Notes on Mechanics of General Physics

## Statics

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

Notes on Statics

## Kinematics

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

## Dynamics

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 The term has largely been superseded by analytical dynamics, or simply dynamics.

### Newtonian mechanics

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

Dynamic equations

Case studies:

• Particles
• Oscillation:
• Rigid body:

### Lagrangian and Hamiltonian mechanics

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.)