  By one of the most fundamental principles in physics, a dynamical system will
exhibit those motions which extremise an action functional. This leads to the
formation of the Euler-Lagrange equations, which serve as a model of how the
system will behave in time. If the dynamics exhibit additional symmetries, then
the motion fulfils additional conservation laws, such as conservation of energy
(time invariance), momentum (translation invariance), or angular momentum
(rotational invariance). To learn a system representation, one could learn the
discrete Euler-Lagrange equations, or alternatively, learn the discrete
Lagrangian function $\mathcal{L}_d$ which defines them. Based on ideas from Lie
group theory, in this work we introduce a framework to learn a discrete
Lagrangian along with its symmetry group from discrete observations of motions
and, therefore, identify conserved quantities. The learning process does not
restrict the form of the Lagrangian, does not require velocity or momentum
observations or predictions and incorporates a cost term which safeguards
against unwanted solutions and against potential numerical issues in forward
simulations. The learnt discrete quantities are related to their continuous
analogues using variational backward error analysis and numerical results
demonstrate the improvement such models can have both qualitatively and
quantitatively even in the presence of noise.
