Classical Mechanics

Coordinate Systems:

• Two dimensional system-
  (a) Cartesian coordinate system (x, y)
  (b) Poler coordinate system (r, θ)
  (c) Relation between Cartesian and Polar coordinate system
          x = r cos θ               ,            r = √(x² + y²)
          y = r sin θ               ,            θ = tan^-1 y/x
  
• Three dimensional system-
  (a) Cartesian coordinate system (x, y, z)
  (b) Cylindrical coordinate system (r, θ, z)
  (c) Spherical coordinate system (r, θ, ∅)
  (d) Relation between Cartesian and Cylindrical coordinate system
          x = r cos θ                ,           r = √(x² + y² + z²)
          y = r sin θ                ,           θ = tan^-1 y/x
          z = z                          ,          z = z  
  (e) Relation between Cartesian and Spherical coordinate system
          x = r sin θ cos ∅      ,           r = √(x² + y² + z²)
          y = r sin θ sin ∅      ,           θ = tan^-1 √(x² + y²)/z
          z = r cos θ                ,           ∅ = tan^-1 y/x

Frame of Referene: If a coordinate system is attached to a rigid body and we describe the position of any particle relative to it, then such coordinate system is called frame of reference.

• Newton's laws of motion-
  (1) Law of inertia (First law)- A body continues in its state of rest or uniform motion, unless no external force is applied to it.
  (2) Law of force (Second law)- The time-rate of change of momentum is proportional to impressed force.
                         F̅ = dp̅/ dt   =  ma̅,
  where p̅ is momentum, m is mass and a̅ an acceleration.
  (3) Law of action and reaction (Third law)- For every action there is always equal and opposite reaction.
                        F̅_ij = - F̅_ji
• Intertia frames (Galilean frames of reference)- A frame of reference in which law of inertia holds.
  (1) All those frames, which are moving with constant velocity relative to an initial frame, are also inertial.
  (2) An inertial frame is unaccelerated.
  (3) The accelerated frames are called inertial frames.
• Mechanics of a particle-
  (a) Conservation of linear momentum- In absense of external force, linear momentum of a particle is conserved.
      i.e., dp̅/ dt  = 0 ⇒ p̅ = mv̅  = const.,
  v̅  is the velocity of a particle.
  (b) Conservation of angular momentum- In absense of external torque, angular momentum of a particle is constant of motion.
  
                     τ = dJ̅/dt = 0    ⇒ J̅ = constant.

Dynamical System: It's a system of particles.

• Configuration- The set of positions of all the particles.
• Degree of freedom- The minimum number of independent coordinates (or variables) required to specify the system.
• Constraints- Limitations imposed on the motion of a system.
  (a) Holonomic constraints- If the constraints can be expressed as the equation form then holonomic constraints otherwise non-holonomic constraints.
  (b) Rheonomous and scleronomous constraints- Equation of constraints containing time as explicit variable then Rheonomous otherwise scleronomous constraints.
  (c) Conservative and dissipative constraints- In conservative constraints total mechanical energy of the system is conserved during constrained motion and constraint forces do not do any work otherwise dissipative constraints.
• Forces of constraints- Constraints are always related to force which restrict the motion of the system. These forces are called forces of constraints.
• Generalized coordinates- A set of independent coordinates sufficient in number to describe completely the state of configuration of a dynamical system. These coordinates are denoted as  q₁ , q₂ , . . . , q_n .
• Principal of virtual work- The work done is zero in the case of an arbitrary virtual displacement of a system from a position of equilibrium
                   δW =Σ F̅_i . δr̅_i= 0     for all i = 1, 2, . . . , N
  where F̅_i is the total force on ith particle and δr̅_i is the virtual displacement.
• Components of generalized force- Let q₁ , q₂ , . . . , q_n are generalized coordinates and
  r̅_i = r̅_i(q₁ , q₂ , . . . , q_n), then for a system of n-particles, the components of generalized force Q_k associated with q_k is
                   Q_k = Σ F̅_i . ∂r̅_i/∂q_k = - ∂V/∂q_k    (k = 1, . . . , n)
• Lagrangian- L = T - V, where T is kinetic energy and V is potential energy.

Lebesgue Measurable Functionsn and Integration

Lebesgue Measurable Function- An extended real valued  function f is Lebesgue measurable if its domain is measurable and if it satisfies one of the following five; for each real number α,
    (a) {x : f(x) < α} is measurable
          (b) {x : f(x) > α} is measurable
          (c) {x : f(x) ≤ α} is measurable
          (d) {x : f(x) ≥ α} is measurable
          (e) {x : f(x) = α} is measurable.

Almost everywhere property- If a set of points where it fails to hold is a set of measure zero.
   If f = g, almost everywhere if f and g have the same domain and m{x : f(x) ≠ g(x)} = 0.

Simple Function- A real valued function ∅ is simple function, if it is lebesgue measurable and assume only a finite number of values.

Borel measurability- A function  f is Borel measurable if for each α, the set  {x : f(x) > α} is Borel set.

Some Important Results:

• If f is an extended real valued function whose domain is measurable. Then the following statements are equivalent : For each real number α,

1. The set {x : f(x) > α} is Lebesgue measurable.
2. The set {x : f(x) < α} is Lebesgue measurable.
3. The set {x : f(x) ≤ α} is Lebesgue measurable.
4. The set {x : f(x) ≥ α} is Lebesgue measurable.

• If c is a constant and f and g two Lebesgue measurable real valued functions defined on the same domain. Then the function f + c, cf, f + g, g - f and fg are also Lebesgue measurable.
• If f is a measurable function and f = g almost everywhere, then g is measurable.

                             Lebesgue Integration

The Riemann Integral: If f is a bounded real valued function defined on the interval [a, b] and

a = x₀ < x₁ < . . . < x_n = b is a sub-division of [a, b].

                   S = Σ(x_i - x_i-1)M_i ,              ∀ i = 1, 2, . . . , n

                   s = Σ(x_i - x_i-1)m_i ,               ∀ i = 1, 2, . . . , n

where     M_i = sup f(x)   

and         m_i = inf f(x).

Step Function- For the given subdivision

a = x₀ < x₁ < . . . < x_n = b of the interval [a, b], a function Φ is a step function if

                      Φ(x) = c_i ,       x_{i -1} < x < x_{i .}

The Lebesgue Integral of a Bounded Function Over a Set of Finite Measure:

• canonical representation of simple function- If Φ is a simple function and {a₁ , . . . ,a_n} is the set of non-zero values of Φ, then,

                  Φ = Σ a_i χ_Ai  ,      where A_i = {x : Φ(x) = a_i} is called canonical representation of simple function. Here A_i are disjoint and a_i are distinct and non-zero.

• integral of simple function: If simple function Φ vanishes outside a set of finite measure, the integral of Φ is

              ∫Φ(x)dx = Σ a_i(mA_i)
when Φ has a canonical representation Φ = Σ a_i χ_Ai  and mA_i is the Lebesgue measure of A_i.
If E is any measurable set, we have
∫_EΦ = ∫Φ · χ_E

• The Lebesgue integral: If f is a bounded measurable function defined on a measurable set E with mE finite, the Lebesgue integral of f over E is

       ∫_E f(x)dx = inf ∫_E ψ . (x)dx for all simple function ψ ≥ f.

The Integral of a Non-negative Function: If f is a non-negative measurable function defined on a measurable set E, then integral of non-negative measurable function f is
                   ∫_E f = sup ∫_E h   for all h ≤ f
where h is a bounded measurable function such that
m{x : h(x) ≠ 0} < ∞.

• Integrable function: A non-negative measurable function f is integrable over the measurable set E, if

                  ∫_E f < ∞.

The General Lebesgue Integral:

• The positive and negative part of function:
          f ⁺(x) = max {f(x), 0}
          f ^-(x) = max {-f(x), 0}
  and        f = f ⁺ - f ^-
             |f| = f ⁺+ f ^-
  If f is measurable function then f is integrable over E if f ⁺and f ^- are both integrable over E, and
    ∫_E f(x)dx = ∫_E f ⁺(x)dx + ∫_E f ^-(x)dx

Convergence in Measure:

• Convergence in measure: A sequence <f_n> of measurable function is said to converge to f in measure if, given ε > 0, there is an N such that for all n ≥ N. we have
       m {x : |f(x) - f_n(x)| ≥ ε} < ε.
• cauchy sequence in measure: A sequence <f_n> of measurable function is Cauchy sequence in measure if given ε > 0, there is an N such that for all m, n ≥ N.
  we have
       m {x : |f_n(x) - f_m(x)| ≥ ε} < ε.

Element Metric Space

Metric space- Let X be a non-empty set and we have a real valued function ρ:X * X → ℝ such that, for all x,y,z ∈ X.
(a) ρ(x,y) ≥ 0
(b) ρ(x,y) = 0 iff x = y
(c) ρ(x,y) ≤ ρ(x,z)+ρ(z,y)
Then ρ is called metric and <X,ρ> is a metric space.

Pseudo metric- A metric ρ with the relaxation, ρ(x,y) = 0 for some x ≠ y.

Extended metric- A metric ρ such that 
                            ρ:X * X → ℝ ∪ {∞}.

Subspace- <X, ρ> is a metric space, Y ⊂ X, then <Y, ρ> is a subspace of <X, ρ>.

Euclidean metric- x̅, y̅ ∈ X ⊂ℝⁿ,
                 x̅ = (x₁, ... , x_n),  y̅ = (y₁, ... , y_n),
                   ρ(x̅, y̅) = [Σ |x_i - y_i|]^1/n, ∀ i = 1, 2, ... , n.

Convergence- A sequence {x_n} converges to an element
x ∈ X if                  lim ρ(x_n , x) = 0.

Open set- A set G ⊂ X is open if it contains a sphere about each of its points.

Closed set- A set F is closed if X ~ F is open.

Neighbourhood- A neighbourhood of a point x ∈ X is an open set which contains x₀.

Interior point- x₀ is an interior point of set A if A is a neighbourhood of x₀.

Interior set- Interior of set A contains all interior points of A.

Limit point- If A ⊂ X and x₀ ∈ X, then x₀ is limit point of A, if every neighbourhood of x₀ contains points of A distinct from x_0.

Closure- Closure of set A is A̅ which contains all points which are either point of A or limit points of A.

Dense set- A, B ⊂ X, A is dense in B if A̅ ⊃ B.

Everywhere Dense- A is everywhere dense if A̅ = X.

Separable metric space- If metric space X has a countable subset which is everywhere dense, then it is separable metric space.

Open Covering- If C  is a collection of open sets in metric space X with the property that every x ∈ X is a member of atleast one set G ∈C . TheC  is called open covering of X.

Sub-covering- A sub-covering of the open coveringC  is any sub-coveringC ' ⊂C  which is also an open covering of X.

Cauchy sequence- A sequence {x_n} in a metric space
X = (X, ρ) is Cauchy sequence, if for every ε > 0, there is N such that m, n > N
                          ⇒ ρ(x_m, x_n) < ε.

Complete metric space- A metric space is complete, if every Cauchy sequence in metric space converges.

Contraction- If (X, ρ) is a metric space, a mapping
T : X → X is called contraction in X if there is a number K, with 0 < K < 1, such that x, y ∈ X, x ≠ y
                          ⇒ ρ(Tx, Ty) ≤ Kρ(x, y).

Bolzano-Weierstrass property- A space X has Balzano-Weierstrass property if every infinite sequence in X has atleast one limit point.

Nowhere dense- If A ⊂ X is nowhere dense, A̅ the closure of A has no interior point. 

First Category (Merger)- A set A ⊂ X is of first category in X, if it is the union of countably many nowhere dense sets in X.

Second category (Non-merger)- A set A ⊂ X is of second category in X, if it is not of first category.

Total boundedness- A metric space X is totally bounded if for every ε > 0, X contains a finite set, called an ε - net, such that the finite set of open spheres of radius ε and centres in the ε - net covers X.

Connected set- A ⊂ X, set A is connected if it cannot be represented as the union of two sets, each of which is disjoint from the closure of the other.