In mathematics and physics, Herglotz's variational principle, named after German mathematician and physicist Gustav Herglotz, is an extension of the Hamilton's principle, where the Lagrangian L explicitly involves the action as an independent variable, and itself is represented as the solution of an ordinary differential equation (ODE) whose right hand side is the Lagrangian instead of an integration of Herglotz's variational principle is known as the variational principle for nonconservative Lagrange equations and Hamilton equations. It was first proposed in the context of contact geometry.
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As in Lagrangian mechanics, we consider a system with degrees of freedom. Let be its generalized coordinates, and let be its generalized velocity. Let be the Lagrangian function of the physical system. Let be the action.
Lagrangian mechanics is derived using Hamilton's principle. Fix a starting time and configuration and an ending time and configuration Hamilton's principle states that physically real trajectories from are the solutions to the problem of variational calculus:Equivalently, it can be formulated as
Herglotz's variational principle simply generalizes by allowing the Lagrangian to depend on the action as well. It is of form depending on variables.
Hamilton's variational principle gives the EulerâÂÂLangrange equations. Similarly, Herglotz's variational principle gives the EulerâÂÂLagrangeâÂÂHerglotz equations which involves an extra term that can describe the dissipation of the system. The original EulerâÂÂLangrange equations are recovered as a special case when
Similar to how Lagrangian mechanics is equivalent to Hamiltonian mechanics, the Lagrangian form of Herglotz principle is equivalent to a Hamiltonian form.
Define the momentum and Hamiltonian by taking a Legendre transformation Then the equations of motion are
If is written as a function of time and configuration, then it satisfies a HamiltonâÂÂJacobi equation
In order to solve this minimization problem, we impose a variation on and suppose undergoes a variation correspondingly, then and since the initial condition is not changed, The above equation a linear ODE for the function and it can be solved by introducing an integrating factor which is uniquely determined by the ODE By multiplying on both sides of the equation of and moving the term to the left hand side, we get
Note that, since the left hand side equals to and therefore we can do an integration of the equation above from to yielding where the so the left hand side actually only contains one term , and for the right hand side, we can perform the integration-by-part on the term to remove the time derivative on
and when is minimized, for all which indicates that the underlined term in the last line of the equation above has to be zero on the entire interval this gives rise to the EulerâÂÂLagrangeâÂÂHerglotz equation.
Generalizations of Noether's theorem and Noether's second theorem apply to Herglotz's variational principle.
An infinitesimal transformation is where are smooth functions of time and configuration, and is an infinitesimal. The transformation deforms a trajectory to and accordingly deforms the action integral as well.
We say that the infinitesimal transformation is a symmetry of the action iff the change in under the infinitesimal transformation is order Given such an infinitesimal symmetry, the quantity is a constant of motion where is more explicitly written as There is also a version for multiple time dimensions.
The motion of a particle of mass in a potential field with damping coefficient isIt can be produced as the EulerâÂÂLagrangeâÂÂHerglotz for
More generally, consider a particle on a line under the influence of 3 forces: a conservative force due to a potential field, a dissipative force proportional to and another force proportional to Write it as
This equation is the EulerâÂÂLagrangeâÂÂHerglotz equation for the Lagrangian
where is any solution of the ODE Some important special cases: