Milnor’s short text was used in the course “Seminar on Mathematical analysis II” in Beijing Normal University, 2015 which set the exercises as homework problems.
The following is an excerpt from [ The action as a function of coordinate]. I would like to comment on the formal derivation of Hamilton’s equations and the problems of independence of variations.
Note that in the derivation above, the variations and are regarded as independent.* Actually, is arbitrary but is not, even though p, q are both independent variables. Since in connected with and and are not independent.
Notice that before (43.8) is derived, we have applied Ledrendre Transformation which requires that
So the coefficient of is , and is arbitrary, so its coefficient must be , Hence we get another Hamilton’s equation
Notice that we only derive half of Hamilton’s equations from the procedure above.
Since we can not say that we derive Hamilton’s equations by applying Hamilton’s equations. In order to make this induction above complete, we have to give the proof of another half of Hamilton’s equations, that is (1).
(1) is related to the definition of . From the definition of Hamiltonian
and , then
With the definition of p, , we have
Hence the half part of Hamilton’sequations is derived.
In this way of deriving Hamilton’s equation, strictly, we first derive (1) from the definition of p, and then by applying (1) in , (2) can be derived.
*We should notice that variations here are simultaneous variations and it’s for a complete system.
These are parts of the solutions to exercises from:
Carroll, S. M. (2005). Spacetime and geometry: An introduction to general relativity. Addison Wesley.
- First consider mapping the infinite cylinder to a semi-infinite cylinder, then consider projecting points on it to a plane.
is the map desired whose image is an open set .
2. No. must be k-dimension manifolds. Furthermore, the dimension of a manifold is unique.
Suppose that is a subset of a manifold , if there are two charts with different dimension and $latex n$, then there exists , , . Then is a diffeomorphism from to . Hence m must be equal to n.
Note: Here I use such an assertion( J. Milnor, Topology from Differentiable Viewpoint, P4).
Assertion. If is a diffeomorphism between open sets and , then must equal , and the linear mapping
must be nonsingular.
Proof. The composition is the identity map of U; hence is the identity map of . Similarly is the identity map of . Thus has a two-sided inverse, and it follows that .
By the way, can also be seen from the , denote the linear map and respectively.
4. First two can be verified directly.
5. Set , . The commutator will be given by
Set . Then , are nowhere vanishing, and their commutator is nowhere vanishing as well.
9. For a system of discrete point particles the energy-momentum tensor take the form
where the index a labels the different particles. Show that, for a dense collection of particles with isotopically distributed velocities, we can smooth over the individual particle worldlines to obtain the perfect-fluid energy-momentum tensor
Doing average over 4-volume,
Then a) in only delta-functions depend on x, b) metric determinant g is a macroscopic quantity, is constant over selected volume and can also be taken away from the integral.
In the last expression the sum is taken over the particles which have worldlines passing through .
. Because of isotropy, .
. The sum should produce a symmetric macroscopic 3-tensor of second order. But all 3-tensors are defined by 3 eigenvectors. Because no preferred direction exits and no preferred directions correspond to the case when the matrix has all eigenvalues equal, that is when the matrix is proportional to kronecker delta. The coefficient of proportionality is the pressure: .
then we get the result.
10. Using the tensor transformation law applied to , show how the electric and magnetic field 3-vectors E and B transform under
(a) a rotation about the y-axis;
(b) a boost along the z-axis.
13. Consider adding to the Lagrangian for electromagnetism an additional term of the form
(a) Express in terms of E and B.
(b) Show that including does not affect Maxwell’s equations. Can you think of a deep reason for this?
(b) The expression in (a) can be expressed by 4-dimension divergence:
Integrate over 4-dimension space, according to Stokes theorem, we can get the additional term in Lagrangian . This term vanishes when varying action