Cynosure对《Mechanics (Berkeley Physics Course, Vol. 1)》的笔记(1)

第1页

- 现在有点儿能理解有人说的“费曼物理讲义很好，但不太适合做教材”。BPC配以FLP最好。
- 这个版本挺好的，同时讲解高斯单位制和国际单位制

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The homogeneity and isotropy of euclidean space can be expressed by two invariance principles, which, in turn, imply two fundamental conservation principles.

Invariance under Translation: empty space is homogeneous --> conservation of linear momentum

Invariance under Rotation: empty space is isotropic to high precision, so that all directions are equivalent --->conservation of angular momentum

CH2
A^: A hat(or A caret): a unit vector in the direction of vector A

Finite Rotations Are Not Vectors

form invariant: the magnitude of a vector is the same in all cartesian coordinate systems that differ by a rigid rotation of the coordinate axis

|v> = d|r>/dt = dr/dt* r^ + r*dθ/dt *θ^

cgs system: velocity(cm/s), acceleration(cm/s^2), mass(grams, g), force(dynes, dyn)
(cgs: centimeter,  gram, second)
SI system:velocity(m/s), acceleration(m/s^2), mass(kg), force(N)

(A x B) x C = (AC)B - (BC)A
A x (B x C) = (AC)B - (AB)C
(A x B)(C x D) = (AC)(BD) - (AD)(BC)
(A x B) x (C x D) = [A(B x D)]C - [A(B x C)]D
A x [B x (C x D)] = (A xC)(BD) - (A x D)(BC)
---------------------------------------
CH3
Newton's Second Law: F = K*d(Mv)/dt     (We choose our units so that K = 1)

In ultimate analysis all contact forces are field forces, for they arise from electromagnetic interactions between atomic particles.

dimensional analysis:量纲分析

gravitational mass ： the mass in the gravitational equation is called the, inertial mass ：the mass in Newton's Second Law is called the.
---------------------------------------------------
CH 4
It is important to realize that momentum conservation applies even to inelastic collisions, in which the kinetic energy is not conserved.

【inertial frame (or galilean frame)】: a reference frame which is without acceleration or rotation
Our ability to say whether or not a particular reference frame is an inertial frame will depend in a strict sense upon the precision with which we can detect the effects of a small acceleration of the frame. In a practical sense, a reference frame in which no acceleration is observed for a particle believed to be free of any force and constraint is taken to be an inertial frame.

But to obtain a valid law in the simple form of Eq. (4.1) or (4.2) we must refer the acceleration to a
reference system that is unaccelerated, an inertial or galilean frame.

We do not know by observation that the sun is not accelerating much faster than this, nor do we know
that the center of the galaxy is not itself accelerating significantly. We do know in practice that the set of assumptions central to classical mechanics works out exceedingly well. These assumptions are:
1 Space is euclidean.
2 Space is isotropic, so that physical properties are the same in all directions in space. Thus the mass M in F=Ma does not depend on the direction of a.
3 Newton's Laws of Motion hold in an inertial system determined for an observer at rest on the earth by taking account of only the acceleration due to the rotation of the earth about its axis and due to the motion of the earth in its orbit around the sun.
4 Newton's law of universal gravitation is valid. A brief discussion of this law was given in Chap. 3 (pages 65-67), and a more detailed discussion is given in Chap. 9.
These assumptions are difficult to test individually to great
precision. The most precise tests, which relate to the motions
of the planets in the solar system, usually involve the entire
package of all four statements above.


牛顿第二定律的适用范围是惯性参考系。

Galilean invariance:(it is a fundamental hypothesis)
The basic laws of physics are identical in all reference
systems that move with uniform (unaccelerated) velocity
with respect to one another.

如何建立一个惯性参考系：p109-p110

conservation of energy + conservation of mass + Galilean invariance ==> conservation of momentum
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
从P120的推导来看，在 conservation of mass + Galilean invariance的前提下，conservation of energy等价于conservation of momentum，即，假设质量守恒，在不同的惯性坐标系下，动能守恒和动量守恒要么同时成立、要么同时不成立。
但P136-137里说：the conservation of linear momentum could be interpreted as a
direct consequence of the principle of galilean invariance。
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<

CH5----------------------------
If all the forces in a problem are known, and if we are clever enough and have computers of adequate speed and capacity to solve for the trajectories of all the particles, then the conservation laws give us no additional information. But since we do not have all this information and these abilities and facilities, the conservation laws are very powerful tools.
Why?(P136)

the law(conservation of energy) gives us no new information not contained in the equations of motion |F> = M|a>

conservation of energy: dE/dt ≡ d(K+U)/dy = dK/dy – Fy = 0  (|Fy> = M|ay>)

The hamiltonian formulation
of mechanics is one way that is very well suited
to reinterpretation in the language of quantum mechanics.

K≡1/2*mv^2中的1/2是由于微积分运算法则产生的。(P146)

if friction is a manifestation of
fundamental forces and they are conservative, how can friction be nonconservative? This is a matter of the detail of our analysis. If we analyze all motion on the atomic level, that of fundamental forces, we shall find the "motion" conservative; but if we see some of the motion as heat, which is useless in the
mechanical sense, we shall consider that friction has acted.

Fag: the work force against the field forces

|F> = -grad(U) = ∇U
极坐标下： grad≡∇≡|r>*∂/∂r + |θ>*1/r*∂/∂θ

PD: potential difference (or voltage drop)

electron volt (eV): the potential energy
difference of a charge e between two points having a potential
difference of one volt, or as the kinetic energy gained by a charge
e in falling through a potential difference of one volt.

internal forces: interparticle forces

Rc.m. : center mass

The velocity of the center of mass is constant in the absence of external forces: Rc.m.'= const。
In the presence of external forces the vector acceleration of the center of mass is equal to the vector sum of the external forces divided by the total mass
of the system: (∑Mi)*Rc.m.'' = Fext

angular momentum |J>≡ |r> cross |p>    (|r>: 质点到参考点的向量)
torque (or moment of force) |N>≡ |r> cross |F>    (|r>: 质点到参考点的向量)
d|J>/dt = |N>
Angular momentum is constant in the absence of torque; this is a statement of the law of conservation of angular momentum. (N=0 ==> J=const)

d|Jtot>/dt = |Next>
|Jtot> = |Jc.m.> + |Rc.m.> cross |P>

invariance of the potential energy under rotation of the reference frame==>conservation of angular momentum.
For N particles the rotational invariance of the potential is ensured if U depends only on the magnitudes of the separations between the several particles.
CH7---------------------------------------
AC：alternating current: 交流电
harmonic oscillator:
E = <K>+<U>
E = <E>, 
<K>=<U>

We usually omit the <> on P(t) when the meaning is clear.
relaxation time: τ=M/b
b: damping coefficient
Q: quality factor: 2π * energy_stored/<energy_loss_per_period>

CH 8------------------------------
K = ½ * Iz *ω^2 =  ½ * (∑mi*|ri>)*ω^2 
Iz≡∑mi*|ri> : moment of inertia

CH10--------------------------------
Many other experiments suggest, as this one does, that c is the upper limit to the velocity of particles. Thus we believe firmly that c is the maximum signaling speed with either particles or electromagnetic waves: c is the ultimate speed.
.
c is the maximum speed at which energy can be transmitted.

β≡v/c, γ≡1/sqrt(1-β^2)

CH 11---------------------------
It is believed that the special theory of relativity gives a good description of the circular (accelerated) motion of particles in a magnetic field.

|p> = M*|v>*γ =  (M*γ)*|v> ≡ M(v) * |v>
total relativistic energy E≡M*c^2 γ = M(v)*c^2
Mass and energy are just different names for the same quantity

E'^2 – p'^2*c^2 =E^2 – p^2*c^2 = M^2*c^4

conservation of energy: ∑Ei = const
conservation of momentum: ∑|pi> = const

|p> = |v> * E/c^2
CH 14----------------
General Relativity

Mi: inertial mass. Mi≡M(2) = a(1)/a(2)*M(1)
Mg: gravitational mass: Mg ≡ F*Re^2/(G*Me)    (= F/g)

Mi(1)/Mg(1) = Mi(2)/Mg(2) (it is constant as long as this ratio of masses is constant)
So, Mi/Mg = const

Q≡ Mg/Mi

The experimental result that no difference has ever been detected between the inertial mass and the gravitational mass of a body suggests that gravitation in a sense may be equivalent
to acceleration. Consider an observer in an elevator which is freely falling with the acceleration g.
首先，实验结果倾向于认为Mi和Mg是相等的。
然后，电梯实验表示Mi和Mg是无法区别的。所以，基于此，爱因斯坦着手发展GR。


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REF
- O. Struve, B. Lynds, and H. Pillans, "Elementary Astronomy,"
- Larousse Encyclopedia of Astronomy," Prometheus Press,





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