Newtonian Mechanics

Newtonian Mechanics

• Based on Newton's Laws of motion, which assume absolute concepts of distance, time, and mass
• Assumes motion is in an inertial frame, but violates Theory of Relativity

Newton's Laws of Motion

• First law (Law of Inertia): A body remains at rest or in uniform motion unless acted upon by a force
• Second law (Equation of Motion): The time rate of change of momentum equals the force acting on a body
• Third law (Action and Reaction): Forces exerted on each other by two bodies are equal in magnitude and opposite in direction

Inertial Frames of Reference

• A frame of reference where Newton's Laws of motion are valid
• Non-accelerated, homogeneous, and isotropic
• Physical experiments can be carried out in different inertial reference frames
• Galilean transformation converts between two inertial frames moving at a constant relative velocity

First-Order Integrals in Newtonian Mechanics

• Fundamental goal: determine equations of motion for an n-body system
• Newton's second-order equation of motion must be solved to calculate spatial locations, velocities, and accelerations
• First-order integrals facilitate solution of Newton's second-order equations

Conservation Laws in Classical Mechanics

• Combine conservation laws with first integrals for linear momentum, angular momentum, and work-energy
• Conservation laws are fundamental laws of nature, applicable beyond Newtonian mechanics

Motion of Finite-Sized and Many-Body Systems

• Rotational degrees of freedom introduced

Center of Mass of a Many-Body System

• Reference point for describing motion of a finite-sized body
• Center of mass provides this reference point

Total Linear Momentum of a Many-Body System

• Center of mass plays a key role

Angular Momentum of a Many-Body System

• Separated into two components: angular momentum about the center of mass and angular motion of the center of mass about the origin
• Position vector with respect to the center of mass and vector location of the center of mass are used

Work and Kinetic Energy for a Many-Body System

• Path and time-independence of forces relate to conservation of energy and momentum

Virial Theorem

• Important theorem for systems of moving particles in classical and quantum physics
• Useful for considering collections of many particles, especially in central-force motion

Applications of Newton's Equations of Motion

• Many-body and constrained motion

Solution of Many-Body Equations of Motion

• General methods used to solve Newton's many-body equations for practical problems

Newton's Law of Gravitation

• Formulated in 1666, published in the Principia