ and $δ \in V$ , then $R \cdot δ = λ \cdot G_{q} δ = 0$ . Uniqueness of $λ$ is the uniqueness statement of [F1].

1.3F2algebra

Clause 2, symmetry: $C^{T} = (G_{q} M^{- 1} G_{q}^{T})^{T} = G_{q} (M^{- 1})^{T} G_{q}^{T} = G_{q} M^{- 1} G_{q}^{T} = C$ , because $M^{- 1}$ is symmetric by [F2].

1.4F2F3algebra

Clause 2, positive definiteness: let $0 \neq z \in R^{k}$ and put $w : = G_{q}^{T} z$ . If $w = 0$ , then $z \cdot G_{q} v = (G_{q}^{T} z) \cdot v = 0$ for every $v \in R^{n}$ ; since $G_{q}$ is surjective, $z \cdot u = 0$ for every $u \in R^{k}$ , hence $z = 0$ , a contradiction. Thus $w \neq 0$ , and by positive definiteness of $M^{- 1}$ , $z^{T} C z = z^{T} G_{q} M^{- 1} G_{q}^{T} z = w \cdot M^{- 1} w > 0.$ Hence $C$ is positive definite. By the kernel argument ( $C z = 0$ implies $z^{T} C z = 0$ implies $z = 0$ ) and [F3], $C$ is invertible.

2.1F2step 1.4algebra∎

The formula for $λ$ : if $R$ is orthogonal to $V$ , clause 1 gives $R = G_{q}^{T} λ$ ; multiplying by $G_{q} M^{- 1}$ on the left gives $G_{q} M^{- 1} R = C λ$ , hence $λ = C^{- 1} G_{q} M^{- 1} R$ . The right-hand side is a composition of $C^{1}$ maps of $(t, q)$ — the entries of $G_{q}$ and of $C^{- 1}$ are $C^{1}$ by If $\det (A)$ is a unit, then $A^{- 1} = \det (A)^{- 1} adj (A)$ , because $G$ is $C^{2}$ and $\det C \neq 0$ — and is linear in $R$ , so $λ$ depends on $(t, q, R)$ in a $C^{1}$ way.

Remarks

The uniqueness in clause 1 is uniqueness of the multiplier for a given reaction $R$ , not yet uniqueness of the reaction or of the acceleration: which $R$ occurs is determined only after the twice-differentiated constraint is imposed, and that is done in The ideal holonomic system in a possibly time-dependent chart: explicit acceleration, constraint preservation, local existence and reaction power.
The matrix $C$ is the inverse of the reduced mass matrix on the conormal directions; its positive definiteness is the reason the constrained acceleration is determined by the data wherever the rank hypothesis holds. At a point where the rank of $G_{q}$ drops, $C$ degenerates and the construction is not available; the proposition below makes no claim there.

Reactions annihilating the virtual displacements are transpose multiples, with a positive definite multiplier matrix

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources