Newtonian dynamics

Last updated

In physics, the Newtonian dynamics is understood as the dynamics of a particle or a small body according to Newton's laws of motion.

Dynamics is the branch of classical mechanics concerned with the study of forces and their effects on motion. Isaac Newton defined the fundamental physical laws which govern dynamics in physics, especially his second law of motion.

Newton's laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a quantitative measure of the force, and the third asserts that a single isolated force doesn't exist. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows:


Contents

Mathematical generalizations

Typically, the Newtonian dynamics occurs in a three-dimensional Euclidean space, which is flat. However, in mathematics Newton's laws of motion can be generalized to multidimensional and curved spaces. Often the term Newtonian dynamics is narrowed to Newton's second law .

Euclidean space Fundamental space of geometry

Euclidean space is the fundamental space of geometry. Originally, this was the three-dimensional space of Euclidean geometry, but, in modern mathematics, there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the Euclidean plane. It has been introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier Euclidean has been added for distinguishing it from other spaces that are considered in physics and modern mathematics.

Curved space spatial geometry which is not "flat" or Euclidean

Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Curved spaces play an essential role in general relativity, where gravity is often visualized as curved space. The Friedmann-Lemaître-Robertson-Walker metric is a curved metric which forms the current foundation for the description of the expansion of space and shape of the universe.

Newton's second law in a multidimensional space

Consider particles with masses in the regular three-dimensional Euclidean space. Let be their radius-vectors in some inertial coordinate system. Then the motion of these particles is governed by Newton's second law applied to each of them

 

 

 

 

(1)

The three-dimensional radius-vectors can be built into a single -dimensional radius-vector. Similarly, three-dimensional velocity vectors can be built into a single -dimensional velocity vector:

 

 

 

 

(2)

In terms of the multidimensional vectors ( 2 ) the equations ( 1 ) are written as

 

 

 

 

(3)

i.e. they take the form of Newton's second law applied to a single particle with the unit mass .

Definition. The equations ( 3 ) are called the equations of a Newtonian dynamical system in a flat multidimensional Euclidean space, which is called the configuration space of this system. Its points are marked by the radius-vector . The space whose points are marked by the pair of vectors is called the phase space of the dynamical system ( 3 ).

Dynamical system mathematical model which describes the time dependence of a point in a geometrical space

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system.

Phase space Mathematical construction for dynamical systems

In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs.

Euclidean structure

The configuration space and the phase space of the dynamical system ( 3 ) both are Euclidean spaces, i. e. they are equipped with a Euclidean structure. The Euclidean structure of them is defined so that the kinetic energy of the single multidimensional particle with the unit mass is equal to the sum of kinetic energies of the three-dimensional particles with the masses :

Kinetic energy energy possessed by an object by virtue of its motion

In physics, the kinetic energy (KE) of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body when decelerating from its current speed to a state of rest.

.

 

 

 

 

(4)

Constraints and internal coordinates

In some cases the motion of the particles with the masses can be constrained. Typical constraints look like scalar equations of the form

.

 

 

 

 

(5)

Constraints of the form ( 5 ) are called holonomic and scleronomic. In terms of the radius-vector of the Newtonian dynamical system ( 3 ) they are written as

.

 

 

 

 

(6)

Each such constraint reduces by one the number of degrees of freedom of the Newtonian dynamical system ( 3 ). Therefore, the constrained system has degrees of freedom.

Definition. The constraint equations ( 6 ) define an -dimensional manifold within the configuration space of the Newtonian dynamical system ( 3 ). This manifold is called the configuration space of the constrained system. Its tangent bundle is called the phase space of the constrained system.

Let be the internal coordinates of a point of . Their usage is typical for the Lagrangian mechanics. The radius-vector is expressed as some definite function of :

.

 

 

 

 

(7)

The vector-function ( 7 ) resolves the constraint equations ( 6 ) in the sense that upon substituting ( 7 ) into ( 6 ) the equations ( 6 ) are fulfilled identically in .

Internal presentation of the velocity vector

The velocity vector of the constrained Newtonian dynamical system is expressed in terms of the partial derivatives of the vector-function ( 7 ):

.

 

 

 

 

(8)

The quantities are called internal components of the velocity vector. Sometimes they are denoted with the use of a separate symbol

 

 

 

 

(9)

and then treated as independent variables. The quantities

 

 

 

 

(10)

are used as internal coordinates of a point of the phase space of the constrained Newtonian dynamical system.

Embedding and the induced Riemannian metric

Geometrically, the vector-function ( 7 ) implements an embedding of the configuration space of the constrained Newtonian dynamical system into the -dimensional flat configuration space of the unconstrained Newtonian dynamical system ( 3 ). Due to this embedding the Euclidean structure of the ambient space induces the Riemannian metric onto the manifold . The components of the metric tensor of this induced metric are given by the formula

,

 

 

 

 

(11)

where is the scalar product associated with the Euclidean structure ( 4 ).

Kinetic energy of a constrained Newtonian dynamical system

Since the Euclidean structure of an unconstrained system of particles is introduced through their kinetic energy, the induced Riemannian structure on the configuration space of a constrained system preserves this relation to the kinetic energy:

.

 

 

 

 

(12)

The formula ( 12 ) is derived by substituting ( 8 ) into ( 4 ) and taking into account ( 11 ).

Constraint forces

For a constrained Newtonian dynamical system the constraints described by the equations ( 6 ) are usually implemented by some mechanical framework. This framework produces some auxiliary forces including the force that maintains the system within its configuration manifold . Such a maintaining force is perpendicular to . It is called the normal force. The force from ( 6 ) is subdivided into two components

.

 

 

 

 

(13)

The first component in ( 13 ) is tangent to the configuration manifold . The second component is perpendicular to . In coincides with the normal force .
Like the velocity vector ( 8 ), the tangent force has its internal presentation

.

 

 

 

 

(14)

The quantities in ( 14 ) are called the internal components of the force vector.

Newton's second law in a curved space

The Newtonian dynamical system ( 3 ) constrained to the configuration manifold by the constraint equations ( 6 ) is described by the differential equations

,

 

 

 

 

(15)

where are Christoffel symbols of the metric connection produced by the Riemannian metric ( 11 ).

Relation to Lagrange equations

Mechanical systems with constraints are usually described by Lagrange equations:

,

 

 

 

 

(16)

where is the kinetic energy the constrained dynamical system given by the formula ( 12 ). The quantities in ( 16 ) are the inner covariant components of the tangent force vector (see ( 13 ) and ( 14 )). They are produced from the inner contravariant components of the vector by means of the standard index lowering procedure using the metric ( 11 ):

,

 

 

 

 

(17)

The equations ( 16 ) are equivalent to the equations ( 15 ). However, the metric ( 11 ) and other geometric features of the configuration manifold are not explicit in ( 16 ). The metric ( 11 ) can be recovered from the kinetic energy by means of the formula

.

 

 

 

 

(18)

See also

Related Research Articles

In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , but also or to highlight its function as an operator. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

Euclidean distance conventional distance in mathematics and physics

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. With this distance, Euclidean space becomes a metric space. The associated norm is called the Euclidean norm. Older literature refers to the metric as the Pythagorean metric. A generalized term for the Euclidean norm is the L2 norm or L2 distance.

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space(M, g) is a real, smooth manifold M equipped with an inner product gp on the tangent space TpM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then pgp(X|p, Y|p) is a smooth function. The family gp of inner products is called a Riemannian metric. These terms are named after the German mathematician Bernhard Riemann. The study of Riemannian manifolds constitutes the subject called Riemannian geometry.

Normal (geometry) in geometry, an object that is perpendicular to a given object

In geometry, a normal is an object such as a line or vector that is perpendicular to a given object. For example, in two dimensions, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point.

In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics and quantum mechanics.

Minkowski space mathematical space setting which eases explanation of special relativity

In mathematical physics, Minkowski space is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.

In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.

An instanton is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. More precisely, it is a solution to the equations of motion of the classical field theory on a Euclidean spacetime.

Taxicab geometry Type of metric geometry

A taxicab geometry is a form of geometry in which the usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, L1 distance, L1 distance or norm, snake distance, city block distance, Manhattan distance or Manhattan length, with corresponding variations in the name of the geometry. The latter names allude to the grid layout of most streets on the island of Manhattan, which causes the shortest path a car could take between two intersections in the borough to have length equal to the intersections' distance in taxicab geometry.

In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by many scientists and mathematicians during the 18th century and onward, after Newtonian mechanics. Since Newtonian mechanics considers vector quantities of motion, particularly accelerations, momenta, forces, of the constituents of the system, an alternative name for the mechanics governed by Newton's laws and Euler's laws is vectorial mechanics.

In analytical mechanics, specifically the study of the rigid body dynamics of multibody systems, the term generalized coordinates refers to the parameters that describe the configuration of the system relative to some reference configuration. These parameters must uniquely define the configuration of the system relative to the reference configuration. This is done assuming that this can be done with a single chart. The generalized velocities are the time derivatives of the generalized coordinates of the system.

Rigid body dynamics studies the movement of systems of interconnected bodies under the action of external forces; described by the laws of kinematics and by the application of Newtons second law (kinetics) or their derivative form Lagrangian mechanics

Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.

A nonholonomic system in physics and mathematics is a system whose state depends on the path taken in order to achieve it. Such a system is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space but finally returns to the original set of parameter values at the start of the path, the system itself may not have returned to its original state.

A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space. To calculate the first class constraint, one assumes that there are no second class constraints, or that they have been calculated previously, and their Dirac brackets generated.

Real coordinate space coordinate space that allows several (n) real variables to be treated as a single variable

In mathematics, real coordinate space of n dimensions, written Rn is a coordinate space that allows several real variables to be treated as a single variable. With various numbers of dimensions, Rn is used in many areas of pure and applied mathematics, as well as in physics. With component-wise addition and scalar multiplication, it is the prototypical real vector space and is a frequently used representation of Euclidean n-space. Due to the latter fact, geometric metaphors are widely used for Rn, namely a plane for R2 and three-dimensional space for R3.

In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.

In theoretical physics, the Udwadia–Kalaba equation is a method for deriving the equations of motion of a constrained mechanical system. The equation was first described by Firdaus E. Udwadia and Robert E. Kalaba in 1992. The approach is based on Gauss's principle of least constraint. The Udwadia–Kalaba equation applies to a wide class of constraints, both holonomic constraints and nonholonomic ones, as long as they are linear with respect to the accelerations. The equation generalizes to constraint forces that do not obey D'Alembert's principle.

Lagrangian mechanics Formulation of classical mechanics

Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.