Classical Mechanics
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What are the fundamental elements of study? (e.g., particles, rigid bodies, continua, fields)
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What are the principal methods of analysis? (e.g., Newtonian, Lagrangian, Hamiltonian, variational, and geometric methods)
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What major formulations were developed? (e.g., Newtonian laws, Lagrangian mechanics, Hamiltonian mechanics, Hamilton–Jacobi theory)
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What is the ontology underlying the formulation?
Formulation
| Formulation | Core Object | Mathematical Space | Dynamics | Ontological Focus |
|---|---|---|---|---|
| Leibnizian | Relational quantities (distances, rates) | Relational network of bodies (no absolute space) | Change of relations between bodies | Relations, no absolute background |
| Galilean | Inertial motion, relative velocity | Euclidean space + universal time, Galilean frames | Uniform motion in inertial frames, Galilean transformations | Relative space, absolute time |
| Newtonian | Forces & acceleration | ( \mathbb{R}^3 ) + absolute time | ( m \ddot{\mathbf{x}} = \mathbf{F} ) | Causal forces, absolute space |
| Lagrangian | Action, energies | Tangent bundle ( TQ ) | Euler–Lagrange equations | Configuration space geometry |
| Hamiltonian | Energy, symplectic form | Phase space ( T^*Q ) | Hamilton’s equations | Flow on energy manifold |
| Hamilton–Jacobi | Action potential ( S ) | Configuration space + scalar field | ( H(q, \partial S/\partial q, t) + \partial_t S = 0 ) | Scalar field encoding trajectories |
| Geometric | Vector field preserving ( \omega ) | Symplectic manifold ( (M, \omega) ) | ( i_{X_H}\omega = dH ) | Structure-preserving flow |
Organizing Abstraction(s)
Which are the fundamental conceptual abstractions used to structure descriptions of the physical world?
| Abstraction | Description | Case(s) |
|---|---|---|
| Particle | An idealized point-like object with mass but no spatial extent. | Newtonian mechanics of point masses |
| Rigid Body | Object whose internal distances remain fixed; captures translational & rotational motion. | Rotating wheel, spinning top |
| Continuum | Matter treated as continuously distributed; fields describe properties like stress, strain. | Fluid flow, elasticity of solids |
| Field | Spatially distributed quantity that can act on particles or bodies. | Electric, magnetic, gravitational fields |
| System | Collection of interacting components (particles, bodies, fields) with defined boundaries. | Planetary system, gas in a container |
| Wave / Oscillation | Disturbance propagating through a medium or field, carrying energy and information. | Sound waves, electromagnetic waves |
| Quantum State | Mathematical object describing probabilities of measurement outcomes in quantum systems. | Electron wavefunction, spin state |
| Point Mass | Simplified particle model focusing only on mass and position, ignoring size or shape. | Pendulum bob, orbiting satellites |
| Extended Body | Object with internal structure that may deform; generalization of rigid bodies. | Soft materials, bridges under load |