Classical Dynamics Coursework Notes

Classical Dynamics (5CCM231)

Instructors and Module Information

• Course Instructors: Paul P. Cook, Neil Lambert, Chiara Cammarota, Christopher P. Herzog.

• Department: Mathematics, King’s College London.

• Note: Module was previously named Intermediate Dynamics, but the content remains unchanged.

• Email contacts provided for instructors.

Contents Overview

1. Review (Page 5)

   • Scalars, Vectors and their properties.

   • Vector Product and associated properties.

   • Triple Products.

   • Matrices and their applications.

   • Understanding Derivatives vs. Partial Derivatives.

   • Co-ordinate systems in physics.

2. Newton and His Three Laws (Page 17)

   • Introduction to Newton’s Laws of Motion.

   • Application of Newton’s Laws through examples including skiing.

   • Analysis of frictional forces, angular motion, work, and conservation laws.

   • Exploration of celestial motion and Kepler’s Laws.

3. Multi-Particle Systems and Rigid Body Motion (Page 43)

   • Understanding multi-particle systems.

   • Rigid body motion analysis.

   • Gyroscopes and Euler's Equations.

4. Solving the Two-Body Problem (Page 59)

   • Introduction to two-body problem dynamics.

5. Lagrangian Mechanics (Page 65)

   • Principle of Least Action.

   • Understanding generalized coordinates and constructing Lagrangian.

   • Application to different physical systems.

6. Hamiltonian Mechanics (Page 101)

   • Understanding Hamilton's equations and the concept of phase space.

   • Poisson Brackets and Canonical Transformations.

Chapter 1: Review

1.1 Scalars, Vectors and all That

• Scalars: A scalar is defined as a single number. Example: Temperature at a point T = 20°C.

• Vectors: Vectors are quantities defined in $ ext{R}^3$, represented by a triplet of numbers (x, y, z) denoting position. Examples of vector representation includes:

  • Position vector: r = ⎛ x \ y \ z ⎞ statement

  • Index notation: $r_a$ $ ext{,}$ where $a=1,2,3$.

• Vector Space: Is defined by two operations,

  • Scalar multiplication by value a: $ar = ⎛ ax \ ay \ az ⎞$.

  • Vector addition: $r<em>1 + r</em>2 = ⎛ x<em>1+y</em>1 \ y<em>1+y</em>2 \ z<em>1+z</em>2 \ ⎞$.

• Basis of R^3: Vectors can be expressed in terms of a basis, e1, e2, e3, where a corresponding vector r can be represented uniquely as:

  • $r = ae<em>1 + be</em>2 + ce_3$.

• Scalar Product/Dot Product: Defined by:

  • $r<em>1 ⋅ r</em>2 = ext{sum}(r<em>{1a}r</em>{2a})$ where $r<em>{1a}, r</em>{2a}$ are components of vectors.

1.2 Vector Product

• The vector product (cross product) is defined for non-parallel vectors in $ ext{R}^3$ and outputs a vector orthogonal to both input vectors.

  • Formula for vector product output:

  • $(v × w)_a = egin{pmatrix}… ext{formatted equation}… ext{including properties and illustrative examples will go here}.\ ext{Full explanations of interaction, orthogonality effects and signs interpretation.} \ ext{Figures illustrating vector interactions, parallelogram area, etc.}

… continue expanding other chapters in the same detailed format referencing equations, examples, illustrations, and important notes as seen above …

Classical Dynamics (5CCM231)

Instructors and Module Information

• Course Instructors: Paul P. Cook, Neil Lambert, Chiara Cammarota, Christopher P. Herzog.

• Department: Mathematics, King’s College London.

• Note: Module was previously named Intermediate Dynamics, but the content remains unchanged.

• Email contacts provided for instructors.

Contents Overview

1. Review (Page 5)

   • Scalars, Vectors and their properties.

   • Vector Product and associated properties.

   • Triple Products.

   • Matrices and their applications.

   • Understanding Derivatives vs. Partial Derivatives.

   • Co-ordinate systems in physics.

2. Newton and His Three Laws (Page 17)

   • Introduction to Newton’s Laws of Motion.

   • Application of Newton’s Laws through examples including skiing.

   • Analysis of frictional forces, angular motion, work, and conservation laws.

   • Exploration of celestial motion and Kepler’s Laws.

3. Multi-Particle Systems and Rigid Body Motion (Page 43)

   • Understanding multi-particle systems.

   • Rigid body motion analysis.

   • Gyroscopes and Euler's Equations.

4. Solving the Two-Body Problem (Page 59)

   • Introduction to two-body problem dynamics.

5. Lagrangian Mechanics (Page 65)

   • Principle of Least Action.

   • Understanding generalized coordinates and constructing Lagrangian.

   • Application to different physical systems.

6. Hamiltonian Mechanics (Page 101)

   • Understanding Hamilton's equations and the concept of phase space.

   • Poisson Brackets and Canonical Transformations.

Chapter 1: Review

1.1 Scalars, Vectors and all That

• Scalars: A scalar is defined as a single number. Example: Temperature at a point $T = 20^{\circ}\text{C}$.

• Vectors: Vectors are quantities defined in $\text{R}^3$, represented by a triplet of numbers $(x, y, z)$ denoting position. Examples of vector representation includes:

  • Position vector: $r = \begin{pmatrix} x \ y \ z \end{pmatrix}$

  • Index notation: $r_a$, where $a=1,2,3$.

• Vector Space: Is defined by two operations,

  • Scalar multiplication by value $a$: $ar = \begin{pmatrix} ax \ ay \ az \end{pmatrix}$.

  • Vector addition: $r<em>1 + r</em>2 = \begin{pmatrix} x<em>1+x</em>2 \ y<em>1+y</em>2 \ z<em>1+z</em>2 \end{pmatrix}$.

• Basis of R^3: Vectors can be expressed in terms of a basis, $\mathbf{e}<em>1, \mathbf{e}</em>2, \mathbf{e}_3$, where a corresponding vector $\mathbf{r}$ can be represented uniquely as:

  • $\mathbf{r} = a\mathbf{e}<em>1 + b\mathbf{e}</em>2 + c\mathbf{e}_3$.

• Scalar Product/Dot Product: Defined by:

  • $\mathbf{r}<em>1 \cdot \mathbf{r}</em>2 = \sum<em>{a} (r</em>{1a}r<em>{2a})$, where $r</em>{1a}, r_{2a}$ are components of vectors.

1.2 Vector Product

• The vector product (cross product) is defined for non-parallel vectors in $\text{R}^3$ and outputs a vector orthogonal to both input vectors.

  • Formula for vector product output:

  • Given vectors $\mathbf{v} = (v<em>x, v</em>y, v<em>z)$ and $\mathbf{w} = (w</em>x, w<em>y, w</em>z)$, their cross product is:
    $\mathbf{v} \times \mathbf{w} = \begin{pmatrix} v<em>y w</em>z - v<em>z w</em>y \ v<em>z w</em>x - v<em>x w</em>z \ v<em>x w</em>y - v<em>y w</em>x \end{pmatrix}$

  • In index notation, using the Levi-Civita symbol $\epsilon<em>{abc}$: $(\mathbf{v} \times \mathbf{w})</em>a = \sum<em>{b,c} \epsilon</em>{abc} v<em>b w</em>c$

  • Properties:

    • The resultant vector $\mathbf{v} \times \mathbf{w}$ is orthogonal to both $\mathbf{v}$ and $\mathbf{w}$.

    • Its magnitude is $|\mathbf{v} \times \mathbf{w}| = |\mathbf{v}||\mathbf{w}|\sin\theta$, where $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$. This magnitude represents the area of the parallelogram formed by $\mathbf{v}$ and $\mathbf{w}$.

    • Anticommutativity: $\mathbf{v} \times \mathbf{w} = -(\mathbf{w} \times \mathbf{v})$.

    • Distributivity over vector addition: $\mathbf{u} \times (\mathbf{v} + \mathbf{w}) = (\mathbf{u} \times \mathbf{v}) + (\mathbf{u} \times \mathbf{w})$.

… continue expanding other chapters in the same detailed format referencing equations, examples, illustrations, and important notes as seen above …