Skip to main content
\(\require{cancel}\)
Physics LibreTexts

13.2: Principle of Least Action

  • Page ID
    9643
  • Hamilton’s crowning achievement was deriving both Lagrangian mechanics and Hamiltonian mechanics, directly in terms of a general form of his principle of least action \(S\), Equation 13.2. Consider the \(S_i\) action for the extremum path of a system in configuration space, that is, along path \(A\) from coordinates \(q_j(t_1)\) at \(t = t_1\) to \(q_j\) (\(t_2\)) at \(t = t_2\) shown in Figure \(\PageIndex{1}\) where \(j = 1, 2, \ldots, n\) coordinates. Then the action \(S_A\) is given by

    \[S _{ A} = \int _{ t _{ 1}} ^{ t _{ 2}} L ( \mathbf{ q} ( t ) , \dot{ \mathbf{ q}} ( t ) , t ) d t\]

    As used in chapter 5.2 a family of neighboring paths is defined by adding an infinitessimal fraction \(\epsilon\) of a continuous, well-behaved neighboring function \(\eta_j\) where \(\epsilon=0\) for the extremum path.

    \[q _{ j} ( t , \epsilon ) = q _{ j} ( t , 0 ) + \epsilon \eta _{ j} ( t ) \label{13.5}\]

    In contrast to the variational case discussed when deriving Lagrangian mechanics, the variational path used here does not assume that the functions () vanish at the end points. Assume that the neighboring path  has an action  where

    \[S _{ B} = \int _{ t _{ 1} + \Delta t} ^{ t _{ 2} + \Delta t} L ( \mathbf{ q} ( t ) + \delta \mathbf{ q} ( t ) , \dot{ \mathbf{ q}} ( t ) + \delta \dot{ \mathbf{ q}} ( t ) ) d t \label{13.6}\]

    Figure \(\PageIndex{1}\): Extremum path A, plus the neighboring path B, shown in configuration space.

    Expanding the integrand of \(S_B\) in Equation \ref{13.6} gives that, relative to the extremum path , the incremental change in action is

    \[\delta S = S _{ B} - S _{ A} = \int _{ t _{ 1}} ^{ t _{ 2}} \sum _{ j} \left( \frac{ \partial L}{ \partial q _{ j}} \delta q _{ j} + \frac{ \partial L}{ \partial \dot{ q} _{ j}} \delta \dot{ q} _{ j} \right) d t + [ L \Delta t _{ t _{ 1}} ^{ t _{ 2}}\]

    The second term in the integral can be integrated by parts since

    \[\delta \dot{ q} _{ j} = d \left( \frac{ \delta q _{ j}}{ d t} \right)\]

    leading to

    \[\delta S = \int _{ t _{ 1}} ^{ t _{ 2}} \sum _{ j} \left( \frac{ \partial L}{ \partial q _{ j}} - \frac{ d}{ d t} \frac{ \partial L}{ \partial \dot{ q} _{ j}} \right) \delta q _{ j} d t + \left[ \sum _{ j} \frac{ \partial L}{ \partial \dot{ q} _{ j}} \delta q _{ j} + L \Delta t \right] _{ t _{ 1}} ^{ t _{ 2}} \label{13.8}\]

    Note that Equation \ref{13.8} includes contributions from the entire path of the integral as well as the variations at the ends of the curve and the \(∆t\) terms. Equation \ref{13.8} leads to the following two pioneering principles of least action in variational mechanics that were developed by Hamilton.

    13.2.1 Hamilton’s Principle

    Derivation of Lagrangian mechanics in chapter 6 was based on the extremum path for neighboring paths between two given locations q(1) and q(2) that the system occupies at times 1 and 2 respectively. For this special case, where the end points do not vary, that is, when (1)= (2)= 0, and ∆1 = ∆2 = 0, then the least action  for the stationary path (Equation \ref{13.8}) reduces to

    \[\delta S = \int _{ t _{ 1}} ^{ t _{ 2}} \sum _{ j} \left( \frac{ \partial L}{ \partial q _{ j}} - \frac{ d}{ d t} \frac{ \partial L}{ \partial \dot{ q} _{ j}} \right) \delta q _{ j} d t = 0\]

    For independent generalized coordinates  , the integrand in brackets vanishes leading to the Euler-Lagrange equations. Conversely, if the Euler-Lagrange equations in 13.9 are satisfied, then,  = 0 that is, the path is stationary. This leads to the statement that the path in configuration space between two configurations q(1) and q(2) that the system occupies at times 1 and 2 respectively, is that for which the action  is stationary. This is a statement of Hamilton’s Principle.

    13.2.2 Least-action principle in Hamiltonian mechanics

    Consideration of the general variation of the least-action path leads to Hamilton’s basic equations of Hamil- tonian mechanics. For the general path, the integral term in equation 13.8 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

    \[\Delta q _{ j} = \delta q _{ j} + \dot{ q} _{ j} \Delta t \label{13.10}\]

    where  and ˙ are evaluated at 1 and 2. Then Equation \ref{13.8} reduces to

    \[\delta S = \left[ \sum _{ j} \frac{ \partial L}{ \partial \dot{ q} _{ j}} \delta q _{ j} + L \Delta t \right] _{ t _{ 1}} ^{ t _{ 2}} = \left[ \sum _{ j} \frac{ \partial L}{ \partial \dot{ q} _{ j}} \Delta q _{ j} + \left( - \sum _{ j} \frac{ \partial L}{ \partial \dot{ q} _{ j}} \dot{ q} _{ j} + L \right) \Delta t \right] _{ t _{ 1}} ^{ t _{ 2}}\]

    Since the generalized momentum  =  , then equation 13.11 can be expressed in terms of the Hamiltonian and generalized momentum as

    \[\left.\begin{aligned} \delta S & = \left[ \sum _{ j} p _{ j} \Delta q _{ j} - H \Delta t \right] ^{ t _{ 2}} = [ \mathbf{ p} \cdot \Delta \mathbf{ q} - H \Delta t ] _{ t _{ 1}} ^{ t _{ 2}} \\ \frac{ \partial S}{ \partial q _{ j}} & = p _{ j} \end{aligned} \right.\]

    Equation 13.12 contains Hamilton’s Principle of Least-action. Equation 13.13 gives an alternative relation of the generalized momentum  that is in terms of the action functional 

    Integrating the action , equation \ref{13.11}, between the end points gives the action for the path between

     = 1 and  = 2, that is, ( (1) 1  (2) 2) to be

    \[S \left( q _{ j} \left( t _{ 1} \right) , t _{ 1} , q _{ j} \left( t _{ 2} \right) , t _{ 2} \right) = \int _{ 1} ^{ 2} [ \mathbf{ p} \cdot \dot \mathbf{ q} - H ( \mathbf{ q} , \mathbf{ p} , t ) ] d t\]

    The stationary path is obtained by using the variational principle

    \[\delta S = \delta \int _{ 1} ^{ 2} [ \mathbf{ p} \cdot \dot \mathbf{ q} - H ( \mathbf{ q} , \mathbf{ p} , t ) ] d t = 0\]

    The integrand in the modified Hamilton’s principle,  = [p · q˙ − (q p)]  can be used in the  Euler- Lagrange equations for  = 1 2 3   to give

    \[\frac{ d}{ d t} \left( \frac{ \partial I}{ \partial \dot{ q} _{ j}} \right) - \frac{ \partial I}{ \partial q _{ j}} = \dot{ p} _{ j} + \frac{ \partial H}{ \partial q _{ j}} = 0\]

    Similarly, the other \(n\) Euler-Lagrange equations give

    \[\frac{ d}{ d t} \left( \frac{ \partial I}{ \partial \dot{ p} _{ j}} \right) - \frac{ \partial I}{ \partial p _{ j}} = - \dot{ q} _{ j} + \frac{ \partial H}{ \partial p _{ j}} = 0\]

    Thus Hamilton’s principle of least-action leads to Hamilton’s equations of motion, that is equations 13.16 13.17..

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

    \[\frac{ d S}{ d t} = \frac{ \partial S}{ \partial t} + \sum _{ j} ^{ n} \frac{ \partial S}{ \partial q _{ j}} \dot{ q} _{ j} = \frac{ \partial S}{ \partial t} + \mathbf{ p} \cdot \dot{ \mathbf{ q}} _{ j} \label{13.18}\]

    But the total time derivative of equation 13.15 equals

    \[\frac{ d S}{ d t} = \mathbf{ p} \cdot \dot{ q} - H ( \mathbf{ q} , \mathbf{ p} , t ) \label{3.19}\]

    Combining Equations \ref{13.18} and \ref{13.19} gives the Hamilton-Jacobi equation which is discussed in chapter 145.

    \[ \frac{ \partial S}{ \partial t} + H ( \mathbf{ q} , \mathbf{ p} , t ) = 0\]

    In summary, Hamilton’s principle of least action led directly to Hamilton’s equations of motion (13.16 13.17) plus the Hamilton-Jacobi equation (13.20). Note that both Hamilton’s Principle (13.8) and Hamilton’s equa- tions of motion (13.16 13.17) have been derived directly from Hamilton’s concept of Least Action  without explicitly invoking the Lagrangian.

    13.2.3 Abbreviated action

    Hamilton’s 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  =  and Equation \ref{13.14} equals

    \[ \begin{align} S \left( q_j \left(t_1 \right), t_1, q _j \left( t_2 \right), t_2 \right) &= \int _1^2 [ \mathbf{p} \cdot \dot { \mathbf{q}} - E ] dt \\[5pt] &= \int _{1} ^{2} \mathbf{ p} \cdot \delta \mathbf{q} - E \left( t _{ 2} - t _{ 1} \right) \label{13.21} \end{align}\]

    The \(\int _{1} ^{2} \mathbf{ p} \cdot \delta \mathbf{q} \) term in Equation \ref{13.21}, is called the abbreviated action which is defined as

    \[S _{ 0} \equiv \int _{ 1} ^{ 2} \mathbf{ p} \cdot \delta \dot{ q} d t = \int _{ 1} ^{ 2} \mathbf{ p} \cdot \delta \mathbf{ q} \label{13.22}\]

    The abbreviated action can be simplified assuming the standard Lagrangian  =  −  has a velocity- independent potential  , then equation 84 gives.

    \[S _{ 0} \equiv \int _{ 1} ^{ 2} \sum _{ j} ^{ n} p _{ j} \dot{ q} _{ j} d t = \int _{ 1} ^{ 2} ( L + H ) d t = \int _{ 1} ^{ 2} 2 T d t = \int _{ 1} ^{ 2} \mathbf{ p} \cdot \delta \mathbf{ q}\]

    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  = 0 but allow for a variation  in the final time. Then using equation 13.21

    \[\delta S = - H \delta t = - E \delta t\]

    However, Equation \ref{13.21} gives that

    \[\delta S = \delta S _{ 0} - E \delta t\]

    Therefore

    \[\delta S _{ 0} = 0\]

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

    \[\delta S _{ 0} = \delta \int _{ 1} ^{ 2} 2 T d t = 0 \label{13.27}\]

    Equation \ref{13.27} 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 identical to 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

    \[\delta S _{ 0} = \int \sum _{ j} ^{ n} p _{ j} \delta q _{ j} = 0 \label{13.28}\]

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