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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. 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 In 3D vectors

○ Cross product of unit vectors : (Note) 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 is similar to scalar product proof : Start with a simple situation and generalize later

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. 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 p_i q_i′ - L

○ p_i: Generalized momenta

○ q_i: 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 always corresponds to a Hermitian operator

⑸ Laplace-Beltrami Analysis

① Laplace-Beltrami operator

Input : 2016.06.26 21:05