Analytical Dynamics Development of the Canonical Equations
aa r X i v : . [ phy s i c s . c l a ss - ph ] O c t Analytical Dynamics Development of theCanonical Equations
John E. Hurtado ∗ Texas A&M University, College Station, Texas 77843-3141
Abstract
It is most common to construct the Hamiltonian function andHamilton’s canonical equations through a Legendre transformation of theLagrangean function or through the central equation. These common per-spectives, however, seem abstract and detached from classical analytical dy-namics. A new and different approach is presented in which the Hamiltonianfunction is created as one investigates d’Alembert’s equation of motion. Thisformulation directly ties the Hamiltonian function and Hamilton’s canonicalequations to the root of classical analytical dynamics more than any otherapproach.
Keywords
Analytical dynamics, Hamiltonian, canonical equations.Like the Lagrangean and the Appellian, the Hamiltonian is yet anotherdescriptive function to generate the equations of motion of a dynamical sys-tem. As mentioned by Pars ([1] p. 184), “the function contains in itselfa complete description of the motions that are possible for the dynamicalsystem.”It is most common to construct the Hamiltonian function and Hamilton’scanonical equations through a Legendre transformation of the Lagrangeanfunction ([2] § § § ∗ Deputy Director and Chief Technology Officer, Bush Combat Development Complex,Texas A&M University System; and Professor, Department of Aerospace Engineering,College of Engineering; [email protected]. § § m ¨ r − f ) · δ r = 0.(For the simplicity of the current presentation, we consider the motion of asingle particle.) Here, m is the particle’s mass, r is the particle’s positionvector for an inertial observer, and ¨ r , the second inertial time derivative of r ,is the particle’s inertial acceleration vector. The vector f represents the totalof all externally applied forces and δ r represents the virtual displacementvector.The position vector is generally a function of time and a set of coordi-nates, r = r ( t, q ). Here, q = { q , . . . , q n } is a general set of unconstrainedcoordinates where n is the minimum number of coordinates that are neededto completely specify the body’s configuration. The set ˙ q = { ˙ q , . . . , ˙ q n } ,are the corresponding time derivatives and are commonly called generalizedvelocities.As mentioned by Papastavridis ([3] p. 280), the string of kinematic iden-tities ∂ ¨ r /∂ ¨ q k = ∂ ˙ r /∂ ˙ q k = ∂ r /∂q k holds true for any well-behaved func-tion of time and generalized coordinates, like the position vector function r = r ( t, q ). Moreover, the k th holonomic Euler-Lagrange operator equalszero when applied to the time derivatives of such functions, as in E k ( ˙ r ) ≡ d / d t ( ∂ ˙ r /∂ ˙ q k ) − ∂ ˙ r /∂q k = 0. These integrability conditions are crucial toanalytical dynamics.Following Likins ([7] p. 181) and Kane ([8] p. 53), the virtual displacementvector δ r is an imagined displacement away from the particle’s true position2n terms of increments in generalized coordinates and is valid at any instantof time, δ r = ( ∂ r /∂q k ) δq k . Equally, because of the kinematic identities, δ r = ( ∂ ˙ r /∂ ˙ q k ) δq k .Either of these expressions for the virtual displacement vector can beused to define the generalized forces, for example Q k ≡ f · ∂ ˙ r /∂ ˙ q k , andthis in turn leads to a restatement of the d’Alembert-Lagrange equation,( m ¨ r · ∂ ˙ r /∂ ˙ q k − Q k ) δq k = 0. And because each generalized coordinate incre-ment δq k is independent from the others and arbitrary in amount, it mustbe that the parenthetical quantity is identically zero for each value of k . m ¨ r · ∂ ˙ r ∂ ˙ q k = Q k (1)At the heart of analytical dynamics is the realization that m ¨ r · ∂ ˙ r /∂ ˙ q k can be transformed into calculations on a descriptive function. For Lagrange,that function is the kinetic energy function whereas for Appell, that functionis the Appellian. For Hamilton, that function is the Hamiltonian and thejourney from m ¨ r · ∂ ˙ r /∂ ˙ q k to the discovery of, and calculations on this functionis precisely the remaining focus of this note.Consider the calculation of m ¨ r · ∂ ˙ r /∂ ˙ q k . m ¨ r · ∂ ˙ r ∂ ˙ q k = dd t (cid:18) m ˙ r · ∂ ˙ r ∂ ˙ q k (cid:19) − m ˙ r · dd t (cid:18) ∂ ˙ r ∂ ˙ q k (cid:19) = dd t (cid:18) m ˙ r · ∂ ˙ r ∂ ˙ q k (cid:19) − m ˙ r · ∂ ˙ r ∂q k (2)We may rewrite this expression by using the kinetic energy function, T ( t, q, ˙ q )= m ˙ r · ˙ r , and by introducing the concept of conjugate momenta , which hereare defined as the projections of the momentum vector m ˙ r along the partialvelocity vector directions, p k ( t, q, ˙ q ) ≡ m ˙ r · ∂ ˙ r /∂ ˙ q k . m ¨ r · ∂ ˙ r ∂ ˙ q k = ˙ p k − ∂T∂q k (3)Hamilton’s equations are ordinary differential equations in the canonicalvariable set ( q, p ) rather than the traditional variable set ( q, ˙ q ). Thus, inmoving ahead we must establish the explicit relationship between the gen-eralized velocities and conjugate momenta, and then use this relationship totransform Eq. (3). 3o begin, note that the velocity vector may be written in terms of gen-eralized coordinates, generalized velocities and partial derivative vectors,˙ r = ( ∂ ˙ r /∂ ˙ q i ) ˙ q i + ∂ r /∂t . Using this relationship in the definition of con-jugate momenta gives the following: p k ( t, q, ˙ q ) = m ˙ r · ∂ ˙ r ∂ ˙ q k = m ∂ ˙ r ∂ ˙ q i ˙ q i · ∂ ˙ r ∂ ˙ q k + m ∂ r ∂t · ∂ ˙ r ∂ ˙ q k = M ki ˙ q i + G k (4)The definitions of M ik = M ki = M ki ( t, q ) and G k = G k ( t, q ) are clear. Theinverse of this relationship is straightforward, where A jk is the inverse of M ki .˙ q j ( t, q, p ) = A jk ( p k − G k ) (5)Having the relationships between the generalized velocities and conju-gate momenta, we now consider the kinetic energy function in terms of thedifferent variable sets. The kinetic energy function is classically definedas T = m ˙ r · ˙ r . Note that if ˙ r = ˙ r ( t, q, ˙ q ), then T = T ( t, q, ˙ q ); but if˙ r = ˙ r ( t, q, ˙ q ( t, q, p )) = ˙ r ( t, q, p ), then T = T ( t, q, p ). For clarity we will be-gin using a superscript ⋆ notation on a scalar or vector function when thefunction depends on the canonical variables ( q, p ).The explicit forms of the kinetic energy can be determined, and we beginwith the form in terms of traditional variables. T ( t, q, ˙ q ) = 12 m ˙ r · ˙ r = 12 M ij ˙ q i ˙ q j + G i ˙ q i + T where T = 12 m ∂ r ∂t · ∂ r ∂t (6)Using the relationship between the generalized velocities and conjugate mo-menta, it is straightforward to determine the kinetic energy function in termsof canonical variables. T ⋆ ( t, q, p ) = 12 m ˙ r ⋆ · ˙ r ⋆ = 12 A ij p i p j − G i A ij G j + T (7)Having these two forms for the kinetic energy function, we return toEq. (3), where the kinetic energy function depends on traditional variables, T = T ( t, q, ˙ q ), yet Hamilton’s equations are tailored to make use of canonicalvariables ( q, p ). Thus, we use the chain rule of calculus applied to T ⋆ ( t, q, p )to achieve the transition. m ¨ r · ∂ ˙ r ∂ ˙ q k = ˙ p k − ∂T∂q k = ˙ p k − ∂T ⋆ ∂q k − ∂T ⋆ ∂p i ∂p i ∂q k (8)4e can perform the explicit computations in the far right expression ofEq. (8) to complete the transition to the canonical variable set ( q, p ). Inthe end, Eq. (1) becomes the following.˙ p k = − ∂A ij ∂q k p i p j + p i ∂∂q k ( A ij G j ) − ∂∂q k (cid:18) G i A ij G j − T (cid:19) + Q k (9)Equations (5) and (9) are first order differential equations in canonical vari-ables ( q, p ) that governing the system’s motion. But they are far from beingrecognized as Hamilton’s canonical equations.Although we’ve successfully transitioned the equations of motion fromtraditional variables ( q, ˙ q ) to canonical variables ( q, p ), there are a few re-maining steps to arrive at Hamilton’s canonical equations. To focus ourpresentation, we consider the governing equations of motion in canonicalvariables ( q, p ) for the special case of no generalized forces, i.e., Q k = 0.˙ q k = A kj ( p j − G j )˙ p k = ∂T ⋆ ∂q k + ∂T ⋆ ∂p i ∂p i ∂q k = − ∂A ij ∂q k p i p j + p i ∂ ( A ij G j ) ∂q k − ∂ ( G i A ij G j ) ∂q k + ∂T ∂q k Hamilton’s equations involve computations on a single descriptive func-tion called the Hamiltonian function. In our pursuit of this new function, weseek an auxiliary scalar function S ⋆ ( t, q, p ) whose addition to T ⋆ becomes afeasible candidate. This leads us to consider the following.˙ q k = A kj ( p j − G j ) = ∂∂p k ( T ⋆ + S ⋆ ) (10)˙ p k = ∂T ⋆ ∂q k + ∂T ⋆ ∂p i ∂p i ∂q k = − ∂∂q k ( T ⋆ + S ⋆ ) (11)First consider Eq. (10) while recalling that T ⋆ = A ij p i p j − G i A ij G j + T . ∂S ⋆ ∂p k = A kj ( p j − G j ) − ∂T ⋆ ∂p k = A kj ( p j − G j ) − A kj p j = − A kj G j → S ⋆ ( t, q, p ) = − p k A kj G j + s q ( t, q ) (12)Here, s q ( t, q ) is a new function, independent of conjugate momenta, thatarises from integration. 5ext consider Eq. (11). Using the computations we have performed sofar leads to another expression for S ⋆ . ∂S ⋆ ∂q k = − ∂T ⋆ ∂q k − ∂T ⋆ ∂p i ∂p i ∂q k = − p i ∂ ( A ij G j ) ∂q k + ∂ ( G i A ij G j − T ) ∂q k → S ⋆ ( t, q, p ) = − p i A ij G j + G i A ij G j − T + s p ( t, p ) (13)Like s q , the function s p ( t, p ) is a new function, independent of generalizedcoordinates, that arises from integration.Comparing Eqs. (12) and (13) reveals s q ( t, q ) = G i A ij G j − T and s p ( t, p ) = 0, and therefore S ⋆ ( t, q, p ) = − p i A ij G j + G i A ij G j − T . Wenow have our first glimpse of the Hamiltonian function. H ⋆ ( t, q, p ) ≡ T ⋆ + S ⋆ = 12 A ij p i p j − p i A ij G j + 12 G i A ij G j − T (14)Equation (14) is a Hamiltonian function H ⋆ ( t, q, p ), which now allows usto write one form of Hamilton’s equations wherein we reconsider the presenceof generalized forces.˙ q k = A kj ( p j − G j )˙ p k = ∂T ⋆ ∂q k + ∂T ⋆ ∂p i ∂p i ∂q k + Q k → ˙ q k = ∂H ⋆ ∂p k ˙ p k = − ∂H ⋆ ∂q k + Q k (15)As mentioned earlier, the classical definition of the kinetic energy functionis the coordinate-free expression T = m ˙ r · ˙ r , which holds true regardless ofthe variable set. We now show that the auxiliary function S ⋆ can be writtenin a similar coordinate-free manner.Consider the auxiliary function S ⋆ while folding in the relationship be-tween the generalized velocities and conjugate momenta. S ⋆ ( t, q, p ) = − p i A ij G j + G i A ij G j − T = − G j A ji ( p i − G i ) − T = − G j ˙ q j − T = − m ∂ ˙ r ∂ ˙ q j · ∂ r ∂t ˙ q j − m ∂ r ∂t · ∂ r ∂t = − m (cid:18) ∂ ˙ r ∂ ˙ q j ˙ q j + ∂ r ∂t (cid:19) · ∂ r ∂t → S = − m ˙ r · ∂ r ∂t T and auxiliary function S in coordi-nate-free forms allows us to write the Hamiltonian function H in a beautifuland concise similar way. H = 12 m ˙ r · ˙ r − m ˙ r · ∂ r ∂t or H = T − p o (16)Here, we’re suggesting that because m ˙ r · ∂ ˙ r /∂ ˙ q k defines p k , then m ˙ r · ∂ r /∂t could define p o because of the striking similarity in appearance. Furthermore,because p o ≡ m ˙ r · ∂ r /∂t stems from the auxiliary function S and has theunits of energy, it should be called the auxiliary energy .d’Alembert’s equation of motion has led to the current form of Hamilton’sequations. (See the far right set of equations in Eq. (15) and Eq. (16).) Whenthe generalized forces Q k are all derivable from a (collection of) potentialenergy function(s), V = V ( t, q ), so that Q k = − ∂V /∂q k , one can reassign (orre-set) the definition of the Hamiltonian function, H ← H + V , and write Hamilton’s famous canonical equations .˙ p k = − ∂H ⋆ ∂q k ; ˙ q k = ∂H ⋆ ∂p k (17)where H = E − p o = 12 m ˙ r · ˙ r + V − m ˙ r · ∂ r ∂t (18)Here, E = T + V is the system energy and we are following Meirovitch ([9] § any and all generalized forces are derivable from potential energy functions. If only some generalized forces are derivable from potential functions, then H accounts forthe potential energy contributions like before and the remaining nonpotentialgeneralized forces are simply added to the right side of the kinetic equation.It is customary to drop the canonical name in that case.The kinetic energy function, the Appell function, and the Hamiltonianfunction are each descriptive functions that naturally appear as one investi-7ates d’Alembert’s equation of motion.Lagrange: m ¨ r · ∂ ˙ r ∂ ˙ q k = dd t (cid:18) ∂T∂ ˙ q k (cid:19) − ∂T∂q k where T = 12 m ˙ r · ˙ r Appell: m ¨ r · ∂ ¨ r ∂ ¨ q k = ∂ A ∂ ¨ q k where A = 12 m ¨ r · ¨ r Hamilton: m ¨ r · ∂ ˙ r ∂ ˙ q k = ˙ p k = − ∂H ⋆ ∂q k with ˙ q k = ∂H ⋆ ∂p k where H = 12 m ˙ r · ˙ r − m ˙ r · ∂ r ∂t Admittedly, the descriptive functions T and A seem to magically appearin their respective developments whereas more effort is needed to coax theappearance of H ⋆ . Nevertheless, this new approach directly ties the construc-tion of the Hamiltonian function and Hamilton’s equations to d’Alembert’sequation and consequently to classical analytical dynamics.Seeing the Lagrange, Appell, and Hamilton equations side by side, we notethat the Lagrangean and Appellian approaches shown here lead to secondorder differential equations whereas the Hamiltonian approach always andnaturally leads to a set of first order differential equations. We have seen,however, that the Lagrangean and Appellian approaches can also lead tofirst order differential equations when nonholonomic velocity variables ([3]p. 418-419), also called quasi-velocities or generalized speeds ([5] § § f ( t, q ) / d t , in terms of generalized coordinates andgeneralized velocities, L = L ( t, q, ˙ q ); likewise, the Appellian function is theappropriate descriptive function, again to within the addition of a functiond f ( t, q ) / d t , in terms of generalized coordinates, generalized velocities, andgeneralized accelerations, A = A ( t, q, ˙ q, ¨ q ); and the Hamiltonian function isthe appropriate descriptive function in terms of generalized coordinates andother speeds, H ⋆ = H ⋆ ( t, q, p ). 8nterestingly, unlike the Lagrangean and Appellian functions, the Hamil-tonian function will not tolerate the addition of a function d f ( t, q ) / d t : theHamiltonian is unique in that way.Finally, although the development throughout has been for a single par-ticle, it should be clear that the method extends in a straightforward way toa collection of particles, a collection of rigid bodies, or a mix of the two. Funding
This research did not receive any specific grant from funding agencies in thepublic, commercial, or not-for-profit sectors.
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