Physical simulation
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Dynamical simulation, in
computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, ...
, is the
simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
of systems of objects that are free to move, usually in three dimensions according to
Newton's laws Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
of dynamics, or approximations thereof. Dynamical simulation is used in
computer animation Computer animation is the process used for digitally generating animations. The more general term computer-generated imagery (CGI) encompasses both static scenes (still images) and dynamic images (moving images), while computer animation refe ...
to assist animators to produce realistic motion, in
industrial design Industrial design is a process of design applied to physical Product (business), products that are to be manufactured by mass production. It is the creative act of determining and defining a product's form and features, which takes place in advan ...
(for example to simulate crashes as an early step in
crash test A crash test is a form of destructive testing usually performed in order to ensure safe design standards in crashworthiness and crash compatibility for various modes of transportation (see automobile safety) or related systems and componen ...
ing), and in
video game Video games, also known as computer games, are electronic games that involves interaction with a user interface or input device such as a joystick, controller, keyboard, or motion sensing device to generate visual feedback. This fee ...
s. Body movement is calculated using time integration methods.


Physics engines

In
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, a program called a
physics engine A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics (including collision detection), soft body dynamics, and fluid dynamics, of use in the domains of computer gr ...
is used to model the behaviors of objects in space. These engines allow simulation of the way bodies of many types are affected by a variety of physical stimuli. They are also used to create Dynamical simulations without having to know anything about physics. Physics engines are used throughout the video game and movie industry, but not all physics engines are alike. They are generally broken into
real-time Real-time or real time describes various operations in computing or other processes that must guarantee response times within a specified time (deadline), usually a relatively short time. A real-time process is generally one that happens in defined ...
and the high precision, but these are not the only options. Most real-time physics engines are inaccurate and yield only the barest approximation of the real world, whereas most high-precision engines are far too slow for use in everyday applications. To understand how these Physics engines are built, a basic understanding of physics is required. Physics engines are based on the actual behaviors of the world as described by
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
. Engines do not typically account for Modern Mechanics (see
Theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
and
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
) because most visualization deals with large bodies moving relatively slowly, but the most complicated engines perform calculations for Modern Mechanics as well as Classical. The models used in Dynamical simulations determine how accurate these simulations are.


Particle model

The first model which may be used in
physics engines A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics (including collision detection), soft body dynamics, and fluid dynamics, of use in the domains of computer gr ...
governs the motion of infinitesimal objects with finite mass called “particles.” This equation, called Newton’s Second law (see
Newton's laws Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in motion ...
) or the definition of force, is the fundamental behavior governing all motion: : \vec = m \vec This equation will allow us to fully model the behavior of particles, but this is not sufficient for most simulations because it does not account for the rotational motion of
rigid bodies In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
. This is the simplest model that can be used in a physics engine and was used extensively in early video games.


Inertial model

Bodies in the real world deform as forces are applied to them, so we call them “soft,” but often the deformation is negligibly small compared to the motion, and it is very complicated to model, so most physics engines ignore deformation. A body that is assumed to be non-deformable is called a
rigid body In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
.
Rigid body dynamics In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are ''rigid'' (i.e. they do not deform under the action of a ...
deals with the motion of objects that cannot change shape, size, or mass but can change orientation and position. To account for rotational energy and momentum, we must describe how force is applied to the object using a moment, and account for the mass distribution of the object using an
inertia tensor The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
. We describe these complex interactions with an equation somewhat similar to the definition of force above: : \frac = \sum_^N \tau_ where \mathbf is the central
inertia tensor The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
, \vec is the
angular velocity In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
vector, and \tau_ is the moment of the ''j''th external force about the mass center. The
inertia tensor The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
describes the location of each particle of mass in a given object in relation to the object's center of mass. This allows us to determine how an object will rotate dependent on the forces applied to it. This angular motion is quantified by the angular velocity vector. As long as we stay below relativistic speeds (see
Relativistic dynamics ''For classical dynamics at relativistic speeds, see relativistic mechanics#Relativistic dynamics, relativistic mechanics.'' Relativistic dynamics refers to a combination of Theory of relativity, relativistic and quantum concepts to describe the re ...
), this model will accurately simulate all relevant behavior. This method requires the
Physics engine A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics (including collision detection), soft body dynamics, and fluid dynamics, of use in the domains of computer gr ...
to solve six
ordinary differential equations In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
at every instant we want to render, which is a simple task for modern computers.


Euler model

The inertial model is much more complex than we typically need but it is the most simple to use. In this model, we do not need to change our forces or constrain our system. However, if we make a few intelligent changes to our system, simulation will become much easier, and our calculation time will decrease. The first constraint will be to put each torque in terms of the principal axes. This makes each torque much more difficult to program, but it simplifies our equations significantly. When we apply this constraint, we diagonalize the moment of inertia tensor, which simplifies our three equations into a special set of equations called
Euler's equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
. These equations describe all rotational momentum in terms of the principal axes: : \begin I_1\dot_+(I_3-I_2)\omega_2\omega_3 &=& N_\\ I_2\dot_+(I_1-I_3)\omega_3\omega_1 &=& N_\\ I_3\dot_+(I_2-I_1)\omega_1\omega_2 &=& N_ \end * The N terms are applied torques about the principal axes * The I terms are the principal moments of inertia * The terms are angular velocities about the principal axes The drawback to this model is that all the computation is on the front end, so it is still slower than we would like. The real usefulness is not apparent because it still relies on a system of non-linear differential equations. To alleviate this problem, we have to find a method that can remove the second term from the equation. This will allow us to integrate much more easily. The easiest way to do this is to assume a certain amount of symmetry.


Symmetric/torque model

The two types of symmetric objects that will simplify
Euler's equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...
are “symmetric tops” and “symmetric spheres.” The first assumes one degree of symmetry, this makes two of the I terms equal. These objects, like cylinders and tops, can be expressed with one very simple equation and two slightly simpler equations. This does not do us much good, because with one more symmetry we can get a large jump in speed with almost no change in appearance. The symmetric sphere makes all of the I terms equal (the
Moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
scalar), which makes all of these equations simple: : \begin I\dot_ &=& N_\\ I\dot_ &=& N_\\ I\dot_ &=& N_ \end * The N terms are applied torques about the principal axes * The terms are angular velocities about the principal axes * The I term is the scalar
Moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
: :I \ \stackrel{=}\ \int_V l^2(m)\,dm = \iiint_V l^2(v)\,\rho(v)\,dv = \iiint_V l^2(x,y,z)\,\rho(x,y,z)\,dx\,dy\,dz \! :where **V is the volume region of the object, **''r'' is the distance from the axis of rotation, **''m'' is mass, **''v'' is volume, **ρ is the pointwise
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
function of the object, **''x'', ''y'', ''z'' are the Cartesian coordinates. These equations allow us to simulate the behavior of an object that can spin in a way very close to the method simulate motion without spin. This is a simple model but it is accurate enough to produce realistic output in real-time Dynamical simulations. It also allows a
Physics engine A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics (including collision detection), soft body dynamics, and fluid dynamics, of use in the domains of computer gr ...
to focus on the changing forces and torques rather than varying inertia.


See also

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Bounding volume In computer graphics and computational geometry, a bounding volume for a set of objects is a closed volume that completely contains the union of the objects in the set. Bounding volumes are used to improve the efficiency of geometrical operatio ...
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Collision detection Collision detection is the computational problem of detecting the intersection (Euclidean geometry), intersection of two or more objects. Collision detection is a classic issue of computational geometry and has applications in various computing ...
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Euler's equations (rigid body dynamics) In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to th ...
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Moment of inertia The moment of inertia, otherwise known as the mass moment of inertia, angular mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceler ...
* Physics Abstraction Layer *
Physics engine A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics (including collision detection), soft body dynamics, and fluid dynamics, of use in the domains of computer gr ...
*
Rigid body dynamics In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are ''rigid'' (i.e. they do not deform under the action of a ...
Computational physics Computer physics engines