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115-512: There are several distinct branches of mathematical physics, and these roughly correspond to particular historical parts of our world. Applying the techniques of mathematical physics to classical mechanics typically involves the rigorous, abstract, and advanced reformulation of Newtonian mechanics in terms of Lagrangian mechanics and Hamiltonian mechanics (including both approaches in the presence of constraints). Both formulations are embodied in analytical mechanics and lead to an understanding of

230-597: A k ⋅ ∂ r k ∂ q j = d d t ∂ T ∂ q ˙ j − ∂ T ∂ q j . {\displaystyle \sum _{k=1}^{N}m_{k}\mathbf {a} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}.} Now D'Alembert's principle

345-469: A k ) ⋅ δ r k = 0. {\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.} Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion. The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up

460-833: A k ( d d t ∂ T ∂ ξ ˙ k − ∂ T ∂ ξ k ) , ξ ˙ a ≡ d ξ a d t , {\displaystyle F^{a}=m\left({\frac {\mathrm {d} ^{2}\xi ^{a}}{\mathrm {d} t^{2}}}+\Gamma ^{a}{}_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}\right)=g^{ak}\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {\xi }}^{k}}}-{\frac {\partial T}{\partial \xi ^{k}}}\right),\quad {\dot {\xi }}^{a}\equiv {\frac {\mathrm {d} \xi ^{a}}{\mathrm {d} t}},} where F

575-508: A Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles in the absence of an electromagnetic field is given by L = T − V , {\displaystyle L=T-V,} where T = 1 2 ∑ k = 1 N m k v k 2 {\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}}

690-405: A complete system of heliocentric cosmology anchored on the principle of vortex motion, Cartesian physics , whose widespread acceptance helped bring the demise of Aristotelian physics. Descartes used mathematical reasoning as a model for science, and developed analytic geometry , which in time allowed the plotting of locations in 3D space ( Cartesian coordinates ) and marking their progressions along

805-486: A distance —with a gravitational field . The gravitational field is Minkowski spacetime itself, the 4D topology of Einstein aether modeled on a Lorentzian manifold that "curves" geometrically, according to the Riemann curvature tensor . The concept of Newton's gravity: "two masses attract each other" replaced by the geometrical argument: "mass transform curvatures of spacetime and free falling particles with mass move along

920-423: A form similar to the total differential of L , but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An integration by parts with respect to time can transfer the time derivative of δq j to the ∂ L /∂(d q j /d t ), in the process exchanging d( δq j )/d t for δq j , allowing

1035-464: A framework of absolute space —hypothesized by Newton as a physically real entity of Euclidean geometric structure extending infinitely in all directions—while presuming absolute time , supposedly justifying knowledge of absolute motion, the object's motion with respect to absolute space. The principle of Galilean invariance/relativity was merely implicit in Newton's theory of motion. Having ostensibly reduced

1150-416: A geodesic curve in the spacetime" ( Riemannian geometry already existed before the 1850s, by mathematicians Carl Friedrich Gauss and Bernhard Riemann in search for intrinsic geometry and non-Euclidean geometry.), in the vicinity of either mass or energy. (Under special relativity—a special case of general relativity—even massless energy exerts gravitational effect by its mass equivalence locally "curving"

1265-474: A heuristic framework devised by Arnold Sommerfeld (1868–1951) and Niels Bohr (1885–1962), but this was soon replaced by the quantum mechanics developed by Max Born (1882–1970), Louis de Broglie (1892–1987), Werner Heisenberg (1901–1976), Paul Dirac (1902–1984), Erwin Schrödinger (1887–1961), Satyendra Nath Bose (1894–1974), and Wolfgang Pauli (1900–1958). This revolutionary theoretical framework

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1380-421: A magnetic field is present, the expression for the potential energy needs restating. And for dissipative forces (e.g., friction ), another function must be introduced alongside Lagrangian often referred to as a "Rayleigh dissipation function" to account for the loss of energy. One or more of the particles may each be subject to one or more holonomic constraints ; such a constraint is described by an equation of

1495-413: A pair ( M , L ) consisting of a configuration space M and a smooth function L {\textstyle L} within that space called a Lagrangian . For many systems, L = T − V , where T and V are the kinetic and potential energy of the system, respectively. The stationary action principle requires that the action functional of the system derived from L must remain at

1610-560: A particle theory of light, the Dutch Christiaan Huygens (1629–1695) developed the wave theory of light, published in 1690. By 1804, Thomas Young 's double-slit experiment revealed an interference pattern, as though light were a wave, and thus Huygens's wave theory of light, as well as Huygens's inference that light waves were vibrations of the luminiferous aether , was accepted. Jean-Augustin Fresnel modeled hypothetical behavior of

1725-536: A separate field, which includes the theory of phase transitions . It relies upon the Hamiltonian mechanics (or its quantum version) and it is closely related with the more mathematical ergodic theory and some parts of probability theory . There are increasing interactions between combinatorics and physics , in particular statistical physics. The usage of the term "mathematical physics" is sometimes idiosyncratic . Certain parts of mathematics that initially arose from

1840-453: A stationary point (a maximum , minimum , or saddle ) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations. Newton's laws and the concept of forces are the usual starting point for teaching about mechanical systems. This method works well for many problems, but for others the approach is nightmarishly complicated. For example, in calculation of

1955-409: A system of N particles, all of these equations apply to each particle in the system). The equation of motion for a particle of constant mass m is Newton's second law of 1687, in modern vector notation F = m a , {\displaystyle \mathbf {F} =m\mathbf {a} ,} where a is its acceleration and F the resultant force acting on it. Where the mass is varying,

2070-1438: A time increment, since this is a virtual displacement, one along the constraints in an instant of time. The first term in D'Alembert's principle above is the virtual work done by the non-constraint forces N k along the virtual displacements δ r k , and can without loss of generality be converted into the generalized analogues by the definition of generalized forces Q j = ∑ k = 1 N N k ⋅ ∂ r k ∂ q j , {\displaystyle Q_{j}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot {\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}},} so that ∑ k = 1 N N k ⋅ δ r k = ∑ k = 1 N N k ⋅ ∑ j = 1 n ∂ r k ∂ q j δ q j = ∑ j = 1 n Q j δ q j . {\displaystyle \sum _{k=1}^{N}\mathbf {N} _{k}\cdot \delta \mathbf {r} _{k}=\sum _{k=1}^{N}\mathbf {N} _{k}\cdot \sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}=\sum _{j=1}^{n}Q_{j}\delta q_{j}.} This

2185-477: A vector of partial derivatives ∂/∂ with respect to the indicated variables (not a derivative with respect to the entire vector). Each overdot is a shorthand for a time derivative . This procedure does increase the number of equations to solve compared to Newton's laws, from 3 N to 3 N + C , because there are 3 N coupled second-order differential equations in the position coordinates and multipliers, plus C constraint equations. However, when solved alongside

2300-409: Is δ S = 0. {\displaystyle \delta S=0.} Instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is one of several action principles . Historically,

2415-1121: Is explicitly time-dependent . If neither the potential nor the kinetic energy depend on time, then the Lagrangian L ( r 1 , r 2 , ... v 1 , v 2 , ...) is explicitly independent of time . In either case, the Lagrangian always has implicit time dependence through the generalized coordinates. With these definitions, Lagrange's equations of the first kind are ∂ L ∂ r k − d d t ∂ L ∂ r ˙ k + ∑ i = 1 C λ i ∂ f i ∂ r k = 0 , {\displaystyle {\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}+\sum _{i=1}^{C}\lambda _{i}{\frac {\partial f_{i}}{\partial \mathbf {r} _{k}}}=0,} where k = 1, 2, ..., N labels

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2530-519: Is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécanique analytique . Lagrangian mechanics describes a mechanical system as

2645-936: Is a point in the configuration space of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the total derivative of its position with respect to time, is q ˙ j = d q j d t , v k = ∑ j = 1 n ∂ r k ∂ q j q ˙ j + ∂ r k ∂ t . {\displaystyle {\dot {q}}_{j}={\frac {\mathrm {d} q_{j}}{\mathrm {d} t}},\quad \mathbf {v} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}{\dot {q}}_{j}+{\frac {\partial \mathbf {r} _{k}}{\partial t}}.} Given this v k ,

2760-502: Is a tradition of mathematical analysis of nature that goes back to the ancient Greeks; examples include Euclid ( Optics ), Archimedes ( On the Equilibrium of Planes , On Floating Bodies ), and Ptolemy ( Optics , Harmonics ). Later, Islamic and Byzantine scholars built on these works, and these ultimately were reintroduced or became available to the West in the 12th century and during

2875-478: Is a useful simplification to treat it as a point particle . For a system of N point particles with masses m 1 , m 2 , ..., m N , each particle has a position vector , denoted r 1 , r 2 , ..., r N . Cartesian coordinates are often sufficient, so r 1 = ( x 1 , y 1 , z 1 ) , r 2 = ( x 2 , y 2 , z 2 ) and so on. In three-dimensional space , each position vector requires three coordinates to uniquely define

2990-424: Is based on a probabilistic interpretation of states, and evolution and measurements in terms of self-adjoint operators on an infinite-dimensional vector space. That is called Hilbert space (introduced by mathematicians David Hilbert (1862–1943), Erhard Schmidt (1876–1959) and Frigyes Riesz (1880–1956) in search of generalization of Euclidean space and study of integral equations), and rigorously defined within

3105-477: Is half of the conversion to generalized coordinates. It remains to convert the acceleration term into generalized coordinates, which is not immediately obvious. Recalling the Lagrange form of Newton's second law, the partial derivatives of the kinetic energy with respect to the generalized coordinates and velocities can be found to give the desired result: ∑ k = 1 N m k

3220-706: Is in the generalized coordinates as required, ∑ j = 1 n [ Q j − ( d d t ∂ T ∂ q ˙ j − ∂ T ∂ q j ) ] δ q j = 0 , {\displaystyle \sum _{j=1}^{n}\left[Q_{j}-\left({\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}\right)\right]\delta q_{j}=0,} and since these virtual displacements δq j are independent and nonzero,

3335-416: Is minimized along the path that the particle actually takes. This choice eliminates the need for the constraint force to enter into the resultant generalized 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

3450-495: Is some external field or external driving force changing with time, the potential changes with time, so most generally V = V ( r 1 , r 2 , ..., v 1 , v 2 , ..., t ). As already noted, this form of L is applicable to many important classes of system, but not everywhere. For relativistic Lagrangian mechanics it must be replaced as a whole by a function consistent with special relativity (scalar under Lorentz transformations) or general relativity (4-scalar). Where

3565-456: Is still valid even if the coordinates L is expressed in are not independent, here r k , but the constraints are still assumed to be holonomic. As always the end points are fixed δ r k ( t 1 ) = δ r k ( t 2 ) = 0 for all k . What cannot be done is to simply equate the coefficients of δ r k to zero because the δ r k are not independent. Instead, the method of Lagrange multipliers can be used to include

Mathematical physics - Misplaced Pages Continue

3680-530: Is the a -th contravariant component of the resultant force acting on the particle, Γ bc are the Christoffel symbols of the second kind, T = 1 2 m g b c d ξ b d t d ξ c d t {\displaystyle T={\frac {1}{2}}mg_{bc}{\frac {\mathrm {d} \xi ^{b}}{\mathrm {d} t}}{\frac {\mathrm {d} \xi ^{c}}{\mathrm {d} t}}}

3795-487: Is the area of interaction between physics and combinatorics . Combinatorics has always played an important role in quantum field theory and statistical physics . However, combinatorial physics only emerged as a specific field after a seminal work by Alain Connes and Dirk Kreimer , showing that the renormalization of Feynman diagrams can be described by a Hopf algebra . Combinatorial physics can be characterized by

3910-412: Is the kinetic energy of the particle, and g bc the covariant components of the metric tensor of the curvilinear coordinate system. All the indices a , b , c , each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates. It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of

4025-407: Is the shortest paths, but not necessarily). In flat 3D real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation and states that free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forces F ≠ 0 , the particle accelerates due to forces acting on it and deviates away from

4140-399: Is the total kinetic energy of the system, equaling the sum Σ of the kinetic energies of the N {\displaystyle N} particles. Each particle labeled k {\displaystyle k} has mass m k , {\displaystyle m_{k},} and v k = v k · v k is the magnitude squared of its velocity, equivalent to

4255-793: Is therefore n = 3 N − C . We can transform each position vector to a common set of n generalized coordinates , conveniently written as an n -tuple q = ( q 1 , q 2 , ... q n ) , by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time: r k = r k ( q , t ) = ( x k ( q , t ) , y k ( q , t ) , z k ( q , t ) , t ) . {\displaystyle \mathbf {r} _{k}=\mathbf {r} _{k}(\mathbf {q} ,t)={\big (}x_{k}(\mathbf {q} ,t),y_{k}(\mathbf {q} ,t),z_{k}(\mathbf {q} ,t),t{\big )}.} The vector q

4370-551: The Euler–Lagrange equations , or Lagrange's equations of the second kind d d t ( ∂ L ∂ q ˙ j ) = ∂ L ∂ q j {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)={\frac {\partial L}{\partial q_{j}}}} are mathematical results from

4485-506: The Renaissance . In the first decade of the 16th century, amateur astronomer Nicolaus Copernicus proposed heliocentrism , and published a treatise on it in 1543. He retained the Ptolemaic idea of epicycles , and merely sought to simplify astronomy by constructing simpler sets of epicyclic orbits. Epicycles consist of circles upon circles. According to Aristotelian physics , the circle was

4600-471: The calculus of variations , which can also be used in mechanics. Substituting in the Lagrangian L ( q , d q /d t , t ) gives the equations of motion of the system. The number of equations has decreased compared to Newtonian mechanics, from 3 N to n = 3 N − C coupled second-order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for. Although

4715-402: The dot product of the velocity with itself. Kinetic energy T is the energy of the system's motion and is a function only of the velocities v k , not the positions r k , nor time t , so T = T ( v 1 , v 2 , ...). V , the potential energy of the system, reflects the energy of interaction between the particles, i.e. how much energy any one particle has due to all

Mathematical physics - Misplaced Pages Continue

4830-450: The equations of motion are given by Newton's laws . The second law "net force equals mass times acceleration ", ∑ F = m d 2 r d t 2 , {\displaystyle \sum \mathbf {F} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}},} applies to each particle. For an N -particle system in 3 dimensions, there are 3 N second-order ordinary differential equations in

4945-462: The variational principles of mechanics, of Fermat , Maupertuis , Euler , Hamilton , and others. Hamilton's principle can be applied to nonholonomic constraints if the constraint equations can be put into a certain form, a linear combination of first order differentials in the coordinates. The resulting constraint equation can be rearranged into first order differential equation. This will not be given here. The Lagrangian L can be varied in

5060-546: The 1880s, there was a prominent paradox that an observer within Maxwell's electromagnetic field measured it at approximately constant speed, regardless of the observer's speed relative to other objects within the electromagnetic field. Thus, although the observer's speed was continually lost relative to the electromagnetic field, it was preserved relative to other objects in the electromagnetic field. And yet no violation of Galilean invariance within physical interactions among objects

5175-817: The Cartesian r k coordinates, for N particles, ∫ t 1 t 2 ∑ k = 1 N ( ∂ L ∂ r k − d d t ∂ L ∂ r ˙ k ) ⋅ δ r k d t = 0. {\displaystyle \int _{t_{1}}^{t_{2}}\sum _{k=1}^{N}\left({\frac {\partial L}{\partial \mathbf {r} _{k}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {\mathbf {r} }}_{k}}}\right)\cdot \delta \mathbf {r} _{k}\,\mathrm {d} t=0.} Hamilton's principle

5290-468: The Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle, F = 0 , it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation are geodesics , the curves of extremal length between two points in space (these may end up being minimal, that

5405-475: The Euler–Lagrange equations can only account for non-conservative forces if a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations. The Euler–Lagrange equations also follow from the calculus of variations . The variation of

5520-562: The Keplerian celestial laws of motion as well as Galilean terrestrial laws of motion to a unifying force, Newton achieved great mathematical rigor, but with theoretical laxity. In the 18th century, the Swiss Daniel Bernoulli (1700–1782) made contributions to fluid dynamics , and vibrating strings . The Swiss Leonhard Euler (1707–1783) did special work in variational calculus , dynamics, fluid dynamics, and other areas. Also notable

5635-571: The Lagrangian as L = T − V obtains Lagrange's equations of the second kind or the Euler–Lagrange equations of motion ∂ L ∂ q j − d d t ∂ L ∂ q ˙ j = 0. {\displaystyle {\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0.} However,

5750-938: The Lagrangian is δ L = ∑ j = 1 n ( ∂ L ∂ q j δ q j + ∂ L ∂ q ˙ j δ q ˙ j ) , δ q ˙ j ≡ δ d q j d t ≡ d ( δ q j ) d t , {\displaystyle \delta L=\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta {\dot {q}}_{j}\right),\quad \delta {\dot {q}}_{j}\equiv \delta {\frac {\mathrm {d} q_{j}}{\mathrm {d} t}}\equiv {\frac {\mathrm {d} (\delta q_{j})}{\mathrm {d} t}},} which has

5865-554: The Lagrangian is another quantity called the action , defined as S = ∫ t 1 t 2 L d t , {\displaystyle S=\int _{t_{1}}^{t_{2}}L\,\mathrm {d} t,} which is a functional ; it takes in the Lagrangian function for all times between t 1 and t 2 and returns a scalar value. Its dimensions are the same as [ angular momentum ], [energy]·[time], or [length]·[momentum]. With this definition Hamilton's principle

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5980-804: The aether. The English physicist Michael Faraday introduced the theoretical concept of a field—not action at a distance. Mid-19th century, the Scottish James Clerk Maxwell (1831–1879) reduced electricity and magnetism to Maxwell's electromagnetic field theory, whittled down by others to the four Maxwell's equations . Initially, optics was found consequent of Maxwell's field. Later, radiation and then today's known electromagnetic spectrum were found also consequent of this electromagnetic field. The English physicist Lord Rayleigh [1842–1919] worked on sound . The Irishmen William Rowan Hamilton (1805–1865), George Gabriel Stokes (1819–1903) and Lord Kelvin (1824–1907) produced several major works: Stokes

6095-432: The allowed paths the particles can move along, but not where they are or how fast they go at every instant of time. Nonholonomic constraints depend on the particle velocities, accelerations, or higher derivatives of position. Lagrangian mechanics can only be applied to systems whose constraints, if any, are all holonomic . Three examples of nonholonomic constraints are: when the constraint equations are non-integrable, when

6210-410: The axiomatic modern version by John von Neumann in his celebrated book Mathematical Foundations of Quantum Mechanics , where he built up a relevant part of modern functional analysis on Hilbert spaces, the spectral theory (introduced by David Hilbert who investigated quadratic forms with infinitely many variables. Many years later, it had been revealed that his spectral theory is associated with

6325-412: The blending of some mathematical aspect and theoretical physics aspect. Although related to theoretical physics , mathematical physics in this sense emphasizes the mathematical rigour of the similar type as found in mathematics. On the other hand, theoretical physics emphasizes the links to observations and experimental physics , which often requires theoretical physicists (and mathematical physicists in

6440-616: The coefficients can be equated to zero, resulting in Lagrange's equations or the generalized equations of motion , Q j = d d t ∂ T ∂ q ˙ j − ∂ T ∂ q j {\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial T}{\partial {\dot {q}}_{j}}}-{\frac {\partial T}{\partial q_{j}}}} These equations are equivalent to Newton's laws for

6555-673: The condition δq j ( t 1 ) = δq j ( t 2 ) = 0 holds for all j , the terms not integrated are zero. If in addition the entire time integral of δL is zero, then because the δq j are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of δq j must also be zero. Then we obtain the equations of motion. This can be summarized by Hamilton's principle : ∫ t 1 t 2 δ L d t = 0. {\displaystyle \int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t=0.} The time integral of

6670-400: The configuration of the system consistent with the constraint forces acting on the system at an instant of time , i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it. Virtual work is

6785-423: The constraints have inequalities, or when the constraints involve complicated non-conservative forces like friction. Nonholonomic constraints require special treatment, and one may have to revert to Newtonian mechanics or use other methods. If T or V or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian L ( r 1 , r 2 , ... v 1 , v 2 , ... t )

6900-914: The constraints. Multiplying each constraint equation f i ( r k , t ) = 0 by a Lagrange multiplier λ i for i = 1, 2, ..., C , and adding the results to the original Lagrangian, gives the new Lagrangian L ′ = L ( r 1 , r 2 , … , r ˙ 1 , r ˙ 2 , … , t ) + ∑ i = 1 C λ i ( t ) f i ( r k , t ) . {\displaystyle L'=L(\mathbf {r} _{1},\mathbf {r} _{2},\ldots ,{\dot {\mathbf {r} }}_{1},{\dot {\mathbf {r} }}_{2},\ldots ,t)+\sum _{i=1}^{C}\lambda _{i}(t)f_{i}(\mathbf {r} _{k},t).} Combinatorics and physics Combinatorial physics or physical combinatorics

7015-439: The context of physics) and Newton's method to solve problems in mathematics and physics. He was extremely successful in his application of calculus and other methods to the study of motion. Newton's theory of motion, culminating in his Philosophiæ Naturalis Principia Mathematica ( Mathematical Principles of Natural Philosophy ) in 1687, modeled three Galilean laws of motion along with Newton's law of universal gravitation on

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7130-429: The coordinates of the position r k = ( x k , y k , z k ) are linked together by a constraint equation, so are those of the virtual displacements δ r k = ( δx k , δy k , δz k ) . Since the generalized coordinates are independent, we can avoid the complications with the δ r k by converting to virtual displacements in the generalized coordinates. These are related in

7245-772: The curved geometry construction to model 3D space together with the 1D axis of time by treating the temporal axis like a fourth spatial dimension—altogether 4D spacetime—and declared the imminent demise of the separation of space and time. Einstein initially called this "superfluous learnedness", but later used Minkowski spacetime with great elegance in his general theory of relativity , extending invariance to all reference frames—whether perceived as inertial or as accelerated—and credited this to Minkowski, by then deceased. General relativity replaces Cartesian coordinates with Gaussian coordinates , and replaces Newton's claimed empty yet Euclidean space traversed instantly by Newton's vector of hypothetical gravitational force—an instant action at

7360-456: The curved geometry, replacing rectilinear axis by curved ones. Gauss also introduced another key tool of modern physics, the curvature. Gauss's work was limited to two dimensions. Extending it to three or more dimensions introduced a lot of complexity, with the need of the (not yet invented) tensors. It was Riemman the one in charge to extend curved geometry to N dimensions. In 1908, Einstein's former mathematics professor Hermann Minkowski , applied

7475-575: The deep interplay between the notions of symmetry and conserved quantities during the dynamical evolution of mechanical systems, as embodied within the most elementary formulation of Noether's theorem . These approaches and ideas have been extended to other areas of physics, such as statistical mechanics , continuum mechanics , classical field theory , and quantum field theory . Moreover, they have provided multiple examples and ideas in differential geometry (e.g., several notions in symplectic geometry and vector bundles ). Within mathematics proper,

7590-459: The development of physics are not, in fact, considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equations and symplectic geometry are generally viewed as purely mathematical disciplines, whereas dynamical systems and Hamiltonian mechanics belong to mathematical physics. John Herapath used the term for the title of his 1847 text on "mathematical principles of natural philosophy",

7705-440: The development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quantum mechanics , quantum field theory , and quantum statistical mechanics has motivated results in operator algebras . The attempt to construct a rigorous mathematical formulation of quantum field theory has also brought about some progress in fields such as representation theory . There

7820-495: The electromagnetic field's invariance and Galilean invariance by discarding all hypotheses concerning aether, including the existence of aether itself. Refuting the framework of Newton's theory— absolute space and absolute time —special relativity refers to relative space and relative time , whereby length contracts and time dilates along the travel pathway of an object. Cartesian coordinates arbitrarily used rectilinear coordinates. Gauss, inspired by Descartes' work, introduced

7935-539: The equation needs to be generalised to take the time derivative of the momentum. In three spatial dimensions, this is a system of three coupled second-order ordinary differential equations to solve, since there are three components in this vector equation. The solution is the position vector r of the particle at time t , subject to the initial conditions of r and v when t = 0. Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems

8050-450: The equations of Kepler's laws of planetary motion . An enthusiastic atomist, Galileo Galilei in his 1623 book The Assayer asserted that the "book of nature is written in mathematics". His 1632 book, about his telescopic observations, supported heliocentrism. Having introduced experimentation, Galileo then refuted geocentric cosmology by refuting Aristotelian physics itself. Galileo's 1638 book Discourse on Two New Sciences established

8165-518: The equations of motion can become complicated. In a set of curvilinear coordinates ξ = ( ξ , ξ , ξ ), the law in tensor index notation is the "Lagrangian form" F a = m ( d 2 ξ a d t 2 + Γ a b c d ξ b d t d ξ c d t ) = g

8280-501: The equations of motion in an arbitrary coordinate system since the displacements δ r k might be connected by a constraint equation, which prevents us from setting the N individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion. If there are constraints on particle k , then since

8395-485: The equations of motion include partial derivatives , the results of the partial derivatives are still ordinary differential equations in the position coordinates of the particles. The total time derivative denoted d/d t often involves implicit differentiation . Both equations are linear in the Lagrangian, but generally are nonlinear coupled equations in the coordinates. For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for

8510-408: The equations of motion, so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion. A fundamental result in analytical mechanics is D'Alembert's principle , introduced in 1708 by Jacques Bernoulli to understand static equilibrium , and developed by D'Alembert in 1743 to solve dynamical problems. The principle asserts for N particles

8625-538: The fields of electromagnetism , waves, fluids , and sound. In the United States, the pioneering work of Josiah Willard Gibbs (1839–1903) became the basis for statistical mechanics . Fundamental theoretical results in this area were achieved by the German Ludwig Boltzmann (1844–1906). Together, these individuals laid the foundations of electromagnetic theory, fluid dynamics, and statistical mechanics. By

8740-412: The first theoretical physicist and one of the founders of modern mathematical physics. The prevailing framework for science in the 16th and early 17th centuries was one borrowed from Ancient Greek mathematics , where geometrical shapes formed the building blocks to describe and think about space, and time was often thought as a separate entity. With the introduction of algebra into geometry, and with it

8855-456: The flow of time. Christiaan Huygens , a talented mathematician and physicist and older contemporary of Newton, was the first to successfully idealize a physical problem by a set of parameters in his Horologium Oscillatorum (1673), and the first to fully mathematize a mechanistic explanation of an unobservable physical phenomenon in Traité de la Lumière (1690). For these reasons, he is considered

8970-477: The form f ( r , t ) = 0. If the number of constraints in the system is C , then each constraint has an equation f 1 ( r , t ) = 0, f 2 ( r , t ) = 0, ..., f C ( r , t ) = 0, each of which could apply to any of the particles. If particle k is subject to constraint i , then f i ( r k , t ) = 0. At any instant of time, the coordinates of a constrained particle are linked together and not independent. The constraint equations determine

9085-802: The formulation of modern theories in physics, including field theory and quantum mechanics. The French mathematical physicist Joseph Fourier (1768 – 1830) introduced the notion of Fourier series to solve the heat equation , giving rise to a new approach to solving partial differential equations by means of integral transforms . Into the early 19th century, following mathematicians in France, Germany and England had contributed to mathematical physics. The French Pierre-Simon Laplace (1749–1827) made paramount contributions to mathematical astronomy , potential theory . Siméon Denis Poisson (1781–1840) worked in analytical mechanics and potential theory . In Germany, Carl Friedrich Gauss (1777–1855) made key contributions to

9200-523: The generalized forces Q i can be derived from a potential V such that Q j = d d t ∂ V ∂ q ˙ j − ∂ V ∂ q j , {\displaystyle Q_{j}={\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial V}{\partial {\dot {q}}_{j}}}-{\frac {\partial V}{\partial q_{j}}},} equating to Lagrange's equations and defining

9315-424: The geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3D space to 4D curved spacetime , the above form of Newton's law also carries over to Einstein 's general relativity , in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense. However, we still need to know the total resultant force F acting on

9430-505: The geometry of the four, unified dimensions of space and time.) Another revolutionary development of the 20th century was quantum theory , which emerged from the seminal contributions of Max Planck (1856–1947) (on black-body radiation ) and Einstein's work on the photoelectric effect . In 1912, a mathematician Henri Poincare published Sur la théorie des quanta . He introduced the first non-naïve definition of quantization in this paper. The development of early quantum physics followed by

9545-717: The idea of a coordinate system, time and space could now be though as axes belonging to the same plane. This essential mathematical framework is at the base of all modern physics and used in all further mathematical frameworks developed in next centuries. By the middle of the 17th century, important concepts such as the fundamental theorem of calculus (proved in 1668 by Scottish mathematician James Gregory ) and finding extrema and minima of functions via differentiation using Fermat's theorem (by French mathematician Pierre de Fermat ) were already known before Leibniz and Newton. Isaac Newton (1642–1727) developed calculus (although Gottfried Wilhelm Leibniz developed similar concepts outside

9660-531: The idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the calculus of variations to mechanical problems, such as the Brachistochrone problem solved by Jean Bernoulli in 1696, as well as Leibniz , Daniel Bernoulli , L'Hôpital around the same time, and Newton the following year. Newton himself was thinking along the lines of the variational calculus, but did not publish. These ideas in turn lead to

9775-2187: The independent virtual displacements to be factorized from the derivatives of the Lagrangian, ∫ t 1 t 2 δ L d t = ∫ t 1 t 2 ∑ j = 1 n ( ∂ L ∂ q j δ q j + d d t ( ∂ L ∂ q ˙ j δ q j ) − d d t ∂ L ∂ q ˙ j δ q j ) d t = ∑ j = 1 n [ ∂ L ∂ q ˙ j δ q j ] t 1 t 2 + ∫ t 1 t 2 ∑ j = 1 n ( ∂ L ∂ q j − d d t ∂ L ∂ q ˙ j ) δ q j d t . {\displaystyle {\begin{aligned}\int _{t_{1}}^{t_{2}}\delta L\,\mathrm {d} t&=\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}\delta q_{j}+{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right)\,\mathrm {d} t\\&=\sum _{j=1}^{n}\left[{\frac {\partial L}{\partial {\dot {q}}_{j}}}\delta q_{j}\right]_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\sum _{j=1}^{n}\left({\frac {\partial L}{\partial q_{j}}}-{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}\right)\delta q_{j}\,\mathrm {d} t.\end{aligned}}} Now, if

9890-402: The kinetic energy in generalized coordinates depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so T = T ( q , q ˙ , t ) . {\displaystyle T=T(\mathbf {q} ,{\dot {\mathbf {q} }},t).} With these definitions,

10005-536: The law of equal free fall as well as the principles of inertial motion, founding the central concepts of what would become today's classical mechanics . By the Galilean law of inertia as well as the principle of Galilean invariance , also called Galilean relativity, for any object experiencing inertia, there is empirical justification for knowing only that it is at relative rest or relative motion—rest or motion with respect to another object. René Descartes developed

10120-862: The location of a point, so there are 3 N coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles; a general point in space is written r = ( x , y , z ) . The velocity of each particle is how fast the particle moves along its path of motion, and is the time derivative of its position, thus v 1 = d r 1 d t , v 2 = d r 2 d t , … , v N = d r N d t . {\displaystyle \mathbf {v} _{1}={\frac {d\mathbf {r} _{1}}{dt}},\mathbf {v} _{2}={\frac {d\mathbf {r} _{2}}{dt}},\ldots ,\mathbf {v} _{N}={\frac {d\mathbf {r} _{N}}{dt}}.} In Newtonian mechanics,

10235-430: The mathematical fields of linear algebra , the spectral theory of operators , operator algebras and, more broadly, functional analysis . Nonrelativistic quantum mechanics includes Schrödinger operators, and it has connections to atomic and molecular physics . Quantum information theory is another subspecialty. The special and general theories of relativity require a rather different type of mathematics. This

10350-429: The more general sense) to use heuristic , intuitive , or approximate arguments. Such arguments are not considered rigorous by mathematicians. Such mathematical physicists primarily expand and elucidate physical theories . Because of the required level of mathematical rigour, these researchers often deal with questions that theoretical physicists have considered to be already solved. However, they can sometimes show that

10465-481: The motion of a torus rolling on a horizontal surface with a pearl sliding inside, the time-varying constraint forces like the angular velocity of the torus, motion of the pearl in relation to the torus made it difficult to determine the motion of the torus with Newton's equations. Lagrangian mechanics adopts energy rather than force as its basic ingredient, leading to more abstract equations capable of tackling more complex problems. Particularly, Lagrange's approach

10580-474: The non-constraint forces . The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle. For a non-conservative force which depends on velocity, it may be possible to find a potential energy function V that depends on positions and velocities. If

10695-435: The others, together with any external influences. For conservative forces (e.g. Newtonian gravity ), it is a function of the position vectors of the particles only, so V = V ( r 1 , r 2 , ...). For those non-conservative forces which can be derived from an appropriate potential (e.g. electromagnetic potential ), the velocities will appear also, V = V ( r 1 , r 2 , ..., v 1 , v 2 , ...). If there

10810-456: The partial derivative of L with respect to the z velocity component of particle 2, defined by v z ,2 = dz 2 / dt , is just ∂ L /∂ v z ,2 ; no awkward chain rules or total derivatives need to be used to relate the velocity component to the corresponding coordinate z 2 ). In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. The number of independent coordinates

10925-500: The particle, which in turn requires the resultant non-constraint force N plus the resultant constraint force C , F = C + N . {\displaystyle \mathbf {F} =\mathbf {C} +\mathbf {N} .} The constraint forces can be complicated, since they generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations. The constraint forces can either be eliminated from

11040-1198: The particles, there is a Lagrange multiplier λ i for each constraint equation f i , and ∂ ∂ r k ≡ ( ∂ ∂ x k , ∂ ∂ y k , ∂ ∂ z k ) , ∂ ∂ r ˙ k ≡ ( ∂ ∂ x ˙ k , ∂ ∂ y ˙ k , ∂ ∂ z ˙ k ) {\displaystyle {\frac {\partial }{\partial \mathbf {r} _{k}}}\equiv \left({\frac {\partial }{\partial x_{k}}},{\frac {\partial }{\partial y_{k}}},{\frac {\partial }{\partial z_{k}}}\right),\quad {\frac {\partial }{\partial {\dot {\mathbf {r} }}_{k}}}\equiv \left({\frac {\partial }{\partial {\dot {x}}_{k}}},{\frac {\partial }{\partial {\dot {y}}_{k}}},{\frac {\partial }{\partial {\dot {z}}_{k}}}\right)} are each shorthands for

11155-516: The perfect form of motion, and was the intrinsic motion of Aristotle's fifth element —the quintessence or universal essence known in Greek as aether for the English pure air —that was the pure substance beyond the sublunary sphere , and thus was celestial entities' pure composition. The German Johannes Kepler [1571–1630], Tycho Brahe 's assistant, modified Copernican orbits to ellipses , formalized in

11270-424: The position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations. In the Lagrangian, the position coordinates and velocity components are all independent variables , and derivatives of the Lagrangian are taken with respect to these separately according to the usual differentiation rules (e.g.

11385-509: 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 units of energy, but 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

11500-521: The previous solution was incomplete, incorrect, or simply too naïve. Issues about attempts to infer the second law of thermodynamics from statistical mechanics are examples. Other examples concern the subtleties involved with synchronisation procedures in special and general relativity ( Sagnac effect and Einstein synchronisation ). The effort to put physical theories on a mathematically rigorous footing not only developed physics but also has influenced developments of some mathematical areas. For example,

11615-531: The principle of Galilean invariance across all inertial frames of reference , while Newton's theory of motion was spared. Austrian theoretical physicist and philosopher Ernst Mach criticized Newton's postulated absolute space. Mathematician Jules-Henri Poincaré (1854–1912) questioned even absolute time. In 1905, Pierre Duhem published a devastating criticism of the foundation of Newton's theory of motion. Also in 1905, Albert Einstein (1879–1955) published his special theory of relativity , newly explaining both

11730-459: The same form as a total differential , δ r k = ∑ j = 1 n ∂ r k ∂ q j δ q j . {\displaystyle \delta \mathbf {r} _{k}=\sum _{j=1}^{n}{\frac {\partial \mathbf {r} _{k}}{\partial q_{j}}}\delta q_{j}.} There is no partial time derivative with respect to time multiplied by

11845-406: The scope at that time being "the causes of heat, gaseous elasticity, gravitation, and other great phenomena of nature". The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems in physics or thought experiments within a mathematically rigorous framework. In this sense, mathematical physics covers a very broad academic realm distinguished only by

11960-441: The spectrum of the hydrogen atom. He was surprised by this application.) in particular. Paul Dirac used algebraic constructions to produce a relativistic model for the electron , predicting its magnetic moment and the existence of its antiparticle, the positron . Prominent contributors to the 20th century's mathematical physics include (ordered by birth date): Lagrangian mechanics In physics , Lagrangian mechanics

12075-578: The theoretical foundations of electricity , magnetism , mechanics , and fluid dynamics . In England, George Green (1793–1841) published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in 1828, which in addition to its significant contributions to mathematics made early progress towards laying down the mathematical foundations of electricity and magnetism. A couple of decades ahead of Newton's publication of

12190-681: The theory of partial differential equation , variational calculus , Fourier analysis , potential theory , and vector analysis are perhaps most closely associated with mathematical physics. These fields were developed intensively from the second half of the 18th century (by, for example, D'Alembert , Euler , and Lagrange ) until the 1930s. Physical applications of these developments include hydrodynamics , celestial mechanics , continuum mechanics , elasticity theory , acoustics , thermodynamics , electricity , magnetism , and aerodynamics . The theory of atomic spectra (and, later, quantum mechanics ) developed almost concurrently with some parts of

12305-477: The use of algebraic concepts to interpret and solve physical problems involving combinatorics. It gives rise to a particularly harmonious collaboration between mathematicians and physicists. Among the significant physical results of combinatorial physics, we may mention the reinterpretation of renormalization as a Riemann–Hilbert problem , the fact that the Slavnov–Taylor identities of gauge theories generate

12420-536: The virtual work, i.e. the work along a virtual displacement, δ r k , is zero: ∑ k = 1 N ( N k + C k − m k a k ) ⋅ δ r k = 0. {\displaystyle \sum _{k=1}^{N}(\mathbf {N} _{k}+\mathbf {C} _{k}-m_{k}\mathbf {a} _{k})\cdot \delta \mathbf {r} _{k}=0.} The virtual displacements , δ r k , are by definition infinitesimal changes in

12535-636: The work done along a virtual displacement for any force (constraint or non-constraint). Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero: ∑ k = 1 N C k ⋅ δ r k = 0 , {\displaystyle \sum _{k=1}^{N}\mathbf {C} _{k}\cdot \delta \mathbf {r} _{k}=0,} so that ∑ k = 1 N ( N k − m k

12650-458: Was group theory , which played an important role in both quantum field theory and differential geometry . This was, however, gradually supplemented by topology and functional analysis in the mathematical description of cosmological as well as quantum field theory phenomena. In the mathematical description of these physical areas, some concepts in homological algebra and category theory are also important. Statistical mechanics forms

12765-498: Was a leader in optics and fluid dynamics; Kelvin made substantial discoveries in thermodynamics ; Hamilton did notable work on analytical mechanics , discovering a new and powerful approach nowadays known as Hamiltonian mechanics . Very relevant contributions to this approach are due to his German colleague mathematician Carl Gustav Jacobi (1804–1851) in particular referring to canonical transformations . The German Hermann von Helmholtz (1821–1894) made substantial contributions in

12880-495: Was detected. As Maxwell's electromagnetic field was modeled as oscillations of the aether , physicists inferred that motion within the aether resulted in aether drift , shifting the electromagnetic field, explaining the observer's missing speed relative to it. The Galilean transformation had been the mathematical process used to translate the positions in one reference frame to predictions of positions in another reference frame, all plotted on Cartesian coordinates , but this process

12995-512: Was replaced by Lorentz transformation , modeled by the Dutch Hendrik Lorentz [1853–1928]. In 1887, experimentalists Michelson and Morley failed to detect aether drift, however. It was hypothesized that motion into the aether prompted aether's shortening, too, as modeled in the Lorentz contraction . It was hypothesized that the aether thus kept Maxwell's electromagnetic field aligned with

13110-531: Was the Italian-born Frenchman, Joseph-Louis Lagrange (1736–1813) for work in analytical mechanics : he formulated Lagrangian mechanics ) and variational methods. A major contribution to the formulation of Analytical Dynamics called Hamiltonian dynamics was also made by the Irish physicist, astronomer and mathematician, William Rowan Hamilton (1805–1865). Hamiltonian dynamics had played an important role in

13225-401: Was to set up independent generalized coordinates for the position and speed of every object, which allows the writing down of a general form of lagrangian (total kinetic energy minus potential energy of the system) and summing this over all possible paths of motion of the particles yielded a formula for the 'action', which he minimized to give a generalized set of equations. This summed quantity

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