Notes on Mechanics of General Physics

Statics, the study of (mechanical) equilibrium and its relation to forces.

Kinematics, the study of motion (position, velocity, acceleration).

Historically, the study of motion and its relation to forces has been called *kinetics*.
The term has largely been superseded by analytical dynamics, or simply dynamics.

Newton’s law includes nonconservative forces. It is only right non-relativistically.

Case studies:

- Particles
- Oscillation:
- Rigid body:

The fundamental laws in theoretical physics can be put in the form of a principle of stationary action.

Principle of stationary action(平稳作用量原理): The evolution of a system between two specified states at two specified times is a stationary point of the action functional. $\mathcal{S}: \mathbf{q}(t) \rightarrow \int_{t_1}^{t_2} \mathcal{L}(\mathbf{q},\dot{\mathbf{q}},t) \mathrm{d}t$. $$\frac{\delta \mathcal{S}}{\delta \mathbf{q}(t)}=0$$

**Hamilton's Principle** is Hamilton's formulation of the Principle of Stationary Action for classical mechanics, where the Lagrangian is the kinetic energy minus the potential energy.
$\mathcal{L} = \text{KE} - \text{PE}$

Generalized coordinates, generalized velocities, and **Lagrange's equations**:
Notes from Intermediate Classical Mechanics

$$\frac{d}{dt} \frac {\partial L}{\partial \dot{q}} = \frac {\partial L}{\partial q}$$

Or equivalently,

$$\mathrm{d}\mathcal{L} = \dot{p} \mathrm{d}q + p \mathrm{d}\dot{q} + \frac{\partial \mathcal{L}}{\partial t} \mathrm{d}t$$

Canonical coordinates, canonical momenta, and **Hamilton's equations**:
Notes from Advanced Dynamics

$$\mathrm{d}\mathcal{H} = -\dot{p} \mathrm{d}q + \dot{q} \mathrm{d}p + \frac{\partial \mathcal{H}}{\partial t} \mathrm{d}t$$

The **Hamiltonian** $\mathcal{H}(q,p,t)$ is the Legendre transformation of the **Lagrangian** $\mathcal{L}(q,\dot{q},t)$, while the coordinates and time are held constant and the canonical momenta are conjugate to the generalized velocities.

$$\mathcal{H} + \mathcal{L} = p \dot{q}$$

Hamiltonian often corresponds to the total energy of the system: $\mathcal{H} = \text{KE} + \text{PE}$.

Advantages of Hamiltonian mechanics over Lagrangian mechanics:

- Algebraic structure: symplectic manifold.
- Symmetry: The canonical coordinates and momenta hold nearly symmetric roles.
- Symmetry and conservation law:
- Symmetry in space: If a coordinate does not occur in the Hamiltonian, the corresponding momentum is conserved;
- Symmetry in time: If time does not occur in the Hamiltonian, the Hamiltonian is conserved.

(The parameters that define the configuration of a system are called generalized coordinates. The set of actual configurations of the system (i.e. configuration space) is a (embedded) manifold in the space of generalized coordinates. The set of position and momenta of a mechanical system forms the cotangent bundle of the configuration manifold.)