Exercises for Symplectic Geometry I

Exercises for Symplectic Geometry I

LMU M¨unchen, Summer term 09 Frederik Witt / Vincent Humili`ere Problem set 2, due for 07/05/09

Classical mechanics

1. The Sliding Particle

Suppose that a particle of unit weight and mass slides without friction on a smooth hypersurfaceqⁿ⁺¹= F(q¹, ..., qⁿ) subject only to the force of gravity (which is directed downward along theqⁿ⁺¹-axis).

(a) The Lagrangian for this motion has the form ”kinetic-minus-potential”. Show that in the x- coordinates it is written

L= 1 2

µ

( ˙q¹)²+...+ ( ˙qⁿ)²+ (∂F

∂qⁱq˙ⁱ)²

¶

−F(q¹, ..., qⁿ).

(b) Show that this is a non-degenerate Lagrangian.

(c) From now on, we suppose thatF is invariant under rotation, i.e., that F(q¹, ..., qⁿ) =f((q¹)²+...+ (qⁿ)²)

for some smooth function f. Show that the projection of the particle onto Rⁿ remains in a fixed 2-plane.

2. The Motion of Rigid Bodies I

Arigid body is a finite set of pointsq1, . . . , qN inR^N with massesm1, . . . , mN such that all the distances dij =|qi−qj|are fixed. The free motion of such a body is governed by the ”kinetic energy” Lagrangian

L= m1

2 |q˙1|²+. . .+mN

2 |q˙N|², where ˙qi denotes as usual the velocity of thei’th point mass.

LetGbe the matrix group

G=

½ µA b 0 1

¶¯¯

¯¯A∈O(n), b∈Rⁿ

¾ .

(a) Check thatGacts as the space of isometries ofRⁿ.

(b) This action induces a diagonal action on (Rⁿ)^N defined by g·(q1, . . . , qN) = (g·q1, . . . , g·qN), whereg·qi denotes the action (by isometry) ofg∈Gonqi∈R^N. Show that the set

M ={q∈(Rⁿ)^N| ∀i, j,|qi−qj|=dij} is an orbit for the diagonal action ofGon (Rⁿ)^N.

Since G is a Lie group (a group together with a differentiable structure for which composition and inversion are smooth) this implies thatM is a manifold called ”configuration space of the rigid body”

3. The Motion of Rigid Bodies II

We use the notations of the previous exercise. Let ¯q∈M be a ”reference configuration” whose center of mass is supposed for convenience to be at the origin:

m1q¯1+. . .+mNq¯N = 0.

A motion of the rigid body is a curve γ : [a, b]→ M. For someA(t), b(t), such a curve can be written γ(t) =g(t)·q¯with

g(t) =

µA(t) b(t)

0 1

¶

∈G.

Also denoteα=A⁻¹A˙ andβ =A⁻¹b.˙

(a) Show that the LagrangianL can be written is the simple form

L( ˙γ(t)) =−tr¡

(α(t)²µ)¢ +1

2m|β(t)|²,

wherem=m1+. . .+mN is the total mass of the body, andµis the positive semi-definite symmetric matrix given by

µ= 1 2

XN k=1

m_kq¯_k^tq¯_k.

(b) Show that there exists some diagonal matrixδsuch that L( ˙γ(t)) =−tr¡

(α(t)²δ)¢ +1

2m|β(t)|².

Meditate the following interpretation: ”the motion of a rigid body is equivalent to the motion of its

’ellipsoid of inertia’ ”.

(c) Show that Lis non-degenerate if and only if the matrix µhas at least one zero eigenvalue. Show thatLis degenerate if and only if the rigid body lies in a subspace of dimension at mostn−2.