Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations using Lagrange multipliers. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, time, contains the information about the dynamics of the system. No new physics are introduced in applying Lagrangian mechanics compared to Newtonian mechanics, it is, more mathematically sophisticated and systematic. Newton's laws can include non-conservative forces like friction. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system. Dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler–Lagrange equations.
Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which may simplify solving for the motion of the system. Lagrangian mechanics reveals conserved quantities and their symmetries in a direct way, as a special case of Noether's theorem. Lagrangian mechanics is important not just for its broad applications, but for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton developed Hamilton's principle that can be used to derive the Lagrange equation and was recognized to be applicable to much of fundamental theoretical physics as well quantum mechanics and the theory of relativity, it can be applied to other systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is used to solve mechanical problems in physics and when Newton's formulation of classical mechanics is not convenient.
Lagrangian mechanics applies to the dynamics of particles, while fields are described using a Lagrangian density. Lagrange's equations are used in optimization problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind. Suppose we have a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion. For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of independent generalized coordinates that characterize the possible motion of the particle; this choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.
For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a point particle. For a system of N point particles with masses m1, m2... mN, each particle has a position vector, denoted r1, r2... rN. Cartesian coordinates are sufficient, so r1 =, r2 = and so on. In three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system; these are all specific points in space to locate the particles. The velocity of each particle is how fast the particle moves along its path of motion, is the time derivative of its position, thusIn Newtonian mechanics, the equations of motion are given by Newton's laws; the second law "net force equals mass times acceleration",applies to each particle. For an N particle system in 3 dimensions, there are 3N second order ordinary differential equations in the positions of the particles to solve for.
Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian, it is possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles can be defined by L = T − V where T = 1 2 ∑ k = 1 N m k v k 2 is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, V is the potential energy of the system. Kinetic energy is the energy of the system's motion, vk2 = vk
In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass i.e. to accelerate. Force can be described intuitively as a push or a pull. A force has both direction, making it a vector quantity, it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, is inversely proportional to the mass of the object. Concepts related to force include: thrust. In an extended body, each part applies forces on the adjacent parts; such internal mechanical stresses cause no acceleration of that body as the forces balance one another. Pressure, the distribution of many small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate.
Stress causes deformation of solid materials, or flow in fluids. Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part this was due to an incomplete understanding of the sometimes non-obvious force of friction, a inadequate view of the nature of natural motion. A fundamental error was the belief that a force is required to maintain motion at a constant velocity. Most of the previous misunderstandings about motion and force were corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved for nearly three hundred years. By the early 20th century, Einstein developed a theory of relativity that predicted the action of forces on objects with increasing momenta near the speed of light, provided insight into the forces produced by gravitation and inertia.
With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong, electromagnetic and gravitational. High-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines; the mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces culminated in the work of Archimedes, famous for formulating a treatment of buoyant forces inherent in fluids.
Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed of the elements earth and water, to be in their natural place on the ground and that they will stay that way if left alone, he distinguished between the innate tendency of objects to find their "natural place", which led to "natural motion", unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows; the place where the archer moves the projectile was at the start of the flight, while the projectile sailed through the air, no discernible efficient cause acts on it.
Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation demands a continuum like air for change of place in general. Aristotelian physics began facing criticism in medieval science, first by John Philoponus in the 6th century; the shortcomings of Aristotelian physics would not be corrected until the 17th century work of Galileo Galilei, influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion, he showed that the bodies were accelerated by gravity to an extent, independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction. Sir Isaac Newton described the motion of all objects using the concepts of inertia and force, in doing so he found they obey certain conservation laws.
In 1687, Newton published his thesis Philosophiæ Naturalis Principia Mathematica. In this work Newton set out three laws of motion that to this day are t
In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a vector quantity, impulse is a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum in the same direction; the SI unit of impulse is the newton second, the dimensionally equivalent unit of momentum is the kilogram meter per second. The corresponding English engineering units are the slug-foot per second. A resultant force causes acceleration and a change in the velocity of the body for as long as it acts. A resultant force applied over a longer time therefore produces a bigger change in linear momentum than the same force applied briefly: the change in momentum is equal to the product of the average force and duration. Conversely, a small force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly. J = F average The impulse is the integral of the resultant force with respect to time: J = ∫ F d t Impulse J produced from time t1 to t2 is defined to be J = ∫ t 1 t 2 F d t where F is the resultant force applied from t1 to t2.
From Newton's second law, force is related to momentum p by F = d p d t Therefore, J = ∫ t 1 t 2 d p d t d t = ∫ p 1 p 2 d p = p 2 − p 1 = Δ p where Δp is the change in linear momentum from time t1 to t2. This is called the impulse-momentum theorem; as a result, an impulse may be regarded as the change in momentum of an object to which a resultant force is applied. The impulse may be expressed in a simpler form when the mass is constant: J = ∫ t 1 t 2 F d t = Δ p = m v 2 − m v 1 where F is the resultant force applied, t1 and t2 are times when the impulse begins and ends m is the mass of the object, v2 is the final velocity of the object at the end of the time interval, v1 is the initial velocity of the object when the time interval begins. Impulse has the same dimensions as momentum. In the International System of Units, these are kg⋅m/s = N⋅s. In English engineering units, they are slug⋅ft/s = lbf⋅s; the term "impulse" is used to refer to a fast-acting force or impact. This type of impulse is idealized so that the change in momentum produced by the force happens with no change in time.
This sort of change is a step change, is not physically possible. However, this is a useful model for computing the effects of ideal collisions. Additionally, in rocketry, the term "total impulse" is used and is considered synonymous with the term "impulse"; the application of Newton's second law for variable mass allows impulse and momentum to be used as analysis tools for jet- or rocket-propelled vehicles. In the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse; this fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicle's propulsive change in velocity to the engine's specific impulse and the vehicle's propellant-mass ratio. Wave–particle duality defines the impulse of a wave collision; the preservation of momentum in the collision is called phase matching. Applications include: Compton effect Nonlinear optics Acousto-optic modulator Electron phonon scattering Serway, Raymond A..
Physics for Scientists and Engineers. Brooks/Cole. ISBN 0-534-40842-7. Tipler, Paul. Physics for Scientists and Engineers: Mechanics and Waves, Thermodynamics. W. H. Freeman. ISBN 0-7167-0809-4. Dynamics
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is conceived in three linear dimensions, although modern physicists consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime; the concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. Debates concerning the nature and the mode of existence of space date back to antiquity. Many of these classical philosophical questions were discussed in the Renaissance and reformulated in the 17th century during the early development of classical mechanics. In Isaac Newton's view, space was absolute—in the sense that it existed permanently and independently of whether there was any matter in the space. Other natural philosophers, notably Gottfried Leibniz, thought instead that space was in fact a collection of relations between objects, given by their distance and direction from one another.
In the 18th century, the philosopher and theologian George Berkeley attempted to refute the "visibility of spatial depth" in his Essay Towards a New Theory of Vision. The metaphysician Immanuel Kant said that the concepts of space and time are not empirical ones derived from experiences of the outside world—they are elements of an given systematic framework that humans possess and use to structure all experiences. Kant referred to the experience of "space" in his Critique of Pure Reason as being a subjective "pure a priori form of intuition". In the 19th and 20th centuries mathematicians began to examine geometries that are non-Euclidean, in which space is conceived as curved, rather than flat. According to Albert Einstein's theory of general relativity, space around gravitational fields deviates from Euclidean space. Experimental tests of general relativity have confirmed that non-Euclidean geometries provide a better model for the shape of space. Galilean and Cartesian theories about space and motion are at the foundation of the Scientific Revolution, understood to have culminated with the publication of Newton's Principia in 1687.
Newton's theories about space and time helped. While his theory of space is considered the most influential in Physics, it emerged from his predecessors' ideas about the same; as one of the pioneers of modern science, Galilei revised the established Aristotelian and Ptolemaic ideas about a geocentric cosmos. He backed the Copernican theory that the universe was heliocentric, with a stationary sun at the center and the planets—including the Earth—revolving around the sun. If the Earth moved, the Aristotelian belief that its natural tendency was to remain at rest was in question. Galilei wanted to prove instead that the sun moved around its axis, that motion was as natural to an object as the state of rest. In other words, for Galilei, celestial bodies, including the Earth, were inclined to move in circles; this view displaced another Aristotelian idea—that all objects gravitated towards their designated natural place-of-belonging. Descartes set out to replace the Aristotelian worldview with a theory about space and motion as determined by natural laws.
In other words, he sought a metaphysical foundation or a mechanical explanation for his theories about matter and motion. Cartesian space was Euclidean in structure—infinite and flat, it was defined as that. The Cartesian notion of space is linked to his theories about the nature of the body and matter, he is famously known for his "cogito ergo sum", or the idea that we can only be certain of the fact that we can doubt, therefore think and therefore exist. His theories belong to the rationalist tradition, which attributes knowledge about the world to our ability to think rather than to our experiences, as the empiricists believe, he posited a clear distinction between the body and mind, referred to as the Cartesian dualism. Following Galilei and Descartes, during the seventeenth century the philosophy of space and time revolved around the ideas of Gottfried Leibniz, a German philosopher–mathematician, Isaac Newton, who set out two opposing theories of what space is. Rather than being an entity that independently exists over and above other matter, Leibniz held that space is no more than the collection of spatial relations between objects in the world: "space is that which results from places taken together".
Unoccupied regions are those that could have objects in them, thus spatial relations with other places. For Leibniz space was an idealised abstraction from the relations between individual entities or their possible locations and therefore could not be continuous but must be discrete. Space could be thought of in a similar way to the relations between family members. Although people in the family are related to one another, the relations do not exist independently of the people. Leibniz argued that space could not exist independently of objects in the world because that implies a difference between two universes alike except for the location of the material world in
In analytical mechanics, a branch of theoretical physics, Routhian mechanics is a hybrid formulation of Lagrangian mechanics and Hamiltonian mechanics developed by Edward John Routh. Correspondingly, the Routhian is the function which replaces both the Lagrangian and Hamiltonian functions; the Routhian, like the Hamiltonian, can be obtained from a Legendre transform of the Lagrangian, has a similar mathematical form to the Hamiltonian, but is not the same. The difference between the Lagrangian and Routhian functions are their variables. For a given set of generalized coordinates representing the degrees of freedom in the system, the Lagrangian is a function of the coordinates and velocities, while the Hamiltonian is a function of the coordinates and momenta; the Routhian differs from these functions in that some coordinates are chosen to have corresponding generalized velocities, the rest to have corresponding generalized momenta. This choice is arbitrary, can be done to simplify the problem.
It has the consequence that the Routhian equations are the Hamiltonian equations for some coordinates and corresponding momenta, the Lagrangian equations for the rest of the coordinates and their velocities. In each case the Lagrangian and Hamiltonian functions are replaced by a single function, the Routhian; the full set thus has the advantages of both sets of equations, with the convenience of splitting one set of coordinates to the Hamilton equations, the rest to the Lagrangian equations. The Routhian approach may offer no advantage, but one notable case where this is useful is when a system has cyclic coordinates, by definition those coordinates which do not appear in the original Lagrangian; the Lagrangian equations are powerful results, used in theory and practice, since the equations of motion in the coordinates are easy to set up. However, if cyclic coordinates occur there will still be equations to solve for all the coordinates, including the cyclic coordinates despite their absence in the Lagrangian.
The Hamiltonian equations are useful theoretical results, but less useful in practice because coordinates and momenta are related together in the solutions - after solving the equations the coordinates and momenta must be eliminated from each other. The Hamiltonian equations are suited to cyclic coordinates because the equations in the cyclic coordinates trivially vanish, leaving only the equations in the non cyclic coordinates; the Routhian approach has the best of both approaches, because cyclic coordinates can be split off to the Hamiltonian equations and eliminated, leaving behind the non cyclic coordinates to be solved from the Lagrangian equations. Overall fewer equations need to be solved compared to the Lagrangian approach; as with the rest of analytical mechanics, Routhian mechanics is equivalent to Newtonian mechanics, all other formulations of classical mechanics, introduces no new physics. It offers an alternative way to solve mechanical problems. In the case of Lagrangian mechanics, the generalized coordinates q1, q2... and the corresponding velocities dq1/dt, dq2/dt... and time t, enter the Lagrangian, L, q ˙ i = d q i d t, where the overdots denote time derivatives.
In Hamiltonian mechanics, the generalized coordinates q1, q2... and the corresponding generalized momenta p1, p2... and time, enter the Hamiltonian, H = ∑ i q ˙ i p i − L, p i = ∂ L ∂ q ˙ i, where the second equation is the definition of the generalized momentum pi corresponding to the coordinate qi. The velocities dqi/dt are expressed as functions of their corresponding momenta by inverting their defining relation. In this context, pi is said to be the momentum "canonically conjugate" to qi; the Routhian is intermediate between L and H. The choice of
In physics, power is the rate of doing work or of transferring heat, i.e. the amount of energy transferred or converted per unit time. Having no direction, it is a scalar quantity. In the International System of Units, the unit of power is the joule per second, known as the watt in honour of James Watt, the eighteenth-century developer of the condenser steam engine. Another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written: power = work time As a physical concept, power requires both a change in the physical system and a specified time in which the change occurs; this is distinct from the concept of work, only measured in terms of a net change in the state of the physical system. The same amount of work is done when carrying a load up a flight of stairs whether the person carrying it walks or runs, but more power is needed for running because the work is done in a shorter amount of time; the output power of an electric motor is the product of the torque that the motor generates and the angular velocity of its output shaft.
The power involved in moving a vehicle is the product of the traction force of the wheels and the velocity of the vehicle. The rate at which a light bulb converts electrical energy into light and heat is measured in watts—the higher the wattage, the more power, or equivalently the more electrical energy is used per unit time; the dimension of power is energy divided by time. The SI unit of power is the watt, equal to one joule per second. Other units of power include ergs per second, metric horsepower, foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550 pounds by one foot in one second, is equivalent to about 746 watts. Other units include a logarithmic measure relative to a reference of 1 milliwatt. Power, as a function of time, is the rate at which work is done, so can be expressed by this equation: P = d W d t where P is power, W is work, t is time; because work is a force F applied over a distance x, W = F ⋅ x for a constant force, power can be rewritten as: P = d W d t = d d t = F ⋅ d x d t = F ⋅ v In fact, this is valid for any force, as a consequence of applying the fundamental theorem of calculus.
As a simple example, burning one kilogram of coal releases much more energy than does detonating a kilogram of TNT, but because the TNT reaction releases energy much more it delivers far more power than the coal. If ΔW is the amount of work performed during a period of time of duration Δt, the average power Pavg over that period is given by the formula P a v g = Δ W Δ t, it is the average amount of energy converted per unit of time. The average power is simply called "power" when the context makes it clear; the instantaneous power is the limiting value of the average power as the time interval Δt approaches zero. P = lim Δ t → 0 P a v g = lim Δ t → 0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration t is given by: W = P t. In the context of energy conversion, it is more customary to use the symbol E rather than W. Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity.
Mechanical power is described as the time derivative of work. In mechanics, the work done by a force F on an object that travels along a curve C is given by the line integral: W C = ∫ C F ⋅ v d t = ∫ C F ⋅ d x, where x defines the path C and v is the velocity along this path. If the force F is derivable from a potential applying the gradi
Kinematics is a branch of classical mechanics that describes the motion of points and systems of bodies without considering the forces that caused the motion. Kinematics, as a field of study, is referred to as the "geometry of motion" and is seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Using arguments from geometry, the position and acceleration of any unknown parts of the system can be determined; the study of how forces act on bodies falls within kinetics, not kinematics. For further details, see analytical dynamics. Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering and biomechanics kinematics is used to describe the motion of systems composed of joined parts such as an engine, a robotic arm or the human skeleton. Geometric transformations called rigid transformations, are used to describe the movement of components in a mechanical system, simplifying the derivation of the equations of motion.
They are central to dynamic analysis. Kinematic analysis is the process of measuring the kinematic quantities used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism and working in reverse, using kinematic synthesis to design a mechanism for a desired range of motion. In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism; the term kinematic is the English version of A. M. Ampère's cinématique, which he constructed from the Greek κίνημα kinema, itself derived from κινεῖν kinein. Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera," once again from the Greek word for movement but the Greek word for writing. Particle kinematics is the study of the trajectory of a particle.
The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is located at your home, such that East is the x-direction and North is the y-direction the coordinate vector to the base of the tower is r =. If the tower is 50 m high the coordinate vector to the top of the tower is r =. In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move in a surface, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without those observations being described with respect to a reference frame; the position vector of a particle is a vector drawn from the origin of the reference frame to the particle. It expresses both the distance of the point from its direction from the origin. In three dimensions, the position of point P can be expressed as P = = x P ı ^ + y P ȷ ^ + z P k ^, where x P, y P, z P are the Cartesian coordinates and ı ^, ȷ ^ and k ^ are the unit vectors along the x, y, z coordinate axes, respectively.
The magnitude of the position vector | P | gives the distance between the origin. | P | = x P 2 + y P 2 + z P 2. The direction cosines of the position vector provide a quantitative measure of direction, it is important to note. The position vector of a given particle is different relative to different frames of reference; the trajectory of a particle is a vector function of time, P, which defines the curve traced by the moving particle, given by P = x P ı ^ + y P ȷ ^ + z P k ^, where