Chapter 1. Dynamics: Vectors and Scalars
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1. Representation of Physical Quantities: Scalar
2. Representation of Physical Quantities: Vector
3. Representation of Physical Quantities: Tensor
4. Conversion of coordinate systems
5. Two Approaches of Newtonian Mechanics
1. Representation of Physical Quantities : Scalar
⑴ Definition: Physical quantity with magnitude only
① Etymology: Latin word “scalae” meaning ‘steps’ or ‘ladder’ → scalaris → scala
② Examples: Time, length, area, volume, speed, mass, temperature, work, energy
⑵ Free in arithmetic operations
2. Representation of Physical Quantities : Vector
⑴ Definition: Physical quantity with both direction and magnitude
① Etymology: Latin word “vehere” meaning ‘to carry’ → vectus → vector
② Examples: Force, displacement, velocity, momentum, impulse, electric field, magnetic field
⑵ Notation of Vectors
① When represented with arrows: Arrow length represents vector magnitude. Arrow direction represents vector direction.
② When represented with symbols
③ Absolute value notation
⑶ Operation 1: Vector Addition
① Methods: Parallelogram law, triangle law
② Property 1: Commutative law
③ Property 2: Associative law
⑷ Operation 2: Vector Subtraction
⑸ Operation 3: Scalar Multiplication of Vectors
⑹ Operation 4: Vector Resolution
① Definition: Decomposing each vector into component vectors along orthogonal coordinate directions
② Formulation
⑺ Operation 5: Scalar Product (Dot Product)
⑻ Operation 6: Vector Product (Cross Product)
① Definition of cross product: Operation satisfying three conditions for v, w ∈ ℝn:
○ v × w is a continuous function of the pair (v, w)
○ v × w is perpendicular to both v and w: (v × w) · v = (v × w) · w = 0
○ If v and w are linearly independent, then v × w ≠ 0
② Direction of cross product: follows right-hand rule.
Figure 1. Direction of cross product
③ In 3D vectors
○ Cross product of unit vectors: when operated in the order i → j → k, the minus sign does not appear
○ Fundamental formula: Cross product of unit vectors can be used for proof
○ Cross product direction: direction of right-hand thumb when curling from A to B
○ Non-commutative property
○ Volume of parallelepiped: Volume of parallelepiped formed by vectors a, b, c is as follows
○ BAC-CAB rule
○ Proof 1. Simple formula expansion
Figure 1. Proof of BAC-CAB rule.
○ Proof 2. The BAC–CAB rule holds for a particular case, and the validity of the identity is preserved under scaling transformations and rotations of the coordinate system. Therefore, the BAC–CAB rule always holds by mathematical induction (a principle proposed by the author).
\[\mathbf{A}=(1,0,0),\quad \mathbf{B}=(p,q,0),\quad \mathbf{C}=(t,0,s)\]○ A particular case: The BAC–CAB rule can be readily verified for the following vectors:
○ Invariance under scaling transformations: The validity of the BAC–CAB rule under scaling transformations follows directly from the fact that both dot and cross products are linear with respect to the magnitude of each vector.
○ Invariance under coordinate rotations: Since $\mathbf{B}\times\mathbf{C}$ and $\mathbf{A}\times(\mathbf{B}\times\mathbf{C})$ transform under coordinate rotations in the same manner as $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$, the validity of the BAC–CAB rule is preserved, thereby completing the proof.
\[\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}\]By applying the BAC–CAB rule, we obtain
③ Multidimensional vectors
○ Satisfying the definition of cross product is only possible when n = 3 or n = 7
○ Reference paper: Massey, W. S. (1983). Cross products of vectors in higher dimensional Euclidean spaces. The American Mathematical Monthly, 90(10), 697-701.
**3. Representation of Physical Quantities: Tensor
4. Conversion of coordinate systems
⑴ 📄 Summary
⑵ 📔 Book
5. Two Approaches of Newtonian Mechanics
⑴ Newtonian Analysis
① Mechanical analysis using vector quantities such as force and acceleration
⑵ Lagrangian Analysis
① Mechanical analysis using scalar quantities such as energy
② L (Lagrangian) = T (kinetic energy) - U (potential energy)
⑶ Hamiltonian Analysis
① Hamiltonian H = Σ pi qi′ - L
○ pi: Generalized momenta
○ qi: Generalized coordinate
② In a single-particle, single-system case, the Hamiltonian can be expressed as H = T (kinetic energy) + U (potential energy)
③ Hamiltonian in quantum mechanics
⑷ Hermitian Analysis
① Hermitian in quantum mechanics
② The Hamiltonian is always a Hermitian operator.
③ The eigenvalues of a Hermitian operator are always real.
\[\langle v,Av\rangle=\langle v,\lambda v\rangle=\lambda\langle v,v\rangle\]○ If $Av=\lambda v$, then
\[\begin{aligned} \langle v,Av\rangle &=\langle A^\dagger v,v\rangle\\\ &=\langle Av,v\rangle\\\ &=\langle\lambda v,v\rangle\\\ &=\lambda^\*\langle v,v\rangle \end{aligned}\]○ On the other hand, since $A$ is Hermitian,
\[\lambda=\lambda^\*\]○ Therefore,
○ Hence, all eigenvalues of a Hermitian operator are real.
⑸ Laplace–Beltrami Analysis
① Laplace–Beltrami Operator
○ Measures how much the value of a function at a point on a curved surface differs from its surrounding values.
\[\Delta f=\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}\]○ In flat Euclidean space, the Laplace–Beltrami operator reduces to the standard Laplacian:
⑹ Applications: Problem sets of operators
Input: 2016.06.26 21:05