Exercises for Symplectic Geometry I
LMU M¨unchen, Summer term 09 Frederik Witt / Vincent Humili`ere Problem set 1, due for 30/04/09
Lagrangian Mechanics
1. Geodesics
What does Energy Conservation mean when the Lagrangian is given by a Riemannian metric ?
2. Lagrangians differing by a divergence term
LetL1, L2:T M →Rbe two Lagrangians,EL1, EL2their respective energies andλL1, λL2 their respective momenta.
(a) Show that
EL1 =EL2 and dλL1 =dλL2
if and only if there is a closed 1-formφonM such thatL1=L2+φ.
(Here,φis interpreted as a function onT M)
(b) Show that such Lagrangians share the same critical values.
3. Natural lift to the tangent bundle
Let X be vector field on a smooth manifoldM, with local flow φt. Each diffeomorphism φt induces a diffeomorphism ofT M given by
Φt(q,q) = (φ˙ t(q), dφt(q)·q),˙ For ˙q∈TqM.
Suppose that in some coordinate chartq:U →Rn, the vector field X has the expression
X=ai(q) ∂
∂qi.
Show that the vector field X0 generating the flow Φt can be written in canonical coordinates on T U associated to this chart,
X0=ai ∂
∂qi + ˙qj∂ai
∂qj
∂
∂q˙j
(Remember that these notations with indicesi, jhide implicit summation. As an example,ai ∂
∂qi actually meansP
iai ∂
∂qi)
4. The Two Body Problem
Consider a pair of point masses (with massesm1 andm2) which move freely subject to a force between them which depends only on the distance between the two bodies and is directed along the line joining the two bodies. It can be represented by a Lagrangian on the manifold M = Rn ×Rn with position coordinates q1, q2:M →Rn of the form
L(q1, q2,q˙1,q˙2) = m1
2 |q˙1|2+m2
2 |q˙2|2−V(|q1−q2|2).
(Here, ( ˙q1,q˙2) are the canonical fiber (velocity) coordinates onT M associated to the coordinate system (q1, q2))
(a) Show that rotations and translations inRn generate a group of symmetries of this Lagrangian and compute the conserved quantities.
(b) What is the interpretation of the conservation law associated to the translations ?
Note: I will give a global mark between 0 and 3 for the whole problem set.
