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.