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9.2: Hamilton's Principle of Stationary Action

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Hamilton’s crowning achievement was his use of the general form of Hamilton’s principle of stationary action S, equation (9.1.2), to derive both Lagrangian mechanics, and Hamiltonian mechanics. Consider the action SA for the extremum path of a system in configuration space, that is, along path A for j=1,2,,n coordinates qj(ti) at initial time ti to qj(tf) at a final time tf as shown in Figure 9.2.1.

10.2.1.PNG
Figure 9.2.1: Extremum path A, plus the neighboring path B, shown in configuration space.

Then the action SA is given by

SA=tftiL(q(t),˙q(t),t)dt

As used in chapter 5.2, a family of neighboring paths is defined by adding an infinitessimal fraction ϵ of a continuous, well-behaved neighboring function ηj where ϵ=0 for the extremum path. That is,

qj(t,ϵ)=qj(t,0)+ϵηj(t)

In contrast to the variational case discussed when deriving Lagrangian mechanics, the variational path used here does not assume that the functions ηi(t) vanish at the end points. Assume that the neighboring path B has an action SB where

SB=tf+Δtti+ΔtL(q(t)+δq(t),˙q(t)+δ˙q(t))dt

Expanding the integrand of SB in Equation ??? gives that, relative to the extremum path A, the incremental change in action is

δS=SBSA=tftij(Lqjδqj+L˙qjδ˙qj)dt+[LΔt]tfti

The second term in the integral can be integrated by parts since δ˙qj=d(δqjdt) leading to

δS=tftij(LqjddtL˙qj)δqjdt+[jL˙qjδqj+LΔt]tfti

Note that Equation ??? includes contributions from the entire path of the integral as well as the variations at the ends of the curve and the Δt terms. Equation ??? leads to the following two pioneering principles of least action in variational mechanics that were developed by Hamilton.

Stationary-action principle in Lagrangian mechanics

Derivation of Lagrangian mechanics in chapter 6 was based on the extremum path for neighboring paths between two given locations q(ti) and q(tf) that the system occupies at the initial and final times ti and tf respectively. For this special case, where the end points do not vary, that is, when δqi(ti)=δqi(tf)=0, and Δti=Δtf=0, then the least action δS for the stationary path ??? reduces to

δS=tftij(LqjddtL˙qj)δqjdt=0

For independent generalized coordinates δqj, the integrand in brackets vanishes leading to the Euler-Lagrange equations. Conversely, if the Euler-Lagrange equations in ??? are satisfied, then, δS=0, that is, the path is stationary. This leads to the statement that the path in configuration space between two configurations q(ti) and q(tf) that the system occupies at times ti and tf respectively, is that for which the action S is stationary. This is a statement of Hamilton’s Principle.

Stationary-action principle in Hamiltonian mechanics

Hamilton used the general variation of the least-action path to derive the basic equations of Hamiltonian mechanics. For the general path, the integral term in Equation ??? vanishes because the Euler-Lagrange equations are obeyed for the stationary path. Thus the only remaining non-zero contributions are due to the end point terms, which can be written by defining the total variation of each end point to be

Δqj=δqj+˙qjΔt

where δqi and ˙qi are evaluated at ti and tf. Then Equation ??? reduces to

δS=[jL˙qjδqj+LΔt]tfti=[jL˙qjΔqj+(jL˙qj˙qj+L)Δt]tfti

Since the generalized momentum pj=L˙qj, then Equation ??? can be expressed in terms of the Hamiltonian and generalized momentum as

δS=[jpjΔqjHΔt]tfti=[pΔqHΔt]tfti

Sqj=L˙qj=pj

Equation ??? contains Hamilton’s Principle of Least-action. Equation ??? gives an alternative relation of the generalized momentum pj that is expressed in terms of the action functional S. Note that equations ??? and ??? were derived directly without invoking reference to the Lagrangian.

Integrating the action δS, Equation ???, between the end points gives the action for the path between t=ti and t=tf, that is, S(qj(ti),t1,qj(tf),t2) to be

S(qj(ti),ti,qj(tf),tf)=fi[p˙qH(q,p,t)]dt

The stationary path is obtained by using the variational principle

δS=δfi[p˙qH(q,p,t)]dt=0

The integrand, I=[p˙qH(q,p,t)], in this modified Hamilton’s principle, can be used in the n Euler-Lagrange equations for j=1,2,3,,n to give

ddt(I˙qj)Iqj=˙pj+Hqj=0

Similarly, the other n Euler-Lagrange equations give

ddt(I˙pj)Ipj=˙qj+Hpj=0

Thus Hamilton’s principle of least-action leads to Hamilton’s equations of motion, that is equations ??? and ???.

The total time derivative of the action S, which is a function of the coordinates and time, is

dSdt=St+njSqj˙qj=St+p˙qj

But the total time derivative of Equation ??? equals

dSdt=p˙qH(q,p,t)

Combining equations ??? and ??? gives the Hamilton-Jacobi equation which is discussed in chapter 15.4. St+H(q,p,t)=0

In summary, Hamilton’s principle of least action leads directly to Hamilton’s equations of motion ???, ??? plus the Hamilton-Jacobi Equation ???. Note that the above discussion has derived both Hamilton’s Principle ???, and Hamilton’s equations of motion ???, ???, directly from Hamilton’s variational concept of stationary action, S, without explicitly invoking the Lagrangian.

Abbreviated action

Hamilton’s Action Principle determines completely the path of the motion and the position on the path as a function of time. If the Lagrangian and the Hamiltonian are time independent, that is, conservative, then H=E and Equation ??? equals

S(qj(t1),t1,qj(t2),t2)=fi[p˙qE]dt=fipδqE(tfti)

The 21pδ˙q term in Equation ???, is called the abbreviated action which is defined as

S0fipδ˙qdt=fipδq

The abbreviated action can be simplified assuming use of the standard Lagrangian L=TU with a velocity-independent potential U, then equation 8.1.4 gives. S0tftinjpj˙qjdt=tfti(L+H)dt=tfti2Tdt=tftipδq

Abbreviated action provides for use of a simplified form of the principle of least action that is based on the kinetic energy, and not potential energy. For conservative systems it determines the path of the motion, but not the time dependence of the motion. Consider virtual motions where the path satisfies energy conservation, and where the end points are held fixed, that is δqi=0, but allow for a variation δt in the final time. Then using the Hamilton-Jacobi equation, ???

δS=Hδt=Eδt

However, Equation ??? gives that

δS=δS0Eδt

Therefore

δS0=0

That is, the abbreviated action has a minimum with respect to all paths that satisfy the conservation of energy which can be written as

δS0=δtfti2Tdt=0

Equation ??? is called the Maupertuis’ least-action principle which he proposed in 1744 based on Fermat’s Principle in optics. Credit for the formulation of least action commonly is given to Maupertuis; however, the Maupertuis principle is similar to the use of least action applied to the "vis viva", as was proposed by Leibniz four decades earlier. Maupertuis used teleological arguments , rather than scientific rigor, because of his limited mathematical capabilities. In 1744 Euler provided a scientifically rigorous argument, presented above, that underlies the Maupertuis principle. Euler derived the correct variational relation for the abbreviated action to be δS0=tftinjpjδqj=0

Hamilton’s use of the principle of least action to derive both Lagrangian and Hamiltonian mechanics is a remarkable accomplishment. It underlies both Lagrangian and Hamiltonian mechanics and confirmed the conjecture of Maupertuis.

Hamilton’s Principle applied using initial boundary conditions

Galley[Gal13] identified a subtle inconsistency in the applications of Hamilton’s Principle of Stationary Action to both Lagrangian and Hamiltonian mechanics. The inconsistency involves the fact that Hamilton’s Principle is defined as the action integral between the initial time ti and the final time tf as boundary conditions, that is, it is assumed to be time symmetric. However, most applications in Lagrangian and Hamiltonian mechanics assume that the action integral is evaluated based on the initial values as the boundary conditions, rather than the initial ti and final times  tf. That is, typical applications require use of a time-asymmetric version of Hamilton’s principle. Galley proposed a framework for transforming Hamilton’s Principle to a time-asymmetric form in order to handle problems where the boundary conditions are based on using only the initial values at the initial time ti, rather than the initial plus final times (ti,tf) that is assumed in the time-symmetric definition of the action in Hamilton’s Principle.

9.2.2.PNG
Figure 9.2.2: The left schematic shows paths between the initial q(ti) and final q(tf) times for conservative mechanics. The solid line designates the path for which the action is stationary, while the dashed lines represent the varied paths. The right schematic shows the paths applied to the doubled degrees of freedom with two initial boundary conditions, that is, q1(ti) and q2(tf) plus assuming that both paths are identical at their intersection and that they intersect at the same final time, that is, q1(tf)=q2(tf).

The following describes the framework proposed by Galley for transforming Hamilton’s Principle to a time-asymmetric form. Let q and ˙q designate sets of N generalized coordinates, plus their velocities, where q and ˙q are the fundamental variables assumed in the definition of the Lagrangian used by Hamilton’s Principle. As illustrated schematically in Figure 9.2.2, Galley proposed doubling the number of degrees of freedom for the system considered, that is, let q(q1,q2) and ˙q(˙q1,˙q2). In addition he defines two identical variational paths 1 and 2, where path 2 is the time reverse of path1. That is, path 1 starts at the initial time ti, and ends at tf, whereas path 2 starts at tf and ends at ti. That is, he assumes that q and ˙q specify the two paths in the space of the doubled degrees of freedom that are identical, and that they intersect at the final time tf. The arrows shown on the paths in Figure 9.2.2 designate the assumed direction of the time integration along these paths.

For the doubled system of degrees of freedom, the total action for the sum of the two paths is given by the time integral of the doubled variables, S(q1,q2) which can be written as

S(q1,q2)=tftiL(q1,˙q1,t)dt+titfL(q2,˙q2,t)dt=tfti[L(q1,˙q1,t)dtL(q2,˙q2,t)]dt

The above relation assumes that the doubled variables (q1,˙q1) and (q2,˙q2) are decoupled from each other. More generally one can assume that the two sets of variables are coupled by some arbitrary function K(q1,˙q1,q2,˙q2,t). Then the action can be written as

S(q1,q2)=tfti[L(q1,˙q1,t)dtL(q2,˙q2,t)+K(q1,˙q1,q2,˙q2,t)]dt

The effective Lagrangian for this doubled system then can be defined as

Λ(q1,q2,˙q1,˙q2,t)[L(q1,˙q1,t)dtL(q2,˙q2,t)+K(q1,˙q1,q2,˙q2,t)]

and the action can be written as S(q1,q2)=tftiΛ(q1,˙q1,q2,˙q2,t)dt

The coupling term K(q1,˙q1,q2,˙q2,t) for the doubled system of degrees of freedom must satisfy the following two properties.

(a) If it can be expressed as the difference of two scalar potentials, ΔU(q1,q2)=U(q1)U(q2), then it can be absorbed into the potential term for each of the doubled variables in the Lagrangian. This implies that K=0, and there is no reason to double the number of degrees of freedom because the system is conservative. Thus K describes generalized forces that are not derivable from potential energy, that is, conservative.

(b) A second property of the coupling term K(q1,˙q1,q2,˙q2,t) is that it must be antisymmetric under interchange of the arbitrary labels 12. That is,

K(q2,˙q2,q1,˙q1,t)=K(q1,˙q1,q2,˙q2,t)

Therefore the antisymmetric function K(q1,˙q1,q2,˙q2,t) vanishes when q2=q1.

The variational condition requires that the action S(q1,q2) has a well defined stationary point for the doubled system. This is achieved by parametrizing both coordinate paths as

q1,2(t,ϵ)=q1,2(t,0)+ϵη1,2(t)

where q1,2(t,0) are the coordinates for which the action is stationary, ϵ1. and where η1,2(t) are arbitrary functions of time denoting virtual displacements of the paths. The doubled system has two independent paths connecting the two initial boundary conditions at ti, and it requires that these paths intersect at tf. The variational system for the two intersecting paths requires specifying four conditions, two per path. Two of the four conditions are determined by requiring that at ti the initial boundary conditions satisfies that η1,2(ti)=0. The remaining two conditions are derived by requiring that the variation of the action S(q1,q2) satisfies

[dSdϵ]ϵ=0=0=tftidt{η1[Λq1dπ1dt]ϵ=0η2[Λq2dπ2dt]ϵ=0}+[η1π1η2π2]t=tf

The canonical momenta π1,2 conjugate to the doubled coordinates q1,2 are defined using the nonconservative Lagrangian Λ to be

πI1(q1,2,˙q1,2)Λ˙qI1(t)=L(q1,˙q1,t)˙qI1(t)+K(q1,˙q1,q2,˙q2,t)˙qI1(t)

where the superscript I designates the solution based on the initial conditions. Note that the conjugate momentum pI1=L(q1,˙q1,t)˙qI1(t) while the K(q1,˙q1,q2,˙q2,t)˙qI1(t) term is part of the total momentum due to the nonconservative interaction. Similarly the momentum for the second path is πI2(q1,2,˙q1,2)Λ˙qI2(t)=L(q1,˙q1,t)˙qI2(t)+K(q1,˙q1,q2,˙q2,t)˙qI2(t)

The last term in Equation ???, that is, the term [η1π1η2π2]t=tf results from integration by parts, which will vanish if

ηI1(tf)πI1(tf)=ηI2(tf)πI2(tf)

The equality condition at the intersection of the two paths at tf requires that

ηI1(tf)=ηI2(tf)

Therefore equations ??? and ??? imply that

πI1(tf)=πI2(tf)

Therefore equations ??? and ??? constitute the equality condition that must be satisfied when the two paths intersect at tf. The equality condition ensures that the boundary term for integration by parts in Equation ??? will vanish for arbitrary variations provided that the two unspecified paths agree at the final time tf. Similarly the conjugate momenta πI1(tf),πI2(tf) must agree, but otherwise are unspecified. As a consequence, the equality condition ensures that the variational principle is consistent with the final state at tf not being specified. That is, the equations of motion are only specified by the initial boundary conditions of the time-asymmetric action for the doubled system.

More physics insight is provided by using a more convenient parametrization of the coordinates in terms of their average and difference. That is, let

qI+qI1+qI22qIqI1qI2

Then the physical limit is

qI+qIqI0

That is, the average history is the relevant physical history, while the difference coordinate simply vanishes. For these coordinates, the nonconservative Lagrangian is Λ(q+,q,˙q+,˙q,t) and the equality conditions reduce to

π(tf)=0η(tf)=0

which implies that the physically relevant average (+) quantities are not specified at the final time tf in order to have a well-defined variational principle.

The canonical momenta are given by

πI+=πI1+πI22=Λ˙qIπI=πI1πI2=Λ˙qI+

The equations of motion can be written as.

ddtΛ˙qI±=ΛqI±

Equation ??? is identically zero for the + subscript, while, in the physical limit (PL), the negative subscript gives that

[ddtΛ˙qIΛqI]PL=0

Substituting for the Lagrangian Λ gives that

ddtL˙qILqI=[KqIddtK˙qI]PLQI(q1,˙q1,t)

where QI is a generalized nonconservative force derived from K.

Note that Equation ??? can be derived equally well by taking the direct functional derivative with respect to qI(t), that is, 0=[δSδqI(t)]PL

The above time-asymmetric formalism applies Hamilton’s action principle to systems that involve initial boundary conditions while the second path corresponds to the final boundary conditions. This framework, proposed recently by Galley, provides a remarkable advance for the handling of nonconservative action in Lagrangian and Hamiltonian mechanics.2 This formalism directly incorporates the variational principle for initial boundary conditions and causal dynamics that are usually required for applications of Lagrangian and Hamiltonian mechanics. Currently, there is limited exploitation of this new formalism because there has been insufficient time for it to become well known, for full recognition of its importance, and for the development and publication of applications. Chapter 10 discusses an application of this formalism to nonconservative systems in classical mechanics.


2This topic goes beyond the planned scope of this book. It is recommended that the reader refer to the work of Galley, Tsang, and Stein[Gal13, Gal14] for further discussion plus examples of applying this formalism to nonconservative systems in classical mechanics, electromagnetic radiation, RLC circuits, fluid dynamics, and field theory.


This page titled 9.2: Hamilton's Principle of Stationary Action is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Douglas Cline via source content that was edited to the style and standards of the LibreTexts platform.

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