Classical mechanics

Classical mechanics is one framework to the study of mechanics.

Vector description

Concise description of mechanical systems can be provided with vectors.

Reference frame

A reference frame is defined by an origin $O$, and the three unit vectors for each dimension of space: $\overrightarrow{x}$, $\overrightarrow{y}$, and $\overrightarrow{z}$.

It can be noted as:

$$R(O;\overrightarrow{x},\overrightarrow{y},\overrightarrow{z})$$

Conventionally, we choose unit vectors perpendicular to each other ($\overrightarrow{x}\bot\overrightarrow{y}\bot\overrightarrow{z}$), so each spatial direction is independent of another.

Objects in the reference frame

Motion: Position, velocity, accelaration

For an object (S) centered at point $P$, its position relative to reference frame $R$ can be described by its position vector $\overrightarrow{OP}=(\overrightarrow{x_P},\overrightarrow{y_P},\overrightarrow{z_P})$.

Then velocity, defined as the instantaneous change in position with respect to time, is then:

$$\overrightarrow{v_S}=\frac{d \overrightarrow{OP}}{dt}=(\overrightarrow{v_x},\overrightarrow{v_y},\overrightarrow{v_z})$$

Finally, acceleration, defined as the instantaneous change in velocity with respect to time, is then:

$$\overrightarrow{a_S}=\frac{d \overrightarrow{v_S}}{dt}=(\overrightarrow{a_x},\overrightarrow{a_y},\overrightarrow{a_z})$$

Mechanical contact: Force and moments

Any force $\overrightarrow{F}$ on object $(S)$ can be decomposed into component forces in each of the three independent spatial direction:

$$\overrightarrow{F}= \overrightarrow{F_x}+\overrightarrow{F_y}+\overrightarrow{F_z}$$

Therefore, it can be described as a vector:

$$\overrightarrow{F}= (F_x,F_y,F_z)$$

where $$F_x=\left\| \overrightarrow{F_x} \right\|$$ and so on.

As a result, if the force $\overrightarrow{F}$ is applied to point $A$ the moment $M(\overrightarrow{F}\to S)_B$ caused by such force at point $B$ is:

$$\overrightarrow{M}(\overrightarrow{F}\to S)_B=\overrightarrow{BA}\wedge \overrightarrow{F}$$

It can also be decomposed into components along three independent spatial directions:

$$\overrightarrow{M}=(M_x,M_y,M_z)$$

Key principles

Newton's 1st law

Newton's 1st law describes the phenomenon inertia: when the overall net external force is zero, the velocity of an object is maintained.

In vector notation, it can be written as:

\sum \overrightarrow{F}_{ext}=0 \Rightarrow \frac{d\overrightarrow{v}}{dt}=0

The state that $\sum \overrightarrow{F}_{ext}=0$ is known mechanical equilibrium.

Newton's 2nd law

Newton's 2nd law describes the relationship between force, mass, and acceleration: the acceleration is proportional to force and the inverse of mass.

In vector notation, it can be written as:

\sum\overrightarrow{F_{ext}}=m\overrightarrow{a}

Newton's 3rd law

Newton's 3rd law describes the reciprocity of force: when object $(S1)$ applies a force on object $(S2)$, $(S2)$ must apply a force opposite to $(S1)$ and equal in magnitude.

In vector notation, it can be written as:

\overrightarrow{F}(S2\to S1)=-\overrightarrow{F}(S1\to S2)

Conservation of momentum

For any closed system, the momentum, defined as the product of mass and velocity (m\overrightarrow{v}), is conserved before and after any mechanical interaction within the system (including collision).

In vector notation, for a closed system of $n$ objects, the principle can be expressed as:

\sum m_n\overrightarrow{v_n}=C

where $C$ is a constant.

Any change to momentum must be a direct result of an external force $\overrightarrow{F_{ext}}$ acting on a member of the system for a duration $t$. This change in momentum, known as impulse, satisfies the following relationship:

\Delta m\overrightarrow{v}=\int \overrightarrow{F_{ext}}dt

Any motion carries kinetic energy

The kinetic energy of an object of mass $m$ and velocity $\overrightarrow{v}$ is:

E_k=\frac{1}{2}m \left\| \overrightarrow{v} \right\|²

Any change in kinetic energy, is the result of an external force acting over a given distance:

\Delta E_k=\int \vec F_{ext} \cdot d\vec r

Conservation of energy

In a closed system, energy is conserved. It can be transformed from one form to another, but not created or destroyed:

\sum E=C

Gravity: any two mass attracts

Two point masses at points $A$ and $B$ exerts a gravitational force on each other:

\vec {F}_g(B\gets A)=G\frac{m_Am_B}{\left\| \overrightarrow{BA} \right\|^2}\widehat{BA}\\ \vec {F}_g(A \gets B)=G\frac{m_Am_B}{\left\| \overrightarrow{AB} \right\|^2}\widehat{AB}

According to shell theorem, spherically symmetric shapes can be approximated by point masses.

For real-world objects, whose shape may be irregular, if the distance between the object is much larger than the size of a given object ($\left\| \overrightarrow{AB} \right\|\gg R_A$), that object may be approximated as a point mass.

Otherwise, for arbitrary real-world objects:

\mathbf F_g(A \gets B) = G \int_A \int_B \rho_A(\mathbf r_A)\rho_B(\mathbf r_B) \frac{\mathbf r_B-\mathbf r_A}{|\mathbf r_B-\mathbf r_A|^3} \,dV_B\,dV_A

Symbol	Meaning
$\mathbf r_A$	position vector of an infinitesimal piece of body $A$.
$\mathbf r_B$	position of an infinitesimal piece of body $B$.
$\rho_A(\mathbf r_A), \rho_B(\mathbf r_B)$	density at those points.
$dV_A=dr_A^3, dV_B=dr_B^3$	infinitesimal volumes.
$dm_A=\rho_A dV_A$, $dm_B=\rho_B dV_B$	infinitesimal masses.
$\lvert \mathbf r_B-\mathbf r_A\rvert$	distance between the two infinitesimal pieces.
$\dfrac{\mathbf r_B-\mathbf r_A}{\lvert \mathbf r_B-\mathbf r_A\rvert^3}$	unit direction from $A$-piece to $B$-piece, divided by distance squared.

The potential energy as a result of gravitational attraction is known as gravitational potential energy.

See also: Two body problem