Analytical Mechanics This page intentionally left blank
Download 10.87 Mb. Pdf ko'rish
|
|
∂f
∂x 1 e 1 + ∂f ∂x 2 e 2 + ∂f ∂x 3 e 3 , where e 1 , e 2 , e
3 are the unit vectors of the canonical of R 3 .
A4.2 Algebraic forms, differential forms, tensors 723 If
∈ Λ 1 (R 3 ), Θ (x) = ϑ
1 (x) dx
1 + ϑ
2 (x) dx
2 + ϑ
3 (x) dx
3 , by identifying Θ (x)
with the vector field Θ = ϑ 1 (x)e
1 + ϑ
2 (x)e
2 + ϑ
3 (x)e
3 , the exterior derivative d Θ :
Θ = ∂ϑ 2 ∂x 1 − ∂ϑ 1 ∂x 2 dx 1 ∧ dx
2 + ∂ϑ 3 ∂x 1 − ∂ϑ 1 ∂x 3 dx 1 ∧ dx
3 + ∂ϑ 3 ∂x 2 − ∂ϑ 2 ∂x 3 dx 2 ∧ dx
3 can be identified with the curl of the vector field Θ :
Θ = ∂ϑ 3 ∂x 2 − ∂ϑ 2 ∂x 3 e 1 + ∂ϑ 1 ∂x 3 − ∂ϑ 3 ∂x 1 e 2 + ∂ϑ 2 ∂x 1 − ∂ϑ 1 ∂x 2 e 3 , since dx 2 ∧ dx
3 = e
2 × e
3 = e
1 , dx
1 ∧ dx
2 = e
1 × e
2 = e
3 and dx
1 ∧ dx
3 = e
1 × e
3 = − e 2 . Finally, if Ω ∈ Λ 2 (R 3 ), Ω (x) = Ω 12 (x) dx 1 ∧ dx 2 − Ω 31 (x) dx
1 ∧ dx
3 + Ω 23 (x) dx
2 ∧ dx
3 , identifying this with the vector field Ω =
23 e 1 + Ω 31 e 2 + Ω 12 e 3 , the exterior derivative d Ω = ∂ Ω 12 ∂x 3 + ∂ Ω 23 ∂x 1 + ∂ Ω 31 ∂x 2 dx 1 ∧ dx
2 ∧ dx
3 can be identified with the divergence of Ω :
Ω = ∂ Ω 12 ∂x 3 + ∂ Ω 31 ∂x 2 + ∂ Ω 23 ∂x 1 . The vanishing of d 2 (the second exterior derivative), in the context of vec- tor analysis in R 3 , summarises the two classical results ∇ × (∇f) = 0 and ∇ · (∇ ×
Θ ) = 0.
D efinition A4.9 A differential k-form Ω ∈
k (M ) is closed if d Ω = 0, and
exact if there exists a (k − 1)-form Θ such that Ω = d
Θ . Remark A4.5 By property (3) (Theorem A4.2) of exterior differentiation, every exact form is closed. The converse is in general false, except in open simply connected sets of R l
T heorem A4.3 (Poincar´e’s lemma) Let A ⊆ R l be an open simply connected set, and Ω be a differential k-form on A. If Ω is closed, then Ω is exact. For the proof, see any one of the books suggested in this appendix. From Poincar´ e’s lemma it immediately follows that a vector field X on R l is a gradient vector field if and only if the Jacobian matrix ∂X i /∂x j of the field is symmetric. We end this brief introduction to forms by studying their behaviour under the action of a diffeomorphism f between two differentiable manifolds M and N . Let
Ω ∈ Λ k (N ), and let f : M → N be a diffeomorphism. Since the differential
724 Algebraic forms, differential forms, tensors A4.3 f
(P ) = df (P ) is an isomorphism of T P M and T f (P )
N (cf. Section 1.7) it is possible to associate with Ω a k-form on Λ k (M ), denoted by f ∗ Ω (the pull-back of Ω ) defined as follows: (f ∗ Ω )(P )(v 1 , . . . , v k ) =
Ω (f (P ))(f ∗ (P )v
1 , . . . , f ∗ (P )v
k ), (A4.39) where P ∈ M and v 1 , . . . , v k ∈ T
P M .
If (y 1 , . . . , y l ) are local coordinates on N and Ω (y) =
i Ω i (y) dy i , it is straightforward to check that (cf. (A4.19)) (f ∗ Ω )(x) =
i,j Ω i (f(y)) ∂f i 1 ∂x j 1 · · ·
∂f i k ∂x j k dx j 1 ∧ . . . ∧ dx j k , (A4.40)
where i = (i 1 , . . . , i k ), j = (j 1 , . . . , j k ). A4.3 Stokes’ theorem The theory of integration of differential forms defined on a manifold is rather rich, and cannot be covered in this brief introduction. We can however devote a little space to Stokes’ theorem, whose statement is very simple and which has useful consequences for a better understanding of some of the topics considered in this book (e.g. the Poincar´ e–Cartan form, see Section 10.3). A differentiable manifold with boundary M is a manifold whose atlas (U α , x α ) α ∈A contains the two following types of charts. If we denote by H l
{x ∈ R l |x l ≥ 0} the upper half-space R l , for some charts x −1 α (U α ) is
an open set of R l homeomorphic to R l , and for others it is homeomorphic to H l
x − 1 (P ) ∈ ∂H
l = {x ∈ R l |x l = 0 } R l − 1 . Clearly the boundary ∂M of M is a smooth manifold of dimension l − 1. Example A4.10 The half-sphere M =
{x ∈ R l +1 |(x 1 ) 2 + · · · + (x l ) 2 + (x l +1 ) 2 = 1, x l +1 ≥ 0} is a differentiable manifold with boundary ∂M =
{x ∈ R l |(x 1 ) 2 + · · · + (x l )
= 1, x l +1 = 0 } S l − 1 . Let M be an l-dimensional oriented manifold with boundary ∂M (hence a manifold of dimension l − 1 coherently oriented). It is not difficult to introduce the notion of the integral of an l-form on M , although a rigorous definition requires the use of a partition of unity (see Singer and Thorpe 1980). We simply A4.3 Algebraic forms, differential forms, tensors 725 note that if (x 1 , . . . , x l ) is a local parametrisation of M , any l-form Ω can be
written as Ω (x) = ω(x) dx 1 . . . dx
l , and hence is identified with a function ω(x) that can be integrated on the corresponding chart U . Evidently the theorem of change of integration variable ensures that the result is independent of the parametrisation. Indeed, if (y 1 , . . . , y l ) is a new parametrisation, and V is the image of the open U in the new local coordinates, from (A4.40) it follows that U ω(x) d l x =
V ω(x(y)) det ∂x i
j d l y (note that det(∂x i /∂y
j ) > 0 since M is oriented). Hence the integral can be extended to M (and we denote it by M Ω ) or to a part of it. T heorem A4.4 (Stokes) Let M be an l-dimensional oriented manifold with boundary, and let Ω ∈ Λ l − 1 (M ); then M d
= ∂M Ω . (A4.41)
Remark A4.6 If M = [a, b], Ω = f : [a, b] → R, equation (A4.41) becomes b a f (x) dx = [f (x)] b a = f (b) − f(a),
and we recover the fundamental theorem of calculus. Similarly it is not difficult to check, using example A4.24, that Stokes’ theorem summarises Green’s theorem, the divergence theorem and the classical Stokes’ theorem of vector calculus (see Giusti 1989). C orollary 4.1 (Stokes’ lemma) If ω is a non-singular 1-form (see Defini- tion 10.9) in R 2l+1
and γ 1 , γ 2 are two closed curves enclosing the same tube of characteristics of ω, then γ 1 ω = γ 2 ω. Proof
By Remark 10.15, dω = 0 on a tube of characteristics. If σ is the portion of the tube of characteristics having as boundary γ 1 − γ
2 (Fig. A4.1) we have γ 1
− γ 2 ω = ∂σ ω = σ dω = 0.
726 Algebraic forms, differential forms, tensors A4.4
1
2
∂s = g 1 – g 2 Fig. 4.1 ∂ σ = γ
1 − γ
2 . A4.4 Tensors We only give the definition of a tensor field on a differentiable manifold, and refer the interested reader to Dubrovin et al. (1991a). Let M be a differentiable manifold of dimension l, P ∈ M. D
τ : T P M × · · · × T P M m times
× T ∗ P M × · · · × T ∗ P
n times
→ R (A4.42)
is an m times covariant, n times contravariant tensor (or of type (m, n) and order m + n). If (x 1
l ) is a local parametrisation in a neighbourhood of P ∈ M and we denote by (e 1 , . . . , e l ) the basis ∂/∂x 1 , . . . , ∂/∂x l of T
P M , and by (e 1∗ , . . . , e l ∗ ) the (dual) basis (dx 1 , . . . , dx l ) of T
∗ P M , a tensor of type (m, n) can be expressed through its components: τ = τ
i m+1
...i m+n
i 1 ...i m e i 1 ∗ ⊗ · · · ⊗ e i m ∗ ⊗ e i m+1 ⊗ · · · ⊗ e i m+n , (A4.43)
where ⊗ denotes the tensor product: if ϑ 1 , ϑ
2 ∈ T
∗ P M then e i ⊗ e j (ϑ 1 , ϑ 2 ) = (ϑ 1 ) i (ϑ 2 ) j , (A4.44) where (ϑ 1 ) i denotes the ith component of ϑ 1 = ϑ
1i e i ∗ . If, on the other hand, v 1 ,
2 ∈ T
P M , we have e i
⊗ e j ∗ (v 1 , v 2 ) = v
i 1 v j 2 . (A4.45) A4.4 Algebraic forms, differential forms, tensors 727 The vector space of tensors of type (m, n) therefore coincides with the tensor product m T ∗ P M n T P M (A4.46)
of m cotangent spaces T ∗ P M with n tangent spaces T P M , and hence has dimension l m +n . Remark A4.7 It is necessary to take care of the fact that the tensor product is distributive with respect to addition, it is associative (as follows from (A4.44) and (A4.45)) but it is not commutative: the spaces T ∗ P M ⊗ T
P M and T
P M ⊗ T ∗ P M are both vector spaces of tensors once covariant and once contravariant, but T ∗ P M ⊗T P M = / T
P M ⊗ T ∗ P M . Let (y 1 , . . . , y l ) be a new choice of local coordinates in a neighbourhood of P ∈ M. Denote by (e 1 , . . . , e l ) = (∂/∂y 1 , . . . , ∂/∂y l ) the associated basis of T P M and by (e 1∗ , . . . , e l ∗
1 , . . . , dy l ) the dual basis. If we denote by J j k = ∂x j ∂y k , (J − 1 ) j k = ∂y j ∂x k (A4.47) the elements of the Jacobian matrix of the transformation of the change of chart and of its inverse, a tensor (A4.43) is transformed into (τ ) i
...i m+n
i 1 ...i m e i 1 ∗ ⊗ · · · ⊗ e i m ∗ ⊗ e i m+1 ⊗ · · · ⊗ e i m+n , (A4.48)
where the old and new components are related by τ i m+1 ...i
m+n i 1 ...i m = (J − 1 ) j 1 i 1 · · · (J
− 1 ) j m i m J i m+1 j m+1 · · · J i m+n j m+n
· (τ ) j m+1 ...j m+n
j 1 ...j m . (A4.49) D efinition A4.11 A tensor field of type (m, n) on a manifold M is a regular map τ : M
→ P ∈M {P } × m T ∗ P M n T P M (A4.50)
that associates with every point P ∈ M a tensor of type (m, n) belonging to m T
P M n T P M and depending regularly on P . In local coordinates (x 1 , . . . , x l ) we have τ (x 1
l ) = τ (x) i m+1
...i m+n
i 1 ...i m dx i 1 . . . dx
i m ∂ ∂x i m+1 · · · ∂ ∂x i m+n
. Vector fields are examples of tensor fields of type (0, 1). The fields of cotangent vectors are tensor fields of type (1, 0). A differential k-form on M is a tensor field of type (k, 0) that is totally skew-symmetric: if we denote by τ i 1
k (x) its
components, then τ i 1 ...i k = ±τ i 1 ...i k , where the sign is positive if i 1 . . . i
k is an
even permutation of i 1 . . . i k , and negative otherwise (hence τ i 1
j i j ...i k = 0). APPENDIX 5: PHYSICAL REALISATION OF CONSTRAINTS The notion of a system subject to an ideal bilateral frictionless holonomic con- straint was introduced in Chapter 1 and further discussed in Chapters 2 and 4. Our discussion, however rigorous from the point of view of mathematical mod- elling, does not consider the more complex issue of the physical phenomenon responsible for the effect of the constraint. Consider, as an example, a point particle of mass m constrained to move on the surface of a sphere of radius R and subject to gravity (spherical pendulum). If (r, ϑ, ϕ) indicate spherical coordinates in R 3 , the Lagrangian of the system is given by L(ϑ, ϕ, ˙ ϑ, ˙ ϕ) =
1 2 mR 2 ( ˙
ϑ 2 + sin 2 ϑ ˙
ϕ 2 ) − mgR cos ϑ. (A5.1)
From a physical point of view, we expect the constraint reactions to be due to the elastic reaction of the materials subject to a small deformation due to contact—in the example under consideration—between the particle and the sphere. Assume then that the constraint is realised physically through a spring of negligible mass, length at rest equal to R and a very large elastic constant k. We then want to use a limiting procedure for k → ∞, in analogy with what was done in Section 2.6 to study unilateral constraints. Setting ξ = r − R, the Lagrangian of the system (that now has three degrees of freedom) can be written as ˆ L = 1 2 m ˙ ξ 2 + 1 2 m(R + ξ) 2 ( ˙
ϑ 2 + sin 2 ϑ ˙
ϕ 2 ) − mg(R + ξ) cos ϑ − 1 2 kξ 2 . (A5.2) The problem is to compare the solutions of the system (A5.2) for very large values of k, corresponding to initial conditions belonging to the sphere of radius R (i.e. ξ = 0, ˙ ξ = 0) with the solutions of the system (A5.1). Note that, in spite of the fact that for t = 0 we have ξ = 0, ˙ ξ = 0, for t > 0 in general ξ = / 0, ˙
ξ = / 0,
since the Lagrange equations associated with (A5.2) couple the variables (ξ, ˙ ξ) with (ϑ, ϕ, ˙ ϑ, ˙ ϕ).
More generally, consider a point particle P of mass m and Cartesian coordinates x = (x
1 , x
2 , x
3 ) subject to a smooth holonomic constraint with equation f (x 1
2 , x
3 ) = 0,
(A5.3) and under the action of a conservative force field with potential energy V (x 1
2 , x
3 ). If f is of class C 2
∇ x f = / 0, equation (A5.3) defines a sur- face S
⊂ R 3 . Assume that the constraint is realised through a recoiling elastic 730 Physical realisation of constraints A5 force proportional to f Download 10.87 Mb. Do'stlaringiz bilan baham: 
|