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118-896: Since Lagrangian mechanics is based on generalized coordinates , transformations of the coordinates q → Q do not affect the form of Lagrange's equations and, hence, do not affect the form of Hamilton's equations if the momentum is simultaneously changed by a Legendre transformation into P i = ∂ L ∂ Q ˙ i   , {\displaystyle P_{i}={\frac {\partial L}{\partial {\dot {Q}}_{i}}}\ ,} where {   ( P 1 , Q 1 ) ,   ( P 2 , Q 2 ) ,   ( P 3 , Q 3 ) ,   …   } {\displaystyle \left\{\ (P_{1},Q_{1}),\ (P_{2},Q_{2}),\ (P_{3},Q_{3}),\ \ldots \ \right\}} are

236-597: 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

354-469: 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

472-833: 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

590-432: A planar linkage . It is also possible to construct the linkage system so that all of the bodies move on concentric spheres, forming a spherical linkage . In both cases, the degrees of freedom of the links in each system is now three rather than six, and the constraints imposed by joints are now c  = 3 −  f . In this case, the mobility formula is given by and the special cases become An example of

708-3160: A symplectic group . The symplectic conditions are equivalent with indirect conditions as they both lead to the equation ε ˙ = J ∇ ε H {\textstyle {\dot {\varepsilon }}=J\nabla _{\varepsilon }H} , which is used in both of the derivations. The Poisson bracket which is defined as: { u , v } η := ∑ i = 1 n ( ∂ u ∂ q i ∂ v ∂ p i − ∂ u ∂ p i ∂ v ∂ q i ) {\displaystyle \{u,v\}_{\eta }:=\sum _{i=1}^{n}\left({\frac {\partial u}{\partial q_{i}}}{\frac {\partial v}{\partial p_{i}}}-{\frac {\partial u}{\partial p_{i}}}{\frac {\partial v}{\partial q_{i}}}\right)} can be represented in matrix form as: { u , v } η := ( ∇ η u ) T J ( ∇ η v ) {\displaystyle \{u,v\}_{\eta }:=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)} Hence using partial derivative relations and symplectic condition gives: { u , v } η = ( ∇ η u ) T J ( ∇ η v ) = ( M T ∇ ε u ) T J ( M T ∇ ε v ) = ( ∇ ε u ) T M J M T ( ∇ ε v ) = ( ∇ ε u ) T J ( ∇ ε v ) = { u , v } ε {\displaystyle \{u,v\}_{\eta }=(\nabla _{\eta }u)^{T}J(\nabla _{\eta }v)=(M^{T}\nabla _{\varepsilon }u)^{T}J(M^{T}\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}MJM^{T}(\nabla _{\varepsilon }v)=(\nabla _{\varepsilon }u)^{T}J(\nabla _{\varepsilon }v)=\{u,v\}_{\varepsilon }} The symplectic condition can also be recovered by taking u = ε i {\textstyle u=\varepsilon _{i}} and v = ε j {\textstyle v=\varepsilon _{j}} which shows that ( M J M T ) i j = J i j {\textstyle (MJM^{T})_{ij}=J_{ij}} . Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that P i j ( ε ) = { ε i , ε j } η = ( M J M T ) i j {\textstyle {\mathcal {P}}_{ij}(\varepsilon )=\{\varepsilon _{i},\varepsilon _{j}\}_{\eta }=(MJM^{T})_{ij}} , which

826-619: A Lagrange matrix of η {\displaystyle \eta } can be constructed as L ( η ) = M T J M {\textstyle {\mathcal {L}}(\eta )=M^{T}JM} . It can be shown that the symplectic condition is also equivalent to M T J M = J {\textstyle M^{T}JM=J} by using the J − 1 = − J {\textstyle J^{-1}=-J} property. The set of all matrices M {\textstyle M} which satisfy symplectic conditions form

944-508: 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}}

1062-399: A car-like robot can reach any position and orientation in 2-D space, so it needs 3 DOFs to describe its pose, but at any point, you can move it only by a forward motion and a steering angle. So it has two control DOFs and three representational DOFs; i.e. it is non-holonomic. A fixed-wing aircraft, with 3–4 control DOFs (forward motion, roll, pitch, and to a limited extent, yaw) in a 3-D space,

1180-452: A collection of many minute particles (infinite number of DOFs), this is often approximated by a finite DOF system. When motion involving large displacements is the main objective of study (e.g. for analyzing the motion of satellites), a deformable body may be approximated as a rigid body (or even a particle) in order to simplify the analysis. The degree of freedom of a system can be viewed as the minimum number of coordinates required to specify

1298-437: A configuration. Applying this definition, we have: A single rigid body has at most six degrees of freedom (6 DOF) 3T3R consisting of three translations 3T and three rotations 3R . See also Euler angles . For example, the motion of a ship at sea has the six degrees of freedom of a rigid body, and is described as: For example, the trajectory of an airplane in flight has three degrees of freedom and its attitude along

SECTION 10

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1416-423: 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

1534-771: A list of N generalized coordinates that need not transform like a vector under rotation and similarly p represents the corresponding generalized momentum , e.g., q ≡ ( q 1 , q 2 , … , q N − 1 , q N ) p ≡ ( p 1 , p 2 , … , p N − 1 , p N ) . {\displaystyle {\begin{aligned}\mathbf {q} &\equiv \left(q_{1},q_{2},\ldots ,q_{N-1},q_{N}\right)\\\mathbf {p} &\equiv \left(p_{1},p_{2},\ldots ,p_{N-1},p_{N}\right).\end{aligned}}} A dot over

1652-541: A local conservation of the symplectic product. The indirect conditions allow us to prove Liouville's theorem , which states that the volume in phase space is conserved under canonical transformations, i.e., ∫ d q d p = ∫ d Q d P {\displaystyle \int \mathrm {d} \mathbf {q} \,\mathrm {d} \mathbf {p} =\int \mathrm {d} \mathbf {Q} \,\mathrm {d} \mathbf {P} } Lagrangian mechanics In physics , Lagrangian mechanics

1770-421: 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

1888-507: A more general set of transformations of phase space corresponding with less permissive transformations of the Hamiltonian, it provides simpler conditions to obtain results that can be further generalized. All of the following conditions, with the exception of bilinear invariance condition, can be generalized for canonical transformations, including time dependance. Since restricted transformations have no explicit time dependence (by definition),

2006-413: 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

2124-1394: A partial time derivative of a function known as generator, which reduces to being only a function of time for restricted canonical transformations. In addition to leaving the form of the Hamiltonian unchanged, it is also permits the use of the unchanged Hamiltonian in the Hamilton's equations of motion due to the above form as: P ˙ = − ∂ K ∂ Q = − ( ∂ H ∂ Q ) Q , P , t Q ˙ = ∂ K ∂ P = ( ∂ H ∂ P ) Q , P , t {\displaystyle {\begin{alignedat}{3}{\dot {\mathbf {P} }}&=-{\frac {\partial K}{\partial \mathbf {Q} }}&&=-\left({\frac {\partial H}{\partial \mathbf {Q} }}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\{\dot {\mathbf {Q} }}&=\,\,\,\,{\frac {\partial K}{\partial \mathbf {P} }}&&=\,\,\,\,\,\left({\frac {\partial H}{\partial \mathbf {P} }}\right)_{\mathbf {Q} ,\mathbf {P} ,t}\\\end{alignedat}}} Although canonical transformations refers to

2242-482: A planar simple closed chain is the planar four-bar linkage , which is a four-bar loop with four one degree-of-freedom joints and therefore has mobility  M  = 1. A system with several bodies would have a combined DOF that is the sum of the DOFs of the bodies, less the internal constraints they may have on relative motion. A mechanism or linkage containing a number of connected rigid bodies may have more than

2360-684: A single numerical value. The coordinates after transformation are similarly labelled with Q for transformed generalized coordinates and P for transformed generalized momentum. Restricted canonical transformations are coordinate transformations where transformed coordinates Q and P do not have explicit time dependence, i.e., Q = Q ( q , p ) {\textstyle \mathbf {Q} =\mathbf {Q} (\mathbf {q} ,\mathbf {p} )} and P = P ( q , p ) {\textstyle \mathbf {P} =\mathbf {P} (\mathbf {q} ,\mathbf {p} )} . The functional form of Hamilton's equations

2478-453: 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

SECTION 20

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2596-409: 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,

2714-1438: 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

2832-854: A transformation ( q , p ) → ( Q , P ) does not preserve the form of Hamilton's equations but in the absence of time dependence in transformation, some simplifications are possible. Following the formal definition for a canonical transformation, it can be shown that for this type of transformation, the new Hamiltonian (sometimes called the Kamiltonian) can be expressed as: K ( Q , P , t ) = H ( q ( Q , P ) , p ( Q , P ) , t ) + ∂ G ∂ t ( t ) {\displaystyle K(\mathbf {Q} ,\mathbf {P} ,t)=H(q(\mathbf {Q} ,\mathbf {P} ),p(\mathbf {Q} ,\mathbf {P} ),t)+{\frac {\partial G}{\partial t}}(t)} where it differs by

2950-890: A variable or list signifies the time derivative, e.g., q ˙ ≡ d q d t {\displaystyle {\dot {\mathbf {q} }}\equiv {\frac {d\mathbf {q} }{dt}}} and the equalities are read to be satisfied for all coordinates, for example: p ˙ = − ∂ f ∂ q ⟺ p i ˙ = − ∂ f ∂ q i ( i = 1 , … , N ) . {\displaystyle {\dot {\mathbf {p} }}=-{\frac {\partial f}{\partial \mathbf {q} }}\quad \Longleftrightarrow \quad {\dot {p_{i}}}=-{\frac {\partial f}{\partial {q_{i}}}}\quad (i=1,\dots ,N).} The dot product notation between two lists of

3068-477: 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

3186-478: Is p ˙ = − ∂ H ∂ q , q ˙ = ∂ H ∂ p {\displaystyle {\begin{aligned}{\dot {\mathbf {p} }}&=-{\frac {\partial H}{\partial \mathbf {q} }}\,,&{\dot {\mathbf {q} }}&={\frac {\partial H}{\partial \mathbf {p} }}\end{aligned}}} In general,

3304-409: 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,

3422-1121: 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

3540-519: 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

3658-449: Is a good example of an automobile's three independent degrees of freedom. The position and orientation of a rigid body in space is defined by three components of translation and three components of rotation , which means that it has six degrees of freedom. The exact constraint mechanical design method manages the degrees of freedom to neither underconstrain nor overconstrain a device. The position of an n -dimensional rigid body

Canonical transformation - Misplaced Pages Continue

3776-936: 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 ,

3894-478: 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

4012-419: Is also non-holonomic, as it cannot move directly up/down or left/right. A summary of formulas and methods for computing the degrees-of-freedom in mechanical systems has been given by Pennestri, Cavacece, and Vita. In electrical engineering degrees of freedom is often used to describe the number of directions in which a phased array antenna can form either beams or nulls . It is equal to one less than

4130-2951: Is also the result of explicitly calculating the matrix element by expanding it. The Lagrange bracket which is defined as: [ u , v ] η := ∑ i = 1 n ( ∂ q i ∂ u ∂ p i ∂ v − ∂ p i ∂ u ∂ q i ∂ v ) {\displaystyle [u,v]_{\eta }:=\sum _{i=1}^{n}\left({\frac {\partial q_{i}}{\partial u}}{\frac {\partial p_{i}}{\partial v}}-{\frac {\partial p_{i}}{\partial u}}{\frac {\partial q_{i}}{\partial v}}\right)} can be represented in matrix form as: [ u , v ] η := ( ∂ η ∂ u ) T J ( ∂ η ∂ v ) {\displaystyle [u,v]_{\eta }:=\left({\frac {\partial \eta }{\partial u}}\right)^{T}J\left({\frac {\partial \eta }{\partial v}}\right)} Using similar derivation, gives: [ u , v ] ε = ( ∂ u ε ) T J ( ∂ v ε ) = ( M ∂ u η ) T J ( M ∂ v η ) = ( ∂ u η ) T M T J M ( ∂ v η ) = ( ∂ u η ) T J ( ∂ v η ) = [ u , v ] η {\displaystyle [u,v]_{\varepsilon }=(\partial _{u}\varepsilon )^{T}\,J\,(\partial _{v}\varepsilon )=(M\,\partial _{u}\eta )^{T}\,J\,(M\,\partial _{v}\eta )=(\partial _{u}\eta )^{T}\,M^{T}JM\,(\partial _{v}\eta )=(\partial _{u}\eta )^{T}\,J\,(\partial _{v}\eta )=[u,v]_{\eta }} The symplectic condition can also be recovered by taking u = η i {\textstyle u=\eta _{i}} and v = η j {\textstyle v=\eta _{j}} which shows that ( M T J M ) i j = J i j {\textstyle (M^{T}JM)_{ij}=J_{ij}} . Thus these conditions are equivalent to symplectic conditions. Furthermore, it can be seen that L i j ( η ) = [ η i , η j ] ε = ( M T J M ) i j {\textstyle {\mathcal {L}}_{ij}(\eta )=[\eta _{i},\eta _{j}]_{\varepsilon }=(M^{T}JM)_{ij}} , which

4248-790: Is also the result of explicitly calculating the matrix element by expanding it. These set of conditions only apply to restricted canonical transformations or canonical transformations that are independent of time variable. Consider arbitrary variations of two kinds, in a single pair of generalized coordinate and the corresponding momentum: d ε = ( d q 1 , d p 1 , 0 , 0 , … ) , δ ε = ( δ q 1 , δ p 1 , 0 , 0 , … ) . {\textstyle d\varepsilon =(dq_{1},dp_{1},0,0,\ldots ),\quad \delta \varepsilon =(\delta q_{1},\delta p_{1},0,0,\ldots ).} The area of

4366-467: Is considered to have seven DOFs. A shoulder gives pitch, yaw, and roll, an elbow allows for pitch, and a wrist allows for pitch, yaw and roll. Only 3 of those movements would be necessary to move the hand to any point in space, but people would lack the ability to grasp things from different angles or directions. A robot (or object) that has mechanisms to control all 6 physical DOF is said to be holonomic . An object with fewer controllable DOFs than total DOFs

4484-469: Is convenient to define the number of constraints c that a joint imposes in terms of the joint's freedom f , where c  = 6 −  f . In the case of a hinge or slider, which are one degree of freedom joints, have f  = 1 and therefore c  = 6 − 1 = 5. The result is that the mobility of a system formed from n moving links and j joints each with freedom f i , i  = 1, ..., j,

4602-443: Is defined by the rigid transformation , [ T ] = [ A ,  d ], where d is an n -dimensional translation and A is an n  ×  n rotation matrix, which has n translational degrees of freedom and n ( n  − 1)/2 rotational degrees of freedom. The number of rotational degrees of freedom comes from the dimension of the rotation group  SO(n) . A non-rigid or deformable body may be thought of as

4720-477: Is given by Recall that N includes the fixed link. There are two important special cases: (i) a simple open chain, and (ii) a simple closed chain. A single open chain consists of n moving links connected end to end by n joints, with one end connected to a ground link. Thus, in this case N  =  j  + 1 and the mobility of the chain is For a simple closed chain, n moving links are connected end-to-end by n  + 1 joints such that

4838-477: 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

Canonical transformation - Misplaced Pages Continue

4956-706: 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,

5074-416: 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

5192-423: Is said to be non-holonomic, and an object with more controllable DOFs than total DOFs (such as the human arm) is said to be redundant. Although keep in mind that it is not redundant in the human arm because the two DOFs; wrist and shoulder, that represent the same movement; roll, supply each other since they can't do a full 360. The degree of freedom are like different movements that can be made. In mobile robotics,

5310-495: 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

5428-456: 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

5546-2502: Is the Poisson bracket . Similarly for the identity for the conjugate momentum, P m using the form of the "Kamiltonian" it follows that: ∂ K ( Q , P , t ) ∂ P m = ∂ K ( Q ( q , p ) , P ( q , p ) , t ) ∂ q ⋅ ∂ q ∂ P m + ∂ K ( Q ( q , p ) , P ( q , p ) , t ) ∂ p ⋅ ∂ p ∂ P m = ∂ H ( q , p , t ) ∂ q ⋅ ∂ q ∂ P m + ∂ H ( q , p , t ) ∂ p ⋅ ∂ p ∂ P m = ∂ H ∂ q ⋅ ∂ q ∂ P m + ∂ H ∂ p ⋅ ∂ p ∂ P m {\displaystyle {\begin{aligned}{\frac {\partial K(\mathbf {Q} ,\mathbf {P} ,t)}{\partial P_{m}}}&={\frac {\partial K(\mathbf {Q} (\mathbf {q} ,\mathbf {p} ),\mathbf {P} (\mathbf {q} ,\mathbf {p} ),t)}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial K(\mathbf {Q} (\mathbf {q} ,\mathbf {p} ),\mathbf {P} (\mathbf {q} ,\mathbf {p} ),t)}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\\[1ex]&={\frac {\partial H(\mathbf {q} ,\mathbf {p} ,t)}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial H(\mathbf {q} ,\mathbf {p} ,t)}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\\[1ex]&={\frac {\partial H}{\partial \mathbf {q} }}\cdot {\frac {\partial \mathbf {q} }{\partial P_{m}}}+{\frac {\partial H}{\partial \mathbf {p} }}\cdot {\frac {\partial \mathbf {p} }{\partial P_{m}}}\end{aligned}}} Due to

5664-530: 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}}}

5782-447: Is the RSSR spatial four-bar linkage. The sum of the freedom of these joints is eight, so the mobility of the linkage is two, where one of the degrees of freedom is the rotation of the coupler around the line joining the two S joints. It is common practice to design the linkage system so that the movement of all of the bodies are constrained to lie on parallel planes, to form what is known as

5900-412: 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

6018-407: 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

SECTION 50

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6136-399: 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

6254-793: 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

6372-859: The M T J M = J {\textstyle M^{T}JM=J} symplectic condition that the infinitesimal area is conserved under canonical transformation: δ a ( 12 ) = ( δ ε ) T J d ε = ( M δ η ) T J M d η = ( δ η ) T M T J M d η = ( δ η ) T J d η = δ A ( 12 ) . {\textstyle \delta a(12)={(\delta \varepsilon )}^{T}\,J\,d\varepsilon ={(M\delta \eta )}^{T}\,J\,Md\eta ={(\delta \eta )}^{T}\,M^{T}JM\,d\eta ={(\delta \eta )}^{T}\,J\,d\eta =\delta A(12).} Note that

6490-551: 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

6608-471: 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

6726-401: The degrees of freedom ( DOF ) of a mechanical system is the number of independent parameters that define its configuration or state. It is important in the analysis of systems of bodies in mechanical engineering , structural engineering , aerospace engineering , robotics , and other fields. The position of a single railcar (engine) moving along a track has one degree of freedom because

6844-402: 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

6962-450: 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

7080-1208: The indirect conditions to check whether a given transformation is canonical. Sometimes the Hamiltonian relations are represented as: η ˙ = J ∇ η H {\displaystyle {\dot {\eta }}=J\nabla _{\eta }H} Where J := ( 0 I n − I n 0 ) , {\textstyle J:={\begin{pmatrix}0&I_{n}\\-I_{n}&0\\\end{pmatrix}},} and η = [ q 1 ⋮ q n p 1 ⋮ p n ] {\textstyle \mathbf {\eta } ={\begin{bmatrix}q_{1}\\\vdots \\q_{n}\\p_{1}\\\vdots \\p_{n}\\\end{bmatrix}}} . Similarly, let ε = [ Q 1 ⋮ Q n P 1 ⋮ P n ] {\textstyle \mathbf {\varepsilon } ={\begin{bmatrix}Q_{1}\\\vdots \\Q_{n}\\P_{1}\\\vdots \\P_{n}\\\end{bmatrix}}} . From

7198-462: 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

SECTION 60

#1732852147132

7316-817: 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

7434-468: 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

7552-475: 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

7670-572: The Hamiltonian equations for ε ˙ {\textstyle {\dot {\varepsilon }}} , ε ˙ = J ∇ ε K = J ∇ ε H {\displaystyle {\dot {\varepsilon }}=J\nabla _{\varepsilon }K=J\nabla _{\varepsilon }H} where ∇ ε K = ∇ ε H {\textstyle \nabla _{\varepsilon }K=\nabla _{\varepsilon }H} can be used due to

7788-571: 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,

7906-938: 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

8024-554: 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

8142-428: The above is obeyed for any arbitrary variations, it would be only possible if the indirect conditions are met. The form of the equation, v T J w {\textstyle {v}^{T}\,J\,w} is also known as a symplectic product of the vectors v {\textstyle {v}} and w {\textstyle w} and the bilinear invariance condition can be stated as

8260-432: 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

8378-411: The body that forms the fixed frame. Then the degree-of-freedom of the unconstrained system of N  =  n  + 1 is because the fixed body has zero degrees of freedom relative to itself. Joints that connect bodies in this system remove degrees of freedom and reduce mobility. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It

8496-616: 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

8614-673: 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

8732-400: 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

8850-423: 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 )

8968-882: 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).} Degrees of freedom (mechanics) In physics ,

9086-429: 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

9204-574: The degrees of freedom for a single rigid body. Here the term degrees of freedom is used to describe the number of parameters needed to specify the spatial pose of a linkage. It is also defined in context of the configuration space, task space and workspace of a robot. A specific type of linkage is the open kinematic chain , where a set of rigid links are connected at joints ; a joint may provide one DOF (hinge/sliding), or two (cylindrical). Such chains occur commonly in robotics , biomechanics , and for satellites and other space structures. A human arm

9322-539: 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

9440-518: 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

9558-501: 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

9676-485: 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

9794-408: 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

9912-477: 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

10030-519: The form of Kamiltonian. Equating the two equations gives the symplectic condition as: M J M T = J {\displaystyle MJM^{T}=J} The left hand side of the above is called the Poisson matrix of ε {\displaystyle \varepsilon } , denoted as P ( ε ) = M J M T {\textstyle {\mathcal {P}}(\varepsilon )=MJM^{T}} . Similarly,

10148-527: The form of the Hamiltonian equations of motion, P ˙ = − ∂ K ∂ Q Q ˙ = ∂ K ∂ P {\displaystyle {\begin{aligned}{\dot {\mathbf {P} }}&=-{\frac {\partial K}{\partial \mathbf {Q} }}\\{\dot {\mathbf {Q} }}&=\,\,\,\,{\frac {\partial K}{\partial \mathbf {P} }}\end{aligned}}} if

10266-523: 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

10384-1045: The generalized momenta P m leads to two other sets of equations: ( ∂ P m ∂ p n ) q , p = ( ∂ q n ∂ Q m ) Q , P ( ∂ P m ∂ q n ) q , p = − ( ∂ p n ∂ Q m ) Q , P {\displaystyle {\begin{aligned}\left({\frac {\partial P_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial q_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial P_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial p_{n}}{\partial Q_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned}}} These are

10502-424: 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

10620-531: 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

10738-2187: 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

10856-451: The infinitesimal parallelogram is given by: δ a ( 12 ) = d q 1 δ p 1 − δ q 1 d p 1 = ( δ ε ) T J d ε . {\textstyle \delta a(12)=dq_{1}\delta p_{1}-\delta q_{1}dp_{1}={(\delta \varepsilon )}^{T}\,J\,d\varepsilon .} It follows from

10974-402: 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,

11092-862: 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,

11210-481: 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

11328-708: The new coordinates need not be completely oriented in one coordinate momentum plane. Hence, the condition is more generally stated as an invariance of the form ( d ε ) T J δ ε {\textstyle {(d\varepsilon )}^{T}\,J\,\delta \varepsilon } under canonical transformation, expanded as: ∑ δ q ⋅ d p − δ p ⋅ d q = ∑ δ Q ⋅ d P − δ P ⋅ d Q {\displaystyle \sum \delta q\cdot dp-\delta p\cdot dq=\sum \delta Q\cdot dP-\delta P\cdot dQ} If

11446-410: The new co‑ordinates, grouped in canonical conjugate pairs of momenta P i {\displaystyle P_{i}} and corresponding positions Q i , {\displaystyle Q_{i},} for i = 1 , 2 , …   N , {\displaystyle i=1,2,\ldots \ N,} with N {\displaystyle N} being

11564-474: 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

11682-424: The number of degrees of freedom in both co‑ordinate systems. Therefore, coordinate transformations (also called point transformations ) are a type of canonical transformation. However, the class of canonical transformations is much broader, since the old generalized coordinates, momenta and even time may be combined to form the new generalized coordinates and momenta. Canonical transformations that do not include

11800-412: The number of parameters that define the configuration of a set of rigid bodies that are constrained by joints connecting these bodies. Consider a system of n rigid bodies moving in space has 6 n degrees of freedom measured relative to a fixed frame. In order to count the degrees of freedom of this system, include the fixed body in the count of bodies, so that mobility is independent of the choice of

11918-435: 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

12036-456: 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

12154-500: 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

12272-1198: 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

12390-424: 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.

12508-538: The position of the car is defined by the distance along the track. A train of rigid cars connected by hinges to an engine still has only one degree of freedom because the positions of the cars behind the engine are constrained by the shape of the track. An automobile with highly stiff suspension can be considered to be a rigid body traveling on a plane (a flat, two-dimensional space). This body has three independent degrees of freedom consisting of two components of translation and one angle of rotation. Skidding or drifting

12626-509: 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

12744-1126: The relation of partial derivatives, converting the η ˙ = J ∇ η H {\displaystyle {\dot {\eta }}=J\nabla _{\eta }H} relation in terms of partial derivatives with new variables gives η ˙ = J ( M T ∇ ε H ) {\displaystyle {\dot {\eta }}=J(M^{T}\nabla _{\varepsilon }H)} where M := ∂ ( Q , P ) ∂ ( q , p ) {\textstyle M:={\frac {\partial (\mathbf {Q} ,\mathbf {P} )}{\partial (\mathbf {q} ,\mathbf {p} )}}} . Similarly for ε ˙ {\textstyle {\dot {\varepsilon }}} , ε ˙ = M η ˙ = M J M T ∇ ε H {\displaystyle {\dot {\varepsilon }}=M{\dot {\eta }}=MJM^{T}\nabla _{\varepsilon }H} Due to form of

12862-459: 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

12980-454: The same number of coordinates is a shorthand for the sum of the products of corresponding components, e.g., p ⋅ q ≡ ∑ k = 1 N p k q k . {\displaystyle \mathbf {p} \cdot \mathbf {q} \equiv \sum _{k=1}^{N}p_{k}q_{k}.} The dot product (also known as an "inner product") maps the two coordinate lists into one variable representing

13098-1228: The time derivative of a new generalized coordinate Q m is Q ˙ m = ∂ Q m ∂ q ⋅ q ˙ + ∂ Q m ∂ p ⋅ p ˙ = ∂ Q m ∂ q ⋅ ∂ H ∂ p − ∂ Q m ∂ p ⋅ ∂ H ∂ q = { Q m , H } {\displaystyle {\begin{aligned}{\dot {Q}}_{m}&={\frac {\partial Q_{m}}{\partial \mathbf {q} }}\cdot {\dot {\mathbf {q} }}+{\frac {\partial Q_{m}}{\partial \mathbf {p} }}\cdot {\dot {\mathbf {p} }}\\&={\frac {\partial Q_{m}}{\partial \mathbf {q} }}\cdot {\frac {\partial H}{\partial \mathbf {p} }}-{\frac {\partial Q_{m}}{\partial \mathbf {p} }}\cdot {\frac {\partial H}{\partial \mathbf {q} }}\\&=\lbrace Q_{m},H\rbrace \end{aligned}}} where {⋅, ⋅}

13216-428: The time explicitly are called restricted canonical transformations (many textbooks consider only this type). Modern mathematical descriptions of canonical transformations are considered under the broader topic of symplectomorphism which covers the subject with advanced mathematical prerequisites such as cotangent bundles , exterior derivatives and symplectic manifolds . Boldface variables such as q represent

13334-410: The trajectory has three degrees of freedom, for a total of six degrees of freedom. Physical constraints may limit the number of degrees of freedom of a single rigid body.  For example, a block sliding around on a flat table has 3 DOF 2T1R consisting of two translations 2T and 1 rotation 1R .  An XYZ positioning robot like SCARA has 3 DOF 3T lower mobility. The mobility formula counts

13452-1091: The transformation is canonical, the two derived results must be equal, resulting in the equations: ( ∂ Q m ∂ p n ) q , p = − ( ∂ q n ∂ P m ) Q , P ( ∂ Q m ∂ q n ) q , p = ( ∂ p n ∂ P m ) Q , P {\displaystyle {\begin{aligned}\left({\frac {\partial Q_{m}}{\partial p_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=-\left({\frac {\partial q_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\\\left({\frac {\partial Q_{m}}{\partial q_{n}}}\right)_{\mathbf {q} ,\mathbf {p} }&=\left({\frac {\partial p_{n}}{\partial P_{m}}}\right)_{\mathbf {Q} ,\mathbf {P} }\end{aligned}}} The analogous argument for

13570-415: The two ends are connected to the ground link forming a loop. In this case, we have N  =  j and the mobility of the chain is An example of a simple open chain is a serial robot manipulator. These robotic systems are constructed from a series of links connected by six one degree-of-freedom revolute or prismatic joints, so the system has six degrees of freedom. An example of a simple closed chain

13688-536: 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

13806-636: 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

13924-401: 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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