Analytical dynamics

# Analytical dynamics

Discussion

Encyclopedia
In classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

, analytical dynamics, or more briefly dynamics, is concerned about the relationship between motion
Motion (physics)
In physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

of bodies and its causes, namely the forces acting on the bodies and the properties of the bodies (particularly mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

and moment of inertia
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...

). The foundation of modern day dynamics is Newtonian mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

and its reformulation as Lagrangian mechanics
Lagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....

and Hamiltonian mechanics
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...

.
The field has a long and important history, as remarked by Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...

:
Some authors (for example, Taylor (2005) and Greenwood (1997)) include special relativity
Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...

within classical dynamics.

## Relationship to statics, kinetics, and kinematics

Historically, there were three branches of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...

: "statics
Statics
Statics is the branch of mechanics concerned with the analysis of loads on physical systems in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity...

" (the study of equilibrium and its relation to forces); "kinetics
Kinetics (physics)
In physics and engineering, kinetics is a term for the branch of classical mechanics that is concerned with the relationship between the motion of bodies and its causes, namely forces and torques...

" (the study of motion and its relation to forces)
and "kinematics
Kinematics
Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....

" (dealing with the implications of observed motions without regard for circumstances causing them).
These three subjects have been connected to dynamics in several ways. One approach combined statics and kinetics under the name dynamics, which became the branch dealing with determination of the motion of bodies resulting from the action of specified forces; another approach separated statics, and combined kinetics and kinematics under the rubric dynamics. This approach is common in engineering books on mechanics, and is still in widespread use among mechanicians.

### Fundamental importance in engineering, diminishing emphasis in physics

Today, dynamics and kinematics continue to be considered the two pillars of classical mechanics. Dynamics is still included in mechanical, aerospace, and other engineering curriculums because of its importance in machine design, the design of land, sea, air, and space vehicles and other applications. However, few modern physicists concern themselves with an independent treatment of "dynamics" or "kinematics", nevermind "statics" or "kinetics". Instead, the entire undifferentiated subject is referred to as classical mechanics. In fact, many undergraduate and graduate text books since mid-20th century on "classical mechanics" lack chapters titled "dynamics" or "kinematics" .
In these books, although the word "dynamics" is used when acceleration is ascribed to a force, the word "kinetics" is never mentioned. However, clear exceptions exist. Prominent examples include The Feynman Lectures on Physics
The Feynman Lectures on Physics
The Feynman Lectures on Physics is a 1964 physics textbook by Richard P. Feynman, Robert B. Leighton and Matthew Sands, based upon the lectures given by Feynman to undergraduate students at the California Institute of Technology in 1961–63. It includes lectures on mathematics, electromagnetism,...

.

## Fundamental Principles

• Newton's laws of motion
Newton's laws of motion
Newton's laws of motion are three physical laws that form the basis for classical mechanics. They describe the relationship between the forces acting on a body and its motion due to those forces...

• Inertia
Inertia
Inertia is the resistance of any physical object to a change in its state of motion or rest, or the tendency of an object to resist any change in its motion. It is proportional to an object's mass. The principle of inertia is one of the fundamental principles of classical physics which are used to...

• Acceleration
Acceleration
In physics, acceleration is the rate of change of velocity with time. In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity. ...

• Momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

• Reaction
Reaction (physics)
The third of Newton's laws of motion of classical mechanics states that forces always occur in pairs. Every action is accompanied by a reaction of equal magnitude but opposite direction. This principle is commonly known in the Latin language as actio et reactio. The attribution of which of the two...

• Newton's law of universal gravitation
Newton's law of universal gravitation
Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them...

• Special theory of relativity

### Axioms and mathematical treatments

• Variational principle
Variational principle
A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...

s and Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

's equations
• Hamilton
William Rowan Hamilton
Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...

's equations
• Canonical transformation
Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates  →  that preserves the form of Hamilton's equations , although it...

s
• Hamilton-Jacobi Theory

## Related engineering branches

• Particle dynamics
• Rigid body dynamics
Rigid body dynamics
In physics, rigid body dynamics is the study of the motion of rigid bodies. Unlike particles, which move only in three degrees of freedom , rigid bodies occupy space and have geometrical properties, such as a center of mass, moments of inertia, etc., that characterize motion in six degrees of...

• Soft body dynamics
Soft body dynamics
Soft body dynamics is a field of computer graphics that focuses on visually realistic physical simulations of the motion and properties of deformable objects . The applications are mostly in video games and film. Unlike in simulation of rigid bodies, the shape of soft bodies can change, meaning...

• Fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

• Hydrodynamics
• Gas dynamics
Gas dynamics
Gas dynamics is a branch of fluid dynamics concerned with studying the motion of gases and its consequent effects. Gas dynamics combines the principles of fluid mechanics and thermodynamics...

• Aerodynamics
Aerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...