Month: August 2016

[Reading Notes] Mechanics by Landau and Lifshitz

The following is an excerpt from [\S 43 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.

Aside. Interpretation of the action in classical mechanics

捕获

 

Note that in the derivation above, the variations \delta p and \delta q are regarded as independent.* Actually, \delta q is arbitrary but \delta p is not, even though p, q are both independent variables. Since p in connected with \dot{q} and  \delta p and \delta \dot{q} are not independent.

Notice that before (43.8) is derived, we have applied Ledrendre  Transformation which requires that

\dot{q}=\frac{\partial H}{\partial p}.                                               (1)

So the coefficient of \delta p is  0,  and \delta q is arbitrary, so its coefficient must be 0, Hence we get another Hamilton’s  equation

\dot{p}=-\frac{\partial H}{\partial q}.                                              (2)

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  p=\frac{\partial L}{\partial \dot{q}}. From the definition of Hamiltonian

H=\Sigma \dot{q}p-L

and \dot{q}=\dot{q}(p,q,t), then

捕获

With the definition of p, p=\frac{\partial L}{\partial \dot{q}}, we have

\dot{q}=\frac{\partial H}{\partial p}.

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 \delta S, (2) can be derived.

*We should notice that variations here are simultaneous variations and it’s for a complete system.

 

 

 

[Solutions] Ch 2 Manifolds in “Spacetime and geometry”

These are parts of the solutions to exercises from:

Carroll, S. M. (2005). Spacetime and geometry: An introduction to general relativity. Addison Wesley.

12345

Solutions:

  1. First consider mapping the infinite cylinder to a semi-infinite cylinder, then consider projecting points on it to a plane.Capture

Let L=1,

(acos\theta,asin\theta,z)\mapsto ^f (acos\theta,asin\theta,e^z)\mapsto ^g (acos\theta/e^z,asin\theta/e^z,1).

g\circ f is the map desired whose image is an open set R^2\backslash (0,0).

2. No. \mathbb{R}^k must be k-dimension manifolds. Furthermore, the dimension of a manifold is unique.

Suppose that U is a subset of a manifold M, if there are two charts with different dimension m and $latex n$, then there exists \Phi_1, \Phi_2\in C^{\infty}, \Phi_1:U\to \mathbb{R}^m, \Phi_2:U\to \mathbb{R}^n. Then \Phi_2\circ \Phi_1 is a diffeomorphism from \Phi_1(U) to \Phi_2(U). Hence m must be equal to n.

Note: Here I use such an assertion( J. Milnor, Topology from Differentiable Viewpoint, P4).

Assertion. If f is a diffeomorphism between open sets U\subset R^k and V\subset R^l, then k must equal l, and the linear mapping

df_x:R^k\to R^l

must be nonsingular.

Proof. The composition f^{-1}\circ f is the identity map of U; hence d(f^{-1})_v\circ df_x is the identity map of R^k. Similarly df_x\circ d(f^{-1})_v is the identity map of R^l. Thus df_x has a two-sided inverse, and it follows that k=l.

By the way, k=l can also be seen from the max\{rank(BA), rank(AB)\}=min\{rank(A),rank(B)\}, A,B denote the linear map d(f^{-1})_v and df_x respectively.

3. Trivial.

4. First two can be verified directly.

Composition formula:

[X,Y]^{\mu}=X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu}.

[X,Y]^{\mu}=[X,Y](x^{\mu})=X(Y(x^{\mu}))-Y(X(x^{\mu}))=X(Y^{\mu})-Y(X^{\mu})\\ =X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu}.

Transformation:

[X,Y]^{\mu'}=X^{\mu'}\partial_{\lambda'}Y^{\mu'}-Y^{\lambda'}\partial_{\lambda'}X^{\mu'}\\ =X^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}}Y^{\mu})-Y^{\lambda}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}}X^{\mu})\\ =X^{\lambda}Y^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}})+\frac{\partial x^{\mu'}}{\partial x^{\mu}}x^{\lambda}\partial_{\lambda}Y^{\mu}-X^{\lambda}Y^{\mu}\partial_{\lambda}(\frac{\partial x^{\mu'}}{\partial x^{\mu}})-\frac{\partial x^{\mu'}}{\partial x^{\mu}}Y^{\lambda}\partial_{\lambda}X^{\mu} \\ = \frac{\partial x^{\mu'}}{\partial x^{\mu}}(X^{\mu}\partial_{\lambda}Y^{\mu}-Y^{\lambda}\partial_{\lambda}X^{\mu})= \frac{\partial x^{\mu'}}{\partial x^{\mu}}([X,Y]^{\mu}).

5. Set  V_{1}=\partial _{x}, V_{2} = A(x, y)\partial _{x} + B(x, y)\partial _{y}. The commutator will be given by

[V_{1} , V_{2} ] = [\partial _{x} A(x, y)]\partial _{x} + [\partial _{x} B(x, y)]\partial _{y}.

Set V_{2} =x \partial _{x} + \partial _{y}. Then V_1, V_2 are nowhere vanishing, and their commutator is nowhere vanishing as well.

6.

(a)

r=\sqrt{x^2+y^2+z^2}=\sqrt{1+\lambda^2}

\theta=arccos \frac{z}{\sqrt{x^2+y^2+z^2}}=arccos\frac{\lambda}{\sqrt{1+\lambda^2}}

\phi=\lambda

(\lambda\in [0,2\pi))

(b)

Cartesian: (-sin\lambda, cos\lambda,1),

Polar:(\frac{\lambda}{\sqrt{1+\lambda^2}},\frac{-1}{1+\lambda^2},1).

7.

(a) \frac{\partial x^{\mu}}{\partial x^{\nu'}}=\frac{\partial(x,z)}{\partial(\chi,\theta)}=\left(\begin{array}{cc} {cosh\chi sin\theta}&{sinh\chi cos\theta}\\{sinh\chi cos\theta}&{-cosh\chi sin\theta} \end{array}\right)
(b)dx=cosh\chi sin\theta dx+sinh\chi cos\theta d\theta,
dz=sinh\chi cos\theta dx-cosh\chi sin\theta d\theta,
ds^2=(sin^2\theta+sinh^2\chi)(dx^2+d\theta^2).

8.
d(\omega\wedge\eta)_{\mu_1\dots\mu_{p+q+1}}\\=\frac{(p+q)!(p+q+1)}{p!q!}\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}\\=\frac{(p+q)!(p+q+1)}{p!q!}[(\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}).\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}+\omega_{[\mu_{2}\dots\mu_{p+1}}.(\partial_{[\mu_1}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]})]\\=\frac{(p+q)!(p+q+1)}{p!q!}[(\partial_{[\mu_1}\omega_{[\mu_2\dots\mu_{p+1}}).\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]}+(-1)^p\omega_{[\mu_{1}\dots\mu_{p}}.(\partial_{[\mu_{p+1}}\eta_{\mu_{p+2}\dots\mu_{p+q+1}]]})]\\=(d\omega)\wedge\eta+(-1)^p\omega\wedge(d\eta)

9.

(a)d*F=3\partial_{[\mu}F_{\nu\rho ]}=1/2[\partial_{\mu}(\epsilon^{\nu_1\nu_2}_{\nu\rho}F_{\nu_1\nu_2})+\partial_{\nu}(\epsilon^{\nu_1\nu_2}_{\rho\mu}F_{\nu_1\nu_2})+\partial_{\rho}(\epsilon^{\nu_1\nu_2}_{\mu\nu}F_{\nu_1\nu_2})]=\epsilon^{\sigma}_{\mu\nu\rho}J_{\rho}=*J
(\partial_{\mu}F^{\nu\mu}=J^{nu})
(b)F=-(**F)
*F=\left( \begin{array}{cccc} 0&0 &0 &0 \\0 &0 &0 &0 \\0  &0 &0 &qsin\theta\\ 0&0 &-qsin\theta&0 \end{array}\right)
F=(- **F)=\left( \begin{array}{cccc} 0&-q/r^2 &0 &0 \\q/r^2 &0 &0 &0 \\0  &0 &0 &0\\ 0&0 &0&0 \end{array}\right)
(c)E_r=q/r^2,E_{\theta}=E_{\phi}=0,
B_{\mu}=0, \mu=r,\theta,\phi
(d)0

 

 

 

[Solutions]Ch 1 Special Relativity and Flat Spacetime

1-6

https://petraaxolotl.wordpress.com/chapter-1-special-relativity-and-flat-spacetime/

9. For a system of discrete point particles the energy-momentum tensor take the form

T^{\mu \nu}=\Sigma_a \frac{p_{\mu}^{(a)} p_{\nu}^{(a)} }{p^{(a)}}\delta^{(3)}(x-x^{(a)}),

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

T^{\mu\nu}=(\rho +p)U^{\mu}U^{\nu} + p\eta^{\mu\nu}.

 

Doing average over 4-volume,

{\bf T}^{\mu\nu}=\dfrac{1}{\Delta V_4}\int_{\Delta V_4}T^{\mu\nu}d V=\dfrac{1}{\sqrt{-g} d^3x^i dx^0}\int_{\Delta V_4}T^{\mu\nu} \sqrt{-g} d^3x^i dx^0 .

Then a) in {\bf T}^{\mu\nu} 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.

{\bf T}^{\mu\nu}=\dfrac{1}{d^3x^i dx^0}\sum_a\dfrac{p_a^\mu p_a^\nu}{p_a^0}\int_{\Delta V_4} \delta^{(3)}({\bf x}-{\bf x}^{(a)}) d^3 x^i dx^0 = \dfrac{1}{d^3x^i}\sum_a\dfrac{p_a^\mu p_a^\nu}{p_a^0}

In the last expression the sum is taken over the particles which have worldlines passing through \Delta V_4.

Consider symmetry,

{\bf T}^{0 0} = \dfrac{1}{d^3x^i}\sum_a p_a^0 \equiv \rho

{\bf T}^{i 0} = \dfrac{1}{d^3x^i}\sum_a p_a^i. Because of isotropy, {\bf T}^{i 0}= 0.

{\bf T}^{i j} = \dfrac{1}{d^3x^i}\sum_a\dfrac{p_a^i p_a^j}{p_a^0}. 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: {\bf T}^{i j} \equiv P \delta^{ij}.

Replace ：

\delta ^{ij}\to \eta^{ij}+U^i U^j

T^{00}\to \rho U^0U^0,

then we get the result.

10. Using the tensor transformation law applied to F_{\mu\nu}, 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.

 

(a) \Lambda^{\mu'}_{\nu}=\left(\begin{array}{cccc} 1&0&0&0 \\ 0&cos\theta&0&sin\theta\\ 0&0&1&0\\ 0&-sin\theta&0&cos\theta \end{array}\right)

(b)\Lambda^{\mu'}_{\nu}=\left(\begin{array}{cccc} cosh\theta&0&0&-sinh\theta \\ 0&1&0&0\\ 0&0&1&0\\ -sinh\theta&0&0&cosh\theta \end{array}\right)

\overline{F}^{\mu\nu}=\Lambda F \Lambda^T=\left(\begin{array}{cccc}                            0& - E1*cosh(x) - B2*sinh(x)& B1*sinh(x) - E2*cosh(x)& E3*sinh(x)^2 - E3*cosh(x)^2\\ E1*cosh(x) + B2*sinh(x)&                     0&                      B3&- B2*cosh(x) - E1*sinh(x)\\ E2*cosh(x) - B1*sinh(x)&                       -B3&                       0&     B1*cosh(x) - E2*sinh(x)\\ E3*cosh(x)^2 - E3*sinh(x)^2&   B2*cosh(x) + E1*sinh(x)&E2*sinh(x) - B1*cosh(x)&                           0 \end{array}\right).

13. Consider adding to the Lagrangian for electromagnetism an additional term of the form \mathcal{L}'=\tilde{\epsilon}_{\mu\nu\rho\sigma}F^{\mu\nu}F{\rho\sigma}

(a) Express \mathcal{L}' in terms of E and B.

(b) Show that including \mathcal{L}'  does not affect Maxwell’s equations. Can you think of a deep reason for this?

 

(a) \tilde{\epsilon}_{\mu\nu\rho\sigma}F^{\mu\nu}F{\rho\sigma}=\tilde{\epsilon}_{ijk}(F^i F^{jk}-F^{i0}F^{jk}+F^{ij}F^{0k}-F^{ij}F^{k0})=4F^{0i}F^{jk}\tilde{\epsilon}^{ijk}=-8\vec{E}\vec{B}

(b) The expression in (a) can be expressed by 4-dimension divergence:

\tilde{\epsilon}_{\mu\nu\rho\sigma}F^{\mu\nu}F{\rho\sigma}=4\partial_i[\tilde{\epsilon}^{ijlm}A_k (\partial_l A_m)]

Integrate \mathcal{L}' over 4-dimension space, according to Stokes theorem, we can get the additional term in Lagrangian L. This term vanishes when varying action S