- 印刷错误不少;
- 数学之美;
- 用变分法推导薛定锷方程;
- 牛顿力学类似用几何方法来解决问题; 分析力学类似用代数方法来解决问题。
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The Variational Principles of Mechanics(1986)
scleronomic:
δL =dL
rheonomic:
δL != dL
“holonomic” condition（以微分形式给出的方程）
“non-holonomic” condition
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cH3
Postulate A:
“The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints.”
Since all the fundamental variational principles of mechanics, the principles of Euler, Lagrange, Jacobi, Hamilton, are but alternative mathematical formulations of d’Alembert’s principle, Postulate A is actually the only postulate of analytical mechanics, and is thus of fundamental importance.
The modification of the potential energy on account of the Lagrangian λ-method represents the potential energy of the forces which are responsible for the maintenance of the given auxiliary conditions.
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CH4
D’Alembert’s principle generalizes Postulate A from the field of statics to the field of dynamics,
Lagrangian function L = T- V (T:kinetic energy, V:potential energy)
if impressed forces is monogenic,
then Hamilton’s principle <==> d’Alembert’s principle
Hamilton's principle holds for arbitrary mechanical systems which are characterized by monogenic forces and holonomic auxiliary conditions.
generalized momentum p_i:
then:
total
energy Λ:
Λ(or H) and L are the most important scalar associated with a mechanical system.
kinosthenic=ignorable variables=speed coordinates=absent coordinates
inertia is an inborn quality of matter which can hardly be reduced to something still simpler.
5.6 summary:
principle of least action,
Jacobi's principle,
Fermat’s principle in optics
holonomic conditions == monogenic forces
scleronomic holonomic conditions == conservative monogenic forces.
non-holonomic conditions == polygenic forces
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active variables
passive variables
kinetic energy ~ velocity
potential energy ~ position
H ~total energy
One kind of bracket determines the other. Hence, if the Lagrange brackets are invariants of a canonical transformation, so are the Poisson brackets.
The great development from classical mechanics to wave mechanics is thus characterized by the following landmarks:
- Delaunay’s treatment of separable multiply-periodic mechanical systems;
- the Sommerfeld-Wilson quantum conditions Einstein’s invariant formulation of the quantum conditions;
- de Broglie’s resonance interpretation of Einstein’s quantum condition;
- Schroedinger’s logarithmic transformation from the phase function S to the wave function ψ
Hence we see that the problem of solving the equations of dynamics and the problem of finding the geodesics of a certain—in general non-Riemannian—manifold are equivalent.
We notice that the perpendicularity of the radius vector to the surface of a sphere is not a characteristic property of Euclidean geometry alone. It is an invariant property of any kind of metrical geometry.
In this geometry, space is homogeneous, i.e. the properties of space around every point are the same. At the same time, space is not isotropic, i.e. the properties of space depend on the direction. Figures can be freely translated, but generally not rotated, in this geometry.
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The time t has changed from an invariant to a covariant quantity, whereas the light velocity c has changed from a covariant to an invariant quantity.
The equivalence of reference systems came about from the fact that the space of physics was considered to be a Euclidean space of three dimensions, in which all points are equivalent (freedom of translation), and all directions are equivalent (freedom of rotation).
The invariance of the distance with respect to rectangular coordinate transformations has the immediate consequence that if s = 0 in one legitimate coordinate system, it remains zero in all legitimate coordinate systems. This means that the propagation of a light wave is an absolute phenomenon which remains the same in all legitimate reference systems. Einstein’s postulate of the invariance of the velocity of light is thus absorbed in the more comprehensive principle of the invariance of the line- element (93.8).
之所以定义为(xi,
yi, zi, t)(而不定义为(x,
y, z, ti)) 是为了和四元数对应起来
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Although Euler must have seen the weakness of Maupertuis’ argument, he refrained from any criticism, and refrained from so much as mentioning his own achievements in this field, putting all his authority in favour of proclaiming Maupertuis as the inventor of the principle of least action. Even knowing Euler’s extraordinarily generous and appreciative character, this self- effacing and self-denying modesty has no parallel in the entire history of science, which abounds in examples to the contrary. 欧拉人品好！
Hamilton called Lagrange the “Shakespeare of mathematics,” on account of the extraordinary beauty, elegance, and depth of the Lagrangian methods.
the basic feature of the differential equations of wave-mechanics is their self-adjoint character, which means that they are derivable from a variational principle.
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Little can be gained and a great deal lost in clarity if we try to operate with the tensor as a whole rather than its components.
This explains why an elastic structure can suddenly collapse; it may seem to do so because the limit of elasticity has been exceeded, but in actual fact it does so because of the highly non-linear nature of the equilibrium equations in the case of large displacements.
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用变分法推导薛定谔方程：
11.20节Noether’s principle.“The Schrodinger equation (cf. p. 279, if we include the time t, is derivable from the following Lagrangian:。。。This represents the conservation of the probabilistic electric charge in Schrödinger’s theory”
此外，8.8节: ““If we introduce de Broglie’s wave-length (88.10) into the amplitude equation (88.14), we obtain Schroedinger’s famous differential equation”。
¹²³⁴⁵⁶⁷⁸⁹ ᵃᵇᵈᵉᶠᵍʰⁱʲᵏˡᵐⁿᵒᵖ ʳˢᵗᵘᵛʷˣʸᶻ ᵅᵝᵞᵟᵠ ₀₁₂₃₄₅₆₇₈₉ ₐ ₑ ₕᵢⱼₖₗₘₙₒₚᵣₛₜᵤᵥ ₓ ᵦ
º¹²³⁴ⁿ₁₂₃₄·∶αβγδεζηθ ικ λ μ νξοπρστυφχψω
∽ ⊥ ∠ ⊙ ⊕ ⊗∈∩∪∑∫∞≡≠±≈＄㏒㎡㎥㎎㎏㎜
ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
ħ∂φ
∈⊂∂Δ∇∀∃e̅Ζ͏͏͏͏͏͏ Z̅
▹◃ ∧†⨯∙↑↓
┘˩⌋⎦┙┚┛
