NEWTONIAN AND HAMILTONIAN MECHANICS

MECHANICS

Study of the motion of a body and its causes

Quantitative laws

Dynamics of an ideal point

KINEMATICS

We just study the motion of a body, not its causes

Fundamental quantities: space - time - velocity - acceleration

Space ⮕ Velocity ⮕ Acceleration

v_{m} = \frac{\Delta S}{\Delta t};

v(t) = \lim\limits_{\Delta t\to 0} \frac{\Delta S}{\Delta t} = \frac{dS}{dt}

a(t) = \lim\limits_{\Delta t\to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}

a_{m} = \frac{\Delta v}{\Delta t};

\Delta S

S(t)

\frac{d}{dt}

Average

Istantaneous

Acceleration ⮕ Velocity ⮕ Space

\Delta v = a_{m} \Delta t

v(t) = v_{0} + \int_{0}^{t} a(t') \,dt'

\Delta S = v_{m}{\Delta t}

a_{m}

a(t)

\int_{0}^{t} dt'

Average

Istanteneous

\int_{0}^{t} dt'

s(t) = s_{0} + \int_{0}^{t} v(t') \,dt'

Uniformly accelerated motion

a(t) = a_{0}

NEWTON'S LAWS

FIRST NEWTON'S LAW

The Principle of Inertia

Every object perseveres in its state of rest, or of uniform motion in a right line, except insofar as it is compelled to change that state by forces impressed thereon.

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

NEWTON'S SECOND LAW

The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.

\, \vec{F}=m\vec{a} = \frac{d\vec{p}}{dt} \,

Law of motion

[\mathrm{Kg \cdot m \cdot s^{-2}}]

[N]

NEWTON'S THIRD LAW

To every action, there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts

\, \vec{F_{12}}=-\vec{F_{21}} \,

Law of action and reaction

WORK OF A CONSTANT FORCE

Along a straight line

\bullet \; L=\vec{F} \cdot \vec{s} = F \cdot x \cdot cos(\theta)

\bullet \; Joule \Rightarrow [Kg \cdot m^{2} \cdot s^{-2}]

WORK: GENERAL CASE

\bullet \; \Delta L_{1} = \vec{F}_{1} \cdot \Delta \vec{s}_{1}

\bullet \; \Delta L_{2} = \vec{F}_{2} \cdot \Delta \vec{s}_{2}

\bullet \; \ldots

\bullet \; \Delta L_{n} = \vec{F}_{n} \cdot \Delta \vec{s}_{n}

L=\lim _{\Delta s \rightarrow 0} \sum_i \vec{F}_i \cdot \Delta \vec{s}_1=\int_1^2 \vec{F} \cdot d \vec{s}

WORK: CONSERVATIVE FORCES

L_{A \rightarrow B}^{(1)}

L_{A \rightarrow B}^{(3)}

L_{A \rightarrow B}^{(2)}

L_{A \rightarrow B}^{(1)}

L_{A \rightarrow B}^{(n)} + L_{B \rightarrow A}^{(m)} = 0

L_{A \rightarrow B}^{(1)}

L_{A \rightarrow B}^{(3)}

L_{A \rightarrow B}^{(2)}

\neq

L_{A \rightarrow B}^{(1)}

L_{A \rightarrow B}^{(n)} + L_{B \rightarrow A}^{(m)} \neq 0

Conservative Forces

Non-Conservative Forces

A Conservative Force: the elestic Force

L_{A \rightarrow B}^{(n)} = L_{A \rightarrow B}^{(m)} = L_{A \rightarrow B}^{(l)}

L_{A \rightarrow B}^{(n)} = -\frac{1}{2}k(x^{2}_{B}-x^{2}_{A})

A non-conservative force: friction

L_{A \rightarrow B}^{(n)} + L_{B \rightarrow A}^{(m)} \neq 0

L_{A \rightarrow B} + L_{B \rightarrow A} = -2 \cdot F_{\mu_{d}} \cdot s

THE IDEA OF ENERGY

Energy, in physics, the capacity for doing work.

Forms of Energy

Kinetic
Gravitational
Elastic
Thermal
Chemical
Nuclear

THE PRINCIPLE OF CONSERVATION OF ENERGY

The total energy of an isolated system remains constant

If energy appears not to be conserved, the explanations lies in a source of dissipation that has not been accounted for

\Delta E_{tot}=0

E_{tot}=cost

KINETIC ENERGY

The kinetic energy of an object is the form of energy that it possesses due to its motion.

T = \frac{1}{2}mv^{2} = \frac{p^2}{2 m}

Work energy theorem

L = \Delta T = \frac{1}{2}m(v^{2}_{f}-v^{2}_{i})

CONSERVATIVE FORCES AND POTENTIAL ENERGY

Potential energy (U) is associated with forces that act on a body in a way that the total work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, whose total work is path independent, are called conservative forces.

L_{A \rightarrow B}=f(A, B) \equiv f(B)-f(A) \equiv

\equiv-(U(B)-U(A))=-\Delta U

POTENTIAL ENERGY AND CONSERVATIVE FORCES

U(x)=\frac{1}{2} k x^2

F_x \Delta x=-\Delta U \Rightarrow F_x=-\frac{\Delta U}{\Delta x} \Rightarrow F_x=-\frac{d U}{d x}

F_x=-\frac{d V}{d x}=-k x

CONSERVATION OF MECHANICAL ENERGY

In presence of conservative forces, the total mechanical energy defined as E=T+U is conserved

L_{A \rightarrow B}=-\Delta U

L_{A \rightarrow B}=\Delta T

\Downarrow

\Delta T+\Delta U=0 \Rightarrow E=T+U=\text { Cost. }

HAMILTONIAN MECHANICS: WHY?

BECAUSE OF ARBITRARY REFERENCE FRAMES AND CONSTRAINTS!!!

HAMILTONIAN MECHANICS

THE HARMONIC OSCILLATOR

k_{s}

E = T + U

E = \frac{1}{2} m v^2 + \frac{1}{2}k_{s}(x-l)^2

H(p,x) = T(p) + U(x)

H(p,x) = \frac{p^2}{2m} + U(x)

HAMILTON'S EQUATIONS OF MOTION

HARMONIC OSCILLATOR

H(p,x) = \frac{p^2}{2m} + U(x)

\begin{cases} \frac{\partial H}{\partial p} = \frac{dx}{dt} \\ \frac{\partial H}{\partial x} = -\frac{dp}{dt} \\ \end{cases}

k_{s}

HAMILTON'S EQUATIONS OF MOTION

DIATOMIC MOLECULE

H(p,x) = \frac{p_1^2}{2m} - \frac{p_2^2}{2m} + \frac{k_s}{2}(x_2-x_1-l)

\begin{cases} \frac{\partial H}{\partial p_1} = \frac{dx_1}{dt}, \, \frac{\partial H}{\partial x_1} = -\frac{dp_1}{dt} \\ \frac{\partial H}{\partial p_2} = \frac{dx_2}{dt}, \, \frac{\partial H}{\partial x_2} = -\frac{dp_2}{dt} \\ \end{cases}

k_{s}